Gauss's Circle Problem

Size: px
Start display at page:

Download "Gauss's Circle Problem"

Transcription

1 Joseph Galante Gauss's Circle Problem Senior Thesis in completion of the B.A. Honors in Mathematics University of Rochester 1

2 1.0 Introduction and Description of the Problem Gauss's Circle problem is a simple counting question with surprisingly complex answers. Simply stated, the problem is to count the number of points with integer coordinates which are inside a circle of given radius centered at the origin. ( Lattice Points in a Circle [MW] ) Why is this interesting? Perhaps because the problem is intimately related to the number of ways to express an integer as the sum of two squares. Furthermore due to its geometric description we can use facts from basic geometry and complex analysis as well as number theory to understand the problem more throughly. We may write that the number of lattice points in a circle of radius R as N(R) = πr + E(R), where E(R) is some error term. Our goal in this paper to is to try to understand the error term. This problem is over 100 years old and a large amount of mathematical content has been built up around it, so we will reproduce and employ some of that machinery here. No real new work is being presented, only restatements of past advances which will be presented in a style so that the reader may pursue the theory at his/her leisure in the future. We will start by using counting methods to find the error term exactly in both R and generalized to R n. Then we will look at Gauss's upper bound on the error, Hardy's lower bound, and summarize further historical advances. Finally we will look at the beautiful connection between the Gauss circle problem and the Leibniz sum formula for π/4, but in order to do so, we will have to derive some facts about the connections between prime numbers and the sum of squares formula..0 Using Counting to Find N(R) Exactly We may use simple counting arguments to get an exact solution for N(R) in both R and more generally in R n. [FrGo], [Mit], and [KeSw] use more efficient counting

3 methods and attempt to program (1960's) computers to get numerical results. With today's modern computers, even inefficient implementations outstrip their complex setups, however it is still comforting to know that an exact solution exists..1 Solution in R Look at just the number of integer points in an interval about the origin. Notice that from 0 to R (excluding zero) there are R points and similarly from 0 to R (excluding zero) there are R points, where R denotes the usual greatest integer function. So, counting the origin, there are (.1.1) N 1 (R) = R + 1 points in the interval. Now the trick is simply to realize that the lattice points in the circle are logically arranged in these intervals. At each integer on the x axis inside the circle, there is an interval of radius R x, and from.1.1 we know how to count the number of points in this interval. The x's we are counting will range from R to R. This leads to sum R (.1.) N R = x= R R N 1 R x = R x 1 x= R which will give the exact solution in R. With the exact solution, we can now plot make a plot of the number of points in the circle verse the radius. The jagged curve is N (R), the smooth curve is the the area of the circle. This indicates that the error is not very large compared to the area. Later we will prove that this is so. (.1.3 Numbers of Points verse Radius) 3

4 . Generalized Solution in R n Its not that hard to continue this logic and generalize our formula for higher dimensions. In R n the problem becomes counting the number of points (x 1,..., x n ) with all the x i integers in the n sphere of radius R centered at the origin. However just as in.1 we realize that the points in the n sphere are arranged so that we can count them in (n 1) spheres of radius R x n formula: with x n ranging from R to R. This leads to a recursive R (..1) N n R = N n 1 R x with N 1 (r) = R + 1 x= R Just as in the planar case, we may write N n (R) = V n (R) + E n (R) where V n (R) is the volume of an n sphere of radius R. We will show later the error term does not outstrip the volume term for large values of R. This is interesting because then an approximate solution to this rather unusual looking recursive formula is easily found..3 Other Generalizations One may wonder about other generalizations as well. If we don't assume that the circle is centered at the origin, then our formula for N 1 becomes more complicated as we cannot use the symmetry about zero. The recursive piece of our formula becomes more complicated as well since we must account for how much each (n 1) sphere is shifted with respect to the origin. However such shiftings are insignificant for spheres with a large radius since most of the lattice points will be away from the boundary of the sphere, and shifting will only include or exclude a comparatively small number of points. Notice as well that the only real information we used about the circle was its equation x + y = R. If we have another convex shape which is symmetric about the x axis or y axis, then we can use N 1 and count points recursively again using its formula rather than that of the circle..4 Another Interpretation of N(r) There is another valuable way to think of N(R) which has a very number theoretic interpretation. N(R) is related to the sum of two squares function r(n) which counts the 4

5 number of ways to write an integer n as the sum of the squares of two integers. The first few values for r(n) are 1,4,4,0,4,8,0,0,4,4,8 corresponding to n = 0,1,...,10. How does this relate geometrically? Well the lattice points in the circle have coordinates (i, j) with i, j integers, and since they are in the circle then we must have that i +j R. The function N(R) will count all such points for R. But also for each n < R, r(n) will count the number of points with i +j = n. Adding up all the representations of n for each n < R will count all the lattice points in the circle which is also given by N(R). So then we get the identity for R an integer R (.4.1) N(R) = r n n=0 which we will use later in the paper. It is this relationship which makes the problem interesting to study since we may use concepts relating to area and geometry to get bounds on number theoretic functions and vice versa. 3.0 Gauss's Upper Bound on the Error We will now take a look bounding the error. Specifically wish to find the big O of the error, i.e., that there exists a constant C and an exponent such that for large value R we have E(R) < CR. Gauss found an elementary way to get 1. This proof was presented in [HiCo]. 3.1 Proof that E(r) = O(r) (3.1.1 A graphical representation of the proof) 5

6 Pick R to be sufficiently large. R must be larger than for this to work at all. Let A(R) denote the area of the squares intersecting the boundary of the circle. Then, (3.1.) E R = N R R A R We know that the maximum distance between any two points in the unit square is, hence all the squares intersecting the boundary of the circle are contained in the annulus of width with radii R+ and R. Let B(R) denote the area of the annulus. We have (3.1.3) B(R) = [(R+ ) (R ) ]π = 4 πr By construction we have that A(R) < B(R). So we have shown that E(R) < 4 πr, which is to say that E(R) = O(R). 3. Generalization to R n In n dimensions, we won't get E n (R) = O(R) from this proof style. The best we can get from this method is simply that E n (R) = O(R n 1 ), but this is alright since V n (R) = O(R n ). Once again pick R to be sufficiently large. R must be larger than n for this to work at all. Let A n (R) denote the volume of n cubes intersecting the boundary of the n sphere. Then (3..1) E n (R) = N n (R) V n (R) A n (R) where V n (R) is the volume of an n sphere of radius R. It is known ([MW]) that (3..) V n R = n/ R n n n/ We know that the maximum distance between any two points in the unit n cube is n, hence all the cubes intersecting the boundary of the n sphere are contained in the n 6

7 annulus of width n with radii R+ n and R n. Let B(R) denote the area of the annulus. So we have (3.3.3) B(R) = V n (R+ n) V n (R n) = n/ n n/ R n n R n n = O(R n 1 ) and also by construction we have that A(R) < B(R), so we have that E n (R) = O(R n 1 ) Why is this multidimensional interpretation important? Well recall that in two dimensions N(R) was related to r(k) which counted the number of ways to write an integer k as the sum of two squares. The n dimensional analog of the Gauss Circle Problem will then count the number of ways to write an integer as the sum of n squares. However more interesting phenomenon occurs when we add more squares. There is the so called Lagrange Four Square Theorem which states that every positive integer is the sum of four squares. This theorem is proved in [StSh] and [PM]. So by studying the Gauss Circle Problem in n dimensions we may gain some insight into sums of n squares and vice versa. While this is interesting, from this point forward in the paper, we will restrict ourselves to two dimensions. 4.0 Hardy's Lower Bound on the Error The great Hardy contributed a significant amount to the understanding of Gauss Circle Problem. In his paper On the Expression of a Number as the Sum of Two Squares [Har], he provides a lower bound of ½ on the possible for which E(R) < CR, provides an explicit analytic formula for the error term using Bessel functions, and offers generalizations of the problem to ellipses. We will examine Hardy's proof of the lower bound of ½ in detail, and summarize the last two results. Specifically, we will show that the error is Big Omega of ½, i.e. that there is a constant C and arbitrarily large values of R so that E(R) > CR 1/ is satisfied. 4.1 Useful Facts from Hardy's Other Works Hardy makes use of the following identities which he and Ramanujan had derived in the a paper entitled Dirichlet's Divisor Problem. We shall give them here without proof. 7

8 (4.1.1) e s u v = s u, v u, v Then we can rewrite as 1 s 4 u v 3/ for s = + i with > 0 (4.1.) f(s)= r n e s n = s 1 s r n s 4 n 3/ for s = + i with > 0 Notice that in this form it is easy to see that 4.1. is holomorphic everywhere except at πi k for k a positive integer, where the function has a branch point. We pick the principal branch for our work here. Hardy describes these points as algebraical infinities of order 3/ since when we approach πi k, we can get behavior like 1/x 3/. Hardy introduces the function (4.1.3) g s = n s n e and notes that this function is holomorphic everywhere except at the origin. He notes that at the origin, when 1, g (s) is of the form (4.1.4) s g s and when = 1, g (s) is of the form (4.1.5) log 1/s g 1 s and in both cases the g part is holomorphic at the origin. We also note that g (s) is a multiple branched function, so we must be careful to define which branch we are using. With this initial setup, we may now start to actually do the mathematics needed to get our desired lower bound. Our method of proof will be as follows. First we will define functions made from the f and g defined above which we understand the behavior of. Then we will relate our understanding of these new functions to that of the Gauss circle problem. Specifically we will look at the behavior at the poles. Finally we will show that the asymptotic behavior at the poles forces the error term to behave in the fashion we desire. 8

9 4. The Function F(s) s n (4..1) F(s) = f(s) πg 0 (s) = r n e Note that F(s) is holomorphic everywhere except at πi k for k a positive integer where it has an isolated singularities of the same nature as Also since g (s) is branching, so is F(s) and so we define it only for the principal branch. Now let us analyze the behavior of F when s = + πi k by letting approach 0 from above. The key here is to use identity 4.1. to get (4..) F( + πi k) = i k 1 i k r n i k 4 n 3/ i k n e Now for 0, the first and second terms goto constants, as does the last term. Furthermore, all the terms for which n k will also approach a constant as 0. So this leaves us with: (4..) F( + πi k) ~ i k r k i k 4 k 3/ ~ r k 4 1/ i 3/ 1/ k r k 1/ 3/4 i 1/ 3/ k 1/4 ~ 1/ e 1/4 i r k k 1/4 3/ by expanding, noting that i k 4 k ~4 i k, simplifying, and noting that 3/ grows faster than 1/. 4.3 The Connection to Gauss's Circle Problem We will now begin making connections with the Gauss circle problem. It is worth noting that in Hardy's proof, he uses the substitution x=r, so the equation N(r) = πr + E(r) becomes N( x) = πx+e( x). To avoid this confusion, we will adopt 9

10 Hardy's notation and use for our circle problem description for the rest of section 4. (4.3.1) R(x) = πx + P(x) where P(x) represents the error term. In light of 4.3.1, formula.4.1 relating the circle problem to r(n) becomes n (4.3.) P(n) = R(n) πn = r i n i=1 We now observe the following identity: s n (4.3.3) F(s)= r n e = = n i=1 n 1 s n r i n r i n 1 e i=1 s n P n P n 1 e = P 1 e s 1 P P 1 e s P 3 P e s 3... (remembering P(0)=0) = P 1 e s 1 e s P e s e s 3 P 3 e s 3 e s 4... = P n e s n e s n 1 = P n e s n where n = n n 1 (The rearrangement of terms is justified since the exponential forces absolute convergence.) We now find a clever way to rewrite the delta term by applying the mean value theorem twice. (Note, I'm somewhat convinced that Hardy's analysis left out a constant, so I'll include it in my analysis. However it ultimately doesn't make that much of a difference for what we are using this estimate for.) (4.3.4) n = n n 1 = ' n 1 where 1 is in (0,1) by the Mean Value Theorem (MVT). = ' n ' n ' n 1 = ' n ' ' n 1 by applying the MVT to the last two terms with 1 in (0,1) and in (0, 1 ) 10

11 s n We use n =e to get (4.3.5) e s n = s e s n 1 s e s n n 4 n s e s n 4 n 3/ We now examine the behavior of for s = + πi k for k a positive integer with approaching zero. We get: (4.3.6) e s n = s e s n O n e n n We now introduce the function G(s) and perform and analysis of it near poles and near the origin. n e s (4.3.7) G(s) = P n n where P(n) is the error term from the Gauss circle problem in We combine 4.3.3, 4.3.6, and and get (4.3.8) F(s) = P n e s n = s P n G s O e n n Using the Gauss's upper bound from section 3 to get P(n) / n = 1 / n n e and n n e dn~ 1 1 n as 0, we get ( ) F(s) = s G(s) / + O(1/ ) = s G(s)/ + o( 3/ ) Applying 4.. we find that (4.3.9) G(s) ~ F(s) / s ~ e 3/4 i r k 3/ 3/4 k 11

12 To summarize, gives the behavior for G(s) near s = + πi k for k a positive integer with approaching zero. Now let us look at what happens near the origin, i.e. when s 0. We approach the origin by s = with 0. From we have (4.3.10) e s n = s e s n n O s s n e O n and plugging this back into gives s (4.3.11) F(s) = G s O s P n n s e s n n 3/ e n O s P n e n =sg(s)/ + O(1) since the 0 and the rest of terms in the sum remain fixed. n 3/ In light of the fact that F(s) is holomorphic at the origin, then we may write (4.3.1) G(s) = O(1/s) = o(s 3/ ) where s = with Bounding P(n) We are now in a position to use the bounds we have found above for F(s) and G(s) to bound the behavior of P(n). In this section we complete the proof. We will introduce yet another function, the H function. P n Cn (4.4.1) H(s) = G(s) Cg 1/4 (s) = 1/4 n for some constant C > 0 to be determined. s n e In light of the given information about g (s) from 4.1.4, we may conclude that for approaching zero, we have (4.4.) g 1/4 ( + πi k) = (4.4.3) g 1/4 ( ) ~ π 3/ 3/ s 3/ g 1/4 s = o( 3/ ) 1

13 Then using our knowledge of the behavior of G(s) found in and 4.3.1, we can now understand the behavior of H(s) for approaching zero. Since both have order 3/ we have (4.4.4) H( + πi k) ~ (4.4.5) H( ) ~ C π 3/ e 3/4 i r k 3/ 3/4 k We can now prove our claim by contradiction. Suppose that for all n > N for sufficiently large N, we have that P(n) Cn 1/4 0. Then we have (4.4.6) H( + πi k) = N 1 P n Cn 1/4 O(1) + O(1) n P n Cn 1/4 n=n O(1) H( ) n P n Cn 1/4 n=n n P n Cn 1/4 n e i k n + n=n n e i e k n P n Cn 1/4 n i e k n e n (by our supposition that P(n) Cn 1/4 0) Now we combine the bounds in 4.4.4, 4.4.5, and to get, (4.4.7) e 3/4 i r k 3/ k C 3/ 3/4 or r k (4.4.8) C k 3/4 A similar argument (changing signs and so forth) gives that if we assume that for all n > N for sufficiently large N, we have that P(n) + Cn 1/4 0, then will also follow. We now derive contradictions. If k is so that r(k)=0, then we would have from that H( + πi k) ~ 0 and from we would get that H( ) O(1) as 0, which is a contradiction. 13

14 If k is so that r(k) 0, then we can pick our constant C to get to contradict by picking r k k 3/4 > C > 0. Thus our assumptions are false and we have that P(x) > Cx 1/4 and P(x) < Cx 1/4 or P(x) < Cx 1/4, or P(x) = (x 1/4 ). Rephrased in our original terminology, this says that E(r)= (r 1/ ), and hence we are done. 4.5 Hardy and Ramanujan's Other Findings As alluded to above, Hardy and Ramanujan made some other significant contributions to the Gauss Circle Problem. In section, we found a formula for N(r) using only counting arguments. Hardy however provides an analytic formula for N(r), or in his terminology R(x). Since analytic formulas are valuable, we will provide them here. r n (4.5.1) R(x) = x 1 x J 1 nx for x not an integer n where J 1 is denotes a Bessel function of order one. When x is an integer, we have (4.5.) R(x) = r(0) + r(1) + r() r(x), which is a restatement of.4.1. Hardy also examined what happens when we shift our geometric picture to an ellipse rather than a circle. Rather than having u +v = x, we would have u + uv+ v =x with,, and integers with > 0 and = 4 > 0. In summary, we get the an analogous statement to and (4.5.3) R(x) = x 1 r n x J 1 4 nx n for x not an integer (For x an integer, R(x) can be found using a simple counting arguments and recursive formulas, as done for the circle in section.) 14

15 5.0 Historical Advances in the Error In summary we have for E(R) CR that ½ < < 1. However more improved bounds have been found in the last 100 years. Much focus has been put on reducing the upper bound. Below is a table to summarize the findings so far. Its a testament to the difficulty of problem that after roughly 80 years of work, only an improvement of about has been made upon the upper bound. The newer proofs become rather long (0 pages plus) and make use of some of the more heavy machinery in analysis and number theory. Theta Approx Citation 46/ Huxley / / Cheng / Vinogradov 37/ Littlewood and Walfisz 194 / Sierpinski 1906, van der Corput 193 (5.0.1 Improvements in the upper bound for [MW]) For interest, the author programmed Mathematica to compute E(R) for the first 1000 integers and find the best to the curve R. Mathematica returned = , which fits reasonably well with the values in the table above, considering the small sample size. Below is a scatter plot of the error for the 1000 points sampled, as well as a precise plot of the error for R < 10. (5.0. E(R) versus R, for integer R) 15

16 (5.0.3 E(R) versus radius R) 6.0 r(n) = 4d 1 (n) 4d 3 (n) Next we will examine a valuable statement in number theory which relates the sum of squares function to divisor functions. This key fact will be the starting point of the beautiful identity which we shall prove in section 7. This proof is found Stein and Shakarchi's Complex Analysis in full detail. We will focus only on the number theoretic aspects and leave other statements requiring only complex analysis as lemmas. Our proof will make use of some facts about the Jacobi Theta functions. 6.1 Definitions (6.1.1) r(n) = the number of ways to write n as the sum of two squares of integers (6.1.) d 1 (n) = the number of divisors of n of the form 4k+1 (6.1.3) d 3 (n) = the number of divisors of n of the form 4k+3 (6.1.4) ( ) = e i n n= (6.1.5) C( ) = n= 1 cos n for in the upper half plane for in the upper half plane 6.. A useful identity For q = e πi with in the upper half plane we have (6..1) ( ) = q n 1 q n = n 1 = n = n 1, n ZxZ q n n 1 = r n q n n=0 16

17 since r(n) counts the number of pairs (n 1, n ) in which to write n = n 1 +n. The rearrangement of terms is justified since is in the upper half plane, so e πi < 1 and we have absolute convergence Another useful identity For q = e πi with in the upper half plane we have 1 (6.3.1) C( ) = n= cos n 1 q = = 1 4 n n= q n q n 1 q n Since is in the upper half plane, then q <1 and we have absolute convergence, so 1 q n q = n (6.3.) 4 q n 1 q n q n. Now using the fact that 1 q n =4 q n q 3n 1 q 4 n n 1 1 q =1 q n 1 q 4 n, we have that We use the geometric series expansion 1 1 q = 4 n m=0 q 4 nm to get that (6.3.3) q n 1 q 4 n = m=0 q n 4 m 1 = d 1 k q k, k=1 since d 1 counts the number of factors of the form 4m+1. A similar argument gives that (6.3.4) q 3 n 1 q 4 n = m=0 q n 4 m 3 = d 3 k q k k=1 Putting everything back together, we get for q = e πi with in the upper half plane (6.3.5) C( ) = 1 4 d 1 k d 3 k q k k=1 17

18 6.4 Outline of the proof that ( ) = C( ) Now if we had that ( ) = C( ), then it would follow that (6.4.1) 1 r n q n or (6.4.) r(n) = 4d 1 (n) 4d 3 (n) = 1 4 d 1 k d 3 k q k k=1 So our goal is simple, to show that ( ) = C( ). We will only outline the proof here as it doesn't involve much analysis directly related to the Gauss circle problem. The proof will involve lots of complex analysis and draw from several areas of the subject, so interested readers should consult [StSh]. Step 1) Show that ( ) has the following properties: ( ) ( ) = 1 q n 1 q n 1 for Im( ) > 0 and q = e πi (6.4.3.) ( 1/ ) = i for Im( ) > 0 ( ) (1 1/ ) ~ i ei /4 as Im( ) goes to infinity in the upper half plane Step ) Show that C( ) has the following properties: ( ) C( ) = C( +) (6.4.4.) C( ) = i/ C( 1/ ) ( ) C( ) 1 as Im( ) in the upper half plane ( ) C(1 ) ~ 4 i e i / as Im( ) in the upper half plane Step 3) Prove the theorem that if f is holomorphic in the upper half plane and satisfies i) f( ) = f( +); ii) f( 1/ ) = f( ); and, iii) f( ) is bounded, then f is constant. 18

19 (Note: This proof is rather messy and involves interesting domains and fractional linear transformations.) Step 4) Define the function f = C/ and show that f meets the conditions of the theorem above and conclude that f is constant and that f must be one. Then we are done. Having shown our desired relation, we may now prove an interesting relation of the Gauss circle problem to the Leibniz sum formula for π/ Leibniz Sum Formula for π/4 In this section we connect the fact that r(n) = 4d 1 (n) 4d 3 (n) back to the Gauss circle problem and the Leibniz sum formula for π/4. This proof was originally presented by Hilbert in [HiCo]. Recall the geometric interpretation of N(R) from section.4. N(R) will count the number of points inside the circle of radius R, but this is also the number of ways to write integers less than or equal to R as the sum of two squares of integers. We have found in.4.1 R that N(R) = r n. Now we apply the identity 6.0 to get n=0 (7.0.1) N(R) = or R (7.0.) ¼ ( N(R) 1 ) = R 1 4 d 1 n d 3 n R d 1 n d 3 n We now find an enlightening way to add the terms in first and second sums. The first term is the sum of all the divisors of the form 4k+1 for all numbers n R. So our factors look like 1,5,9,13... and will stop when we reach R. How many times will a given factor appear in the sum? Hilbert notes that each number appears as many times as a factor as there are multiples of it that are less than R. So for example, 1 will appear R times, 5 will appear R /5, 9 will appear R /9, and so on. So then we have that R (7.0.3) d 1 n = R R 5 R 9 R

20 R (7.0.4) d 3 n = R 3 R 7 R 11 R and adding them back together and rearranging we get (7.0.5) ¼ ( N(R) 1 ) = R R 3 R 5 R 7 R 9 R 11 R 13 R Note that our rearrangement is justified since these are only finite sums, since when the denominator of the fraction becomes larger than R, the greatest integer function will become zero, leaving only a finite number of nonzero terms. Now suppose we truncate the sum when the denominator is R, i.e. at the term R /R = R. All terms after that point will be less than R and will have alternating signs and go to zero, so at most the error from the truncation is O(R). (7.0.6) ¼ ( N(R) 1 ) = R R...± R O R 3 If we remove the brackets from the each of the remaining terms, each term can increase in value by at most (1 1/n) < 1, where n is the denominator of that term, and again the terms are alternating and go to zero, so the error is again at most O(R), which gives (7.0.7) ¼ ( N(R) 1 ) = R R 3 R 5 R 7...±R O R Dividing by R gives (7.0.7) 1 N R 1 4 R = ±1/ R O 1/ R 7 Now let R. Then we have that the error O(1/R) goes to zero and the series will converge by the alternating series test. We recognize the series as the Leibniz sum formula for π/4 and get 1 N R 1 (7.0.8) lim = R 4 R 4 or 0

21 N R (7.0.9) lim = R R Although this statement can be derived from facts in section 3, this proof highlights the deep connections of the Gauss circle problem to number theory. 8.0 Conclusion ( N(r)/r π) The Gauss Circle Problem is a deep problem with an easy statement. We showed that ideas as simple as counting lattice points inside a circle have surprisingly deep connections to the sum of squares function and to divisor functions which occur frequently in number theory. We found that a lot of interest is placed in the error term E(R) and that it is known that E(R) has a growth rate somewhere between (R 1/ ) and O(R). Finding the upper bound of O(R) was elementary, but the lower bound involved a good deal of work. Historically people have focused on refining the upper bound and over 100 years of progress have reduced it to only around Lastly, we started a proof connecting the sum of squares functions to divisor functions, and saw that as a beautiful consequence of this statement that we get the Leibniz sum formula for π/4 back out. The curious reader with appropriate background should now be able to consult the following sources for a more through analysis of the problem. 1

22 9.0 Appendix: Works Sited [CiCo] Cilleruello, J. & Cordoba, A. Trigonometric Polynomials and Lattice Points. Proceedings of the American Mathematical Society. Vol No. 4 (Aug. 199) pg [Cil93] Cilleruello, J. "The Distribution of Lattice Points on Circles." J. Number Th. 43, 198 0, [FrGo] Frasier, W & Gotlieb, C. A Calculation of the Number of Lattice Points in the Circle and Sphere. Mathematics of Computation. Vol 16, No 79. (July 196) pg 8 90 [Guy] Guy, R. K. "Gauß's Lattice Point Problem." F1 in Unsolved Problems in Number Theory, nd ed. New York: Springer Verlag, pp , [Har] Hardy, G. H. "On the Expression of a Number as the Sum of Two Squares." Quart. J. Math. 46, 63 83, [HiCo] Hilbert, D. and Cohn Vossen, S. Geometry and the Imagination. New York: Chelsea, pp , [Hux90] Huxley, M. N. "Exponential Sums and Lattice Points." Proc. London Math. Soc. 60, , [Hux93] Huxley, M. N. "Corrigenda: 'Exponential Sums and Lattice Points."' Proc. London Math. Soc. 66, 70, [Hux03] Huxley, M. N. "Exponential Sums and Lattice Points III." Proc. London Math. Soc. 87, , 003. [IwMo]Iwaniec, H & Mozzochi, C.J. On the Divisor and Circle Problems. Journal of Number Theory. 9. pg [KeSw] Keller, H.B. & Swenson, J.R. Experiments on the Lattice Problem of Gauss. Mathematics of Computation. Vol. 17, No. 83. (July 1963) pg 3 30

23 [Mit] Mitchell, W.C. The Number of Lattice Points in a k Dimensional Hypersphere. Mathematics of Computation. Vol. 0. No. 94. (April 1966) pg [MW] Weisstein, E. Mathworld. Gauss's Circle Problem. Sum of Squares Function. Hypersphere. Available Online. [NiZu] Niven, I & Zuckerman, H. Lattice Points in Regions. Proceedings of the American Mathematical Society. Vol 18. No.. (April 1967), pg [PM] PlanetMath.org Proof of Lagrange's Four Square Theorem. Available Online. [Shi] Shiu. P. Counting Sums of Two Squares: The Meissel Lehmer Method. Mathematics of Computation. Vol. 47. No (July 1986). pg [StSh] Stein, E. & Shakarchi, R. Complex Analysis. Princeton Lectures in Analysis. Chp. 10, 003 [Tit34] Titchmarsh, E. C. "The Lattice Points in a Circle." Proc. London Math. Soc. 8, , [Tit35] Titchmarsh, E. C. "Corrigendum. The Lattice Points in a Circle." Proc. London Math. Soc. 38, 555, [Tsa] Tsang. K. Counting Lattice Points in the Sphere. Bull. London Math. Soc. 3 (000) pg [WiAu] Wintner, Aurel. On the Lattice Problem of Gauss. American Journal of Mathematics. Vol. 63, No. 3. (July 1941),

24 Special Thanks: Special thanks to Professor Allan Greenleaf for providing valuable feedback for this thesis topic and reviewing the paper. Special thanks to Matjaz Kranjc, Dan Kneezel, and Megan Walter for making valuable editorial comments about this paper. Special thanks to Diane Cass for helping locate reference materials. Appendix: Notes on Mathematica The author used Mathematica to do a best fit curve to attempt to find an experimental value for error complexity. This notebook is available upon request from the author. 4

Lectures 5-6: Taylor Series

Lectures 5-6: Taylor Series Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

More information

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

More information

Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

9.2 Summation Notation

9.2 Summation Notation 9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

More information

The Ideal Class Group

The Ideal Class Group Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

More information

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

More information

PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

More information

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

DRAFT. Algebra 1 EOC Item Specifications

DRAFT. Algebra 1 EOC Item Specifications DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

More information

Ideal Class Group and Units

Ideal Class Group and Units Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

More information

Polynomial Operations and Factoring

Polynomial Operations and Factoring Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

More information

1.7 Graphs of Functions

1.7 Graphs of Functions 64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005 Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

More information

Overview of Math Standards

Overview of Math Standards Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x) SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

More information

The Epsilon-Delta Limit Definition:

The Epsilon-Delta Limit Definition: The Epsilon-Delta Limit Definition: A Few Examples Nick Rauh 1. Prove that lim x a x 2 = a 2. (Since we leave a arbitrary, this is the same as showing x 2 is continuous.) Proof: Let > 0. We wish to find

More information

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

More information

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

More information

Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook. Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

For example, estimate the population of the United States as 3 times 10⁸ and the

For example, estimate the population of the United States as 3 times 10⁸ and the CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

More information

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

More information

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships These materials are for nonprofit educational purposes

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

Course Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics

Course Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics Course Notes for Math 16: Mathematical Statistics Approximation Methods in Statistics Adam Merberg and Steven J. Miller August 18, 6 Abstract We introduce some of the approximation methods commonly used

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

a = bq + r where 0 r < b.

a = bq + r where 0 r < b. Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental

More information

2. THE x-y PLANE 7 C7

2. THE x-y PLANE 7 C7 2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

More information

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper. FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

More information

Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

More information

Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom. Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

More information

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

More information

Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them

More information

ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

More information

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w. hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

More information

Doug Ravenel. October 15, 2008

Doug Ravenel. October 15, 2008 Doug Ravenel University of Rochester October 15, 2008 s about Euclid s Some s about primes that every mathematician should know (Euclid, 300 BC) There are infinitely numbers. is very elementary, and we

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}

Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11} Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following

More information

4.3 Lagrange Approximation

4.3 Lagrange Approximation 206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

More information

South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

More information

Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

More information

Complex Function Theory. Second Edition. Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY

Complex Function Theory. Second Edition. Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY Complex Function Theory Second Edition Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY Contents Preface to the Second Edition Preface to the First Edition ix xi Chapter I. Complex Numbers 1 1.1. Definition

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

Continued Fractions. Darren C. Collins

Continued Fractions. Darren C. Collins Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history

More information

Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients

Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca

More information

To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions

To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

More information

MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials

MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial. The first half of the note is derived from

More information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

More information

MATH 132: CALCULUS II SYLLABUS

MATH 132: CALCULUS II SYLLABUS MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

More information

SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

More information

The van Hoeij Algorithm for Factoring Polynomials

The van Hoeij Algorithm for Factoring Polynomials The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

Factoring Polynomials

Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304 MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

More information

In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

More information

Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

More information