Add: = /10 + 0/ / /10 + 7/100

Transcription

1 9-1 DECIM ALS Their PLACE in the W ORLD Math 10 F8 Definition: DECIM AL NUM BERS If x is any w hole number, x. d1 d d3 d 4... represents the number: x + d 1/10 + d /100 + d 3/ d 4/ d 5/ ( which is also known as: x + d1x10 + dx10 + d3x10 + d4x10 + d5x10... ) so, e.g.: = /10 + 0/ / / / fully expanded: 3x10 + x10 + 1x10 + 5x10 + 0x10 + 6x10 + 7x10 + 8x10 The "." (in , e.g.) is called the decimal point. The place values of the digits to the right of the decimal point are tenths, hundredths, thousandths, tenthousandths, hundred-thousandths, millionths, etc. (Not to be confused with tens, hundreds, thousands, etc.) n In general: the place value of the digit in the nth decimal place (n places after the decimal point) is Notice this is not symmetric... the first place to the left of the decimal point is 10, not 10...! Read and expand into long form (w ords): OPERATIONS ON DECIMAL NUMBERS We know how to add, subtract, multiply and divide decimals; w e now look at why. Add: = /10 + 0/100 + / /10 + 7/100 CHIP M ODEL M A Y HELP! Subtract /10 + 0/ ( 3 + 4/10 + 7/100 )...What's w rong w ith this picture? Multiplication & Division by 10' s:.17 x 10 = ( + 1/10 + 7/100 ) x 10 = x10 + (1/10)x10 + (7/100)x10 = x /10 = 1.7 Similarly,.17 x 10 = ( + 1/10 + 7/100 ) x 100 = (1/10) (7/100) 100 = = 17 )1 1.7 x 10 = 1.7 x 1/10 = = ( /10 ) 10 = (7/10) 10 = =.17 Similarly, 1.7 x 10 = 1.7 x 1/100 = =.17 M ULTIPLICATION BY 10 INCREASES THE PLACE VALUE OF EACH DIGIT (BY ONE PLACE). DIVISION BY 10 DOES THE OPPOSITE. Making sense out of the multiplication and division algorithms: 3.5 (35 x 1/10) x.17 x (17 x 1/100) 15. x = (17 x 35 ) x (1/10) x (1/100) = (17 x 35 ) x (1/1000) 3 = (17 x 35 ) / (10 ) Estimate the total!

2 9-1A DECIMALS AROUND the W ORLD Math 10 f8 Rounding decimals: Round to the nearest hundredth. The nearest tw o decimals in hundredths are 4.36 and The nearer is 4.37, since it is.003 greater than 4.367, as opposed to 4.36, w hich is.007 less than This is apparent from a number line view : Now round to the nearest hundredth: either rounds down to 4.36, or up to 4.37: )))))) When a number is rounded, it has been replaced w ith a nearby, but less precise, value. The difference betw een the value before rounding and the rounded-off value is called the rounding error. The problems w ith rounding: How do you round to the nearest hundredth? The nearest numbers terminating in the hundredths' place are 4.36 and 4.37, both.005 from There are three "rounding rules" that may be adopted to cover such instances: 1) Round to nearest even: The most w idely adopted rule. Favored because about half the time you round up, half the time you round dow n; so a long series of rounding tends to average out the errors to zero. ) Round to nearest odd: Same as above, but opposite, for contrary people. 3) Round up: This is the simplest because any decimal w ith "5" in the next place (after the one to w hich the number is being rounded) gets rounded up. The disadvantage to this rule is that by alw ays rounding halfw ay-points up, totals are too high on the average. If you're the tax collector, you round up! Eg. Round to the nearest: thousandth tenth unit hundred thousand Caution: Don' t round "piecemeal": For example, Teddy rounded to 4.98 this way: Help Teddy: If is to be rounded to the nearest hundredth w ill either truncate to 4.97 or round up to 4.98 w hich of these is closer to ? What is the halfw ay point betw een 4.97 and 4.98?...is above or below that "halfw ay point"? Caution: Don't round too soon:...especially w hen using a calculator, don't round until the computation is complete! Calculators carry extra digits " for free", in most instances. This method minimizes error from rounding. ( ) For example, to compute 3.50 x, since 3.50 expresses 3 digits of precision, w e w ish to have three digits in the final result. If w e round the square root of too early, w e risk an erroneous answ er If w e round to 1.41, then multiply by 3.50, w e obtain 4.935, w hich w e round to 4.94; how ever, if w e carry the extra decimals and multiply by 3.50, w e obtain , w hich rounds to 4.95 (even x 3.50 w ill give this result). Conclusion: don't round too soon. Caution: Express correctly! Round 8, to the nearest tenth. Round 8, to the nearest ten. 8, ,500 A number w hich has been rounded to 3400 is betw een 3350 & A number w hich has been rounded to or 3400 is betw een &

3 9-1B COMPUTATIONS WITH SCIENTIFIC NOTATION MATH 10 f8 this material w as discussed w eeks ago, but it makes sense to review it now is difficult to see. We write, alternately, 540,000,000,000,000 to help count the zeroes and figure the place value. Even so, most do not fully comprehend the words "billion" and "trillion". Thus, to many of us, reading or hearing the number above is five hundred forty trillion is not illuminating. Very large numbers (as w ell as very small numbers) w ritten in our traditional numeration system are cumbersome to w rite, read and use. For purposes of expressing, multiplying and dividing such numbers, alternate means p are often employed, most commonly scientific notation w rite as x.ddd... x 10. (w here x may not be 0.) Use of scientific notation requires: recognition of place value expressed in exponential form, ability to round decimal numbers, full command of the arithmetic properties of multiplication and division, understanding of significant digits (this relates to rounding). Write in scientific notation form. (Check by multiplying!) Key idea: Focus on the 6 Examples: 7,000,000 = 7 x 10 leading digit. If it is in 7,650,000 = 7.65 x 10 6 the right place, the rest = 7 x 10 w ill follow!.019 = 1.9 x 10 Try it: 540,000,000,000 = 5.4 x 10 Fill in the = 10.0 x 10 = = 7.13 x 10 exponents... And be able to convert from Scientific Notation to Standard Form: x 10 = 3.70 x 10 = 4.50 x 10 = 4.50 x 10 = 3784 Arithmetic in Scientific Notation: (3.1 x 10 )x(4.5 x10 ) 7 4 = (3.1 x 4.5) x (10 x 10 ) (1.36 x 10 ) * (4.34 x 10 ) / (.78 x 10 ) = (4.34 /.78) x (10 /10 ) 11 = x No knees!: * 5.67 x x x x x 10 1 Use your calculator to assist in computing: * (5.407 x 10 ) x (3.66 x 10 ) ( x 10) = x10 Express the result w ith correct number of significant digits in scientific notation. 4 5 ( x x10 )

4 9 - Rationals & Decimals M ath 1 0 f8 Note there are three forms of decimals: terminating (.05) non-terminating repeating (.3333 ) and non-terminating non-repeating ( ) RATIONAL NUM BERS M AY BE EX PRESSED AS DECIM ALS Converting rational fraction to decimal form: Use the long division algorithm. E.g. 7 = 1 = 3 = 3 = 1 = 9 = 3 = = 5 = 4 = = 5 = 1 = 1 = A rational fraction in reduced form is equivalent to a terminating decimal if, and only if, the denominator (of the reduced fraction) has only ' s and/or 5' s as prime factors. For example, 1/15 terminates in three places because w hich happens because 10 is a multiple of 5 ))) = )))) = ))))))))) = )))) = (... and, thus, 1000 is a multiple of 5 ) If a rational fraction in reduced form has denominator w ith prime factors other than and 5, the decimal does not terminate. For example: Can 1 be written as x?... as x?... x? Rationals w ritten as decimals may be TERMINATING (e.g. 3/5 =.6 ) or NON-TERMINATING (e.g. 3/11 =.77 ). Rationals w hich, reduced, have denominators w ith prime factors & 5 only are TERMINATING decimals because: Other rationals have NON-TERMINATING, REPEATING decimal expansions because: A DECIM AL NUM ERAL M AY BE EX PRESSED AS RATIO OF TW O INTEGERS, IF... THE DECIM AL IS TERM INATING: A terminating decimal has a finite number of digits to the right of the decimal point. Such decimals are easily expressed in rational form (ratio of tw o integers); just read them aloud. E.g.: 3.47 = 3 + 4/10 + 7/100 = 300/ / /100 = ( )/100 = 347/ = You can make even shorter w ork of such conversions: = 5030/ If a decimal number terminates in the nth place after the decimal, it is equivalent to a fraction w hose numerator n is the number formed by the digits w ithout the decimal point, and denominator 10 (w here n is number of digits after the decimal point). THE DECIMAL IS NON-TERMINATING REPEATING: Call the repeating (" unknow n" ) number x. n Multiply x by 10 w here n is the length of the repetend (the number of digits that repeat). n n Subtract x from 10 x to obtain (10-1)x. p p Solve for x; multiply resulting fraction by 10 /10 if necessary to clear a decimal. For example: x = x = x = so x = = = 99x = 99 By the w ay, notice the expression of a rational number as a decimal is not unique..9 = Every terminating decimal has an alternate form. EG = THE SET OF TERM INATING & REPEATING DECIM ALS IS EQUIVALENT TO THE SET OF RATIONAL NUM BERS

5 9-3 REAL NUMBERS!!! 8 1 The Pythagoreans believed that "Number rules the universe"; everything can be described in terms of numbers, and, further, that b c all numbers are rational integers, or ratios of integers. 3 Pythagoras proved w hat is called the Pythagorean Theorem : a a = b + c the sum of the squares of the sides of a right triangle is the square of the hypotenuse. (and this happens only in a right triangle.) Pythagoras' view of the describability of the universe in terms of rational numbers w as contradicted destroyed by the very theorem w hich now bears his name. For if w e construct a right triangle w ith sides of equal length 1, then w e can demonstrate that the hypotenuse (whose length ought to correspond to some number), is, and cannot be expressed as a rational number cannot be expressed as the ratio of two integers. As we argue below: First w e note that every w hole number has a unique prime factorization. r 1 r r 3 r 4 rk w = p 1 p p 3 p 4 pk Then w must have a prime factorization w ith the same primes to even pow ers: r r r r r w = p p p p p k k Suppose = p/q w here p,q Z, q 0 [after all, that's w hat it w ould mean for to be rational].... and w e may assume p/q has been reduced to low est terms (so p and q have NO common factors). Multiplying both sides by q and squaring gives us q = p. Which means must be a factor of p, w hich in turn implies must be a factor of p. So p has an odd pow er of in its prime factorization... This is not possible. Since every part of our argument is true, the only flaw must be in the assumption w e made in the first place, that can be w ritten in rational form.] Pythagoras and his followers were devastated. Even today, we are troubled to discover there are numbers which cannot be written in the friendly form of a ratio of integers but the irritation is diminished by the comforting thought that although such numbers exist, they are relatively few, and may be mostly ignored. NOT! Let's make a small digression. We now know these irrationals exist, and is one of them, and can not be w ritten as a ratio of integers; but w hat is the nature of an irrational? What is the decimal representation of an irrational number? We have previously seen that a rational number can be expressed in decimal form using the division algorithm, and that the decimal form either terminates or repeats. Furthermore, any terminating or repeating decimal can be expressed as a ratio of two integers. Therefore: The set of rational numbers is exactly the set of terminating and repeating decimals. The Irrationals (reals not expressible as ratio of two integers) are non-terminating, non-repeating decimals. Consider, e.g or The sum of tw o rationals is rational. What about the sum of tw o irrationals? This can be a little difficult to explore, since we have no general arithmetic algorithms for irrationals. What is + 3? (Hint: square that!) Is the sum of two irrationals alw ays irrational?...how about + 8? ? What is the sum of a rational number and an irrational? Consider, for example, + ½. Intuitively, w e have a non-terminating non-repeating decimal plus.5, the total of w hich should be non-terminating non-repeating. However, we look for proof... 1 students/disciples of the school of Pythagoras at Crotona in Southern Italy... or so he is credited. Pythagoras may have proved it, perhaps one or some of his student follow ers. 3 though it w as know n to the Babylonians & Egyptians at least centuries earlier

6 We can easily prove + ½ is irrational: Suppose + ½ = a/b... where a/b is rational. Then = a/b ½ which, since it is the difference of two rational numbers, is a rational number. This contradicts the known status of it s irrational. The argument above showing is irrational may be used to show that the square root of any prime (or of any composite number whose prime factors are not all to an even power) is not rational. Furthermore, there are other irrational numbers (pi and e for instance) that " occur" in numerous circumstances. In addition, for every irrational, we can construct a whole family of irrationals by adding any rational. (Irrational + rational is...) Thus w e can easily see there are at least as many irrational numbers as rationals. It was not until the end of the nineteenth century that a means of tackling the cardinality of infinite sets was devised to settle this question: How does the size of the set of irrationals compare with the size of the set of rationals? 4 Georg Cantor w as primarily responsible for the development of theory of sets & classes, w ith particular emphasis on the infinite. He devised the means of comparing cardinality of sets by mapping, or establishing a 1-1 correspondence between two sets to show they are of the same cardinality. (Recall our demonstrations that the cardinality of N is equivalent to that of Z.) Although the rationals are dense in the number line, the cardinality of Q is the same as that of N! But the cardinality of the irrationals is greater! Here is Cantor's ingeniously simple proof: Any set w hose cardinality is that of N can be listed. (The act of listing establishes a 1-1 correspondence with N.) Suppose a " list" of all irrationals is presented. We will show that the list does not CANNOT contain all the irrationals, by constructing an irrational number that is not in the list. Suppose the first few numbers in the alleged list are: m.ddddd e.g n.eeeee o.fffff p.ggggg We select a digit different from d 1 as the first decimal digit of our number; select a digit different from e as the second digit of our number; select a digit different from f 3 as the third digit, and so on. The number so constructed cannot be the first number in the list because it differs in the first decimal place; cannot be the second number in the list because it differs in the second decimal place from that number; and so on. So it is not in the list at all! Thus the list is not complete. Therefore, it is not possible to list all the irrationals, and thus they cannot be put into 1-1 correspondence with the natural numbers. The cardinality of the set of irrationals is greater than the cardinality of N. Thus although the rationals are dense in the number line (no interval gaps), there are other numbers (irrationals) in the number line, and they far outnumber the rationals! Properties of Arithmetic Operations on REAL numbers: Addition and multiplication on the set of all real numbers have the follow ing properties: CLOSURE; COMMUTATIVITY; ASSOCIATIVITY; IDENTITY (0 for + ; 1 for x); INVERSES for + : for any decimal a, -a is the additive inverse. INVERSES for x: for each decimal b other than 0, there is another decimal number, 1/b such that bx(1/b) = 1 Subtraction and Division have the closure property except for division by Cantor's theory of the infinite so rocked the scientific and, particularly, mathematical w orld of his time that some antagonists w ere able to block his advancement; he w as view ed as subversive by some, unbalanced by others. The bitterness of his life, coupled w ith insecurity bred in Cantor, and perhaps some genetic predisposition, led to a number of breakdow ns for w hich Cantor w as hospitalized. In the early tw entieth century his w ork began to be recognized as the profoundly real w ork of a genius; this recognition w as too little and too late.

7 Our new Universe: N W Z Q R N = { 1,, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, } N W = { 0, 1,, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, } W Z = {, 5, 4, 3,, 1, 0, 1,, 3, 4, 5, } Z Q = p { / q p Z, q Z, and q 0 } Q = { 0, / 1, / 1, /, /, / 3, / 3, / 3, / 1, / 1, / 3, } = The set of all terminating and repeating decimals R = The set of all decimals R = The set containing all rationals and all irrationals = Q I = Set of all decimals that terminate or repeat Set of all nonterminating nonrepeating decimals = Set of all decimals that can Set of all decimals that cannot be w ritten as a ratio of integers be w ritten as a ratio of integers Note: If we think of R as our new Universe, I, the set of irrationals, is the complement of Q, the set of rationals. Every real number is rational or irrational one or the other/

8 13-3 Some details & footnotes Math 10. Trichotomy: Given any tw o real numbers a and b, exactly one of the following holds: a < b or a > b or a = b. In particular, given any real number r, then either r is positive (r > 0), or r is negative (r < 0), or r is 0. Some important details: " The irrationals form a set disjoint (separate) from the rationals. " Together the rationals and irrationals make up R, the set of real numbers. " N W Z Q R. (Each of these is contained in is part of the next.) " The set of real numbers is larger than the set of rationals, and is dense in the number line! " The set of real numbers can be thought of as corresponding to the points (all points) on a line. (The set of real numbers may also be thought of as the set of all decimal numbers, including those w hich terminate or repeat (rationals) and those w hich neither terminate nor repeat ( irrationals). ) Radicals & Fractional exponents:, called radical, stands for the principle square root the positive root of a positive number. To indicate the negative of the square root, w e w rite. For instance, 16 = = -4. 1/ x may be written in the form x. Notice that x x = x (provided x exists)... And this can also be written 1/ 1/ x x = x

9 (STRICTLY Optional!!) More Radicals & Fractional exponents: 3 The, or radical, symbol is used to denote other roots of numbers, such as cube root: 7 = 3 n 6 6 We define: denotes the nth root of a number. For instance, 64 = because = 64. 1/ 3 1/3 1/q q Just as x may be written in the form x, x is x and in general: x = x. p/q 1/q p q p...and more generally: x = (x ) = ( x). 1/3 3 /5 1/5 E.g. 8 = 8 = 3 = (3 ) = () = 4 1/3 3 /5 1/5 8 = 1/ 8 = 1/ 3 = 1/(3 ) = 1/() = 1/4 QUIZ YOURSELF! (1-15 simplify; 16-0 answ er rational, irrational, unpredictable)(* = optional!) 1/4 1/5 1/3 1/3 1*. 16 = *. 3 = 3*. 7 = 4*. 64 = 1/ 1/ 1/ 1/ 3/ 5. 4 = = = 8*. 4 = a 1/ 1/ *. (/) b = 10*. (16/9) = 11. x = 1. 7ab = / = x y = 15. { 64x y (x+y)} = 16. The sum of a rational and an irrational is: 17. The product of a rational and irrational is: 18. The sum (difference) of tw o rationals is: 19. The sum (difference) of tw o irrationals is: 0. The product (quotient) of tw o rationals is: 1. The product (quotient) of tw o irrationals is: Answ ers?: / ( / ) or / 10. 3/4 11. (x) = x b 1/ b 3 3 a a ab = 4 9a b b = 3ab b = = x y = 3 (x ) y = 4 3x y y = 1x y y / / 3 4 1/ 15. { 64x y (x+ y) } = { xy (x+ y) } = 8x y (x+ y) {x} 16. irrational (see prior discussion) 17. if the rational is zero, the product is rational because it's 0; otherwise, irrational. (Consider, eg,, and... If q Q and q b = p Q then b = p/q must be rational unless q is zero.) 18. rational (+ or ) 19. unpredictable (+ or ) (Consider + = ; (9 ) + ( 5) = 4) 0. rational ( or ) 1. unpredictable ( or ) (Consider = ; 3 = 6)