STRAND B: Number Theory. UNIT B2 Number Classification and Bases: Text * * * * * Contents. Section. B2.1 Number Classification. B2.


 Alisha Bailey
 10 months ago
 Views:
Transcription
1 STRAND B: Number Theory B2 Number Classification and Bases Text Contents * * * * * Section B2. Number Classification B2.2 Binary Numbers B2.3 Adding and Subtracting Binary Numbers B2.4 Multiplying Binary Numbers B2.5 Other Bases
2 B2 Number Classification and Bases B2. Number Classification The number system is classified into various categories. Natural numbers (or 'counting numbers') These are the positive whole numbers, namely, 2, 3,,,... Integers These are the set of positive and negative whole numbers, e.g., 0, 364, 7, 02. The set of integers is denoted by Z. Rational Numbers A rational number is a number which can be written in the form m, where m and n are integers. n For example, 4 5 is a rational number. A rational number is in its simplest form if m and n have no common factor and n is positive. The set of rational numbers is denoted by Q. Irrational Numbers There are numbers which cannot be written in the form m, where m and n are integers. n Examples of irrational numbers are π, 2, 3, 5. Terminating Decimals These are decimal numbers which stop after a certain number of decimal places. For example, 7 = is a terminating decimal because it stops (terminates) after 3 decimal places. Recurring Decimals These are decimal numbers which keep repeating a digit or group of digits; for example = is a recurring decimal. The six digits repeat in this order. Recurring decimals are written with dots over the first and last digit of the repeating digits, e.g
3 B2. Note All terminating and recurring decimals can be written in the form m, so they are n rational numbers. Real Numbers These are made up of all possible rational and irrational numbers; the set of real numbers is denoted by R. We can use a Venn diagram to illustrate this classification of number, as shown here. R + 2 Q + Q 2π π N Z R : Real numbers N : Natural numbers Z : Integers Q : Rational numbers Q + : Positive rational numbers Some numbers have been inserted to illustrate each set. Worked Example Classify the following numbers as integers, rational, irrational, recurring decimals, terminating decimals. 5, 7, 0.6, , 8,, 0, π 4, 49 2
4 B2. Number Rational Irrational Integer Recurring Terminating Decimal Decimal π 4 49 Worked Example 2 Show that the numbers and are rational. For a terminating decimal the proof is straightforward = = which is rational because m = 69 and n = 200 are integers. For a recurring decimal, we multiply by a power of ten so that after the decimal point we have only the repeating digits = Also = Subtracting the second equation from the first gives = 97 which is rational because m = 97 and n = ( ) = =
5 B2. Worked Example 3 Prove that 3 is irrational. Assume that 3 is rational so we can find integers m and n such that 3 = m n, and m and n have no common factors. Square both sides, 2 3 = m n 2 Multiply both sides by n 2, m 2 2 = 3n Since 3n 2 is divisible by 3, then m 2 is divisible by 3. Since m is an integer, m 2 is an integer and 3 is prime, then m must be divisible by 3. Let m = 3 p, where p is an integer n = m = ( 3p) = 9p 2 2 n = 3p Since 3p 2 is divisible by 3, n 2 is divisible by 3 and hence n is divisible by 3. We have shown that m and n are both divisible by 3. This contradicts the original assumption that 3 = m, where m and n are integers with n no factor common. Our original assumption is wrong. 3 is not rational it is irrational. Exercises. Classify the following numbers as rational or irrational, terminating or recurring decimals. (a) 00 (b) 06. (c) π 02. (d) 3 99 (e) 075. (f) π (g) (h) 6. 4 (i) π 2 (j) 5 (k) sin 45 (l) cos60 4
6 B2. 2. Where possible, write each of the following numbers in the form and n are integers with no common factors (a) 0.49 (b) 03. (c) 49 4 (e) 0.47 (f) 0. (g) 009. (i) 0.25 (j) (d) 7 (h) m n, where m Write each of the following recurring decimals in the form integers with no common factors. m, where m and n are n (a) 04. (b) (c) (d) 08. (e) (f) 05. (g) Given the rational numbers a = 2 3, b = 7 9, c = 5, d = 8 show that a+ b, c+ d, a+ c+ d, ab, cd, bc and abc are rational numbers. 5. Prove that the following numbers are irrational, 2, 5, 6. What happens if you try to prove that 4 is irrational using the proof by a contradiction method. 7. Write down two rational numbers and two irrational numbers which lie in each of the following intervals. (a) 0< x < (b) < x < 9 (c) 4< x < 2 (d) 98 < x < Which of these expressions can be written as recurring decimals and which can be written as nonrecurring decimals? 2 7, π, 7, 7, x = a + b State whether x is rational or irrational in each of the following cases, and show sufficient working to justify each answer. (a) a = 5 and b = 2 (b) a = 5 and b = 6 (c) a = 2 and b = 7 (d) a = 3 7 and b = 4 7 5
7 B2. 0. (a) Write down two irrational numbers that multiply together to give a rational number. (b) Which of the following numbers are rational? 2 2, 2 2, 4 2, 4 2, π 2, π 2. (a) A Mathematics student attempted to define an irrational number as follows: "An irrational number is a number which, in its decimal form, goes on and on." (i) Give an example to show that this definition is not correct. (ii) What must be added to the definition to make it correct? (b) Which of the following are rational and which are irrational? 4, 6, , 3 Express each of the rational numbers in the form integers. 3 2 p, where p and q are q 2. (a) Write down an irrational number that lies between 4 and 5. (b) N is a rational number which is not equal to zero. Show clearly why N must also be rational. 3. n is a positive integer such that n = 5. 4 correct to decimal place. (a) (i) Find a value of n. (b) (ii) Explain why n is irrational. Write down a number between 0 and that has a rational square root. B2.2 Binary Numbers We normally work with numbers in base 0. In this section we consider numbers in base 2, often called binary numbers. In base 0 we use the digits 0,, 2, 3, 4, 5, 6, 7, 8, and 9. In base 2 we use only the digits 0 and. Binary numbers are at the heart of all computing systems since, in an electrical circuit, 0 represents no current flowing whereas represents a current flowing. 6
8 B2.2 In base 0 we use a system of place values as shown below: Note that, to obtain the place value for the next digit to the left, we multiply by 0. If we were to add another digit to the front (left) of the numbers above, that number would represent 0 000s. In base 2 we use a system of place values as shown below: = = 73 Note that the place values begin with and are multiplied by 2 as you move to the left. Once you know how the place value system works, you can convert binary numbers to base 0, and vice versa. Worked Example Convert the following binary numbers to base 0: (a) (b) 0 (c) 000 For each number, consider the place value of every digit. (a) = 7 The binary number is 7 in base 0. (b) = 5 The binary number 0 is 5 in base 0. (c) = 02 The binary number 000 is 02 in base 0. 7
9 B2.2 Worked Example 2 Convert the following base 0 numbers into binary numbers: (a) 3 (b) (c) 40 We need to write these numbers in terms of the binary place value headings, 2, 4, 8, 6, 32, 64, 28,..., etc. (a) 2 3 = 2 + The base 0 number 3 is written as in base 2. (b) = The base 0 number is written as 0 in base 2. (c) = The base 0 number 40 is written as in base 2. Exercises. Convert the following binary numbers to base 0: (a) 0 (b) (c) 00 (d) 0 (e) 000 (f) 0 (g) (h) 000 (i) 0000 (j) 000 (k) 000 (l) Convert the following base 0 numbers to binary numbers: (a) 9 (b) 8 (c) 4 (d) 7 (e) 8 (f) 30 (g) 47 (h) 52 (i) 67 (j) 84 (k) 200 (l) Convert the following base 0 numbers to binary numbers: (a) 5 (b) 9 (c) 7 (d) 33 Describe any pattern that you notice in these binary numbers. What will be the next base 0 number that will fit this pattern? 4. Convert the following base 0 numbers to binary numbers: (a) 3 (b) 7 (c) 5 (d) 3 What is the next base 0 number that will continue your binary pattern? 8
10 B A particular binary number has 3 digits. (a) What are the largest and smallest possible binary numbers? (b) Convert these numbers to base When a particular base 0 number is converted it gives a 4digit binary number. What could the original base 0 number be? 7. A 4digit binary number has 2 zeros and 2 ones. (a) List all the possible binary numbers with these digits. (b) Convert these numbers to base A binary number has 8 digits and is to be converted to base 0. (a) What is the largest possible base 0 answer? (b) What is the smallest possible base 0 answer? 9. The base 0 number 999 is to be converted to binary. How many more digits does the binary number have than the number in base 0? 0. Calculate the difference between the base 0 number and the binary number, giving your answer in base 0. B2.3 Adding and Subtracting Binary Numbers It is possible to add and subtract binary numbers in a similar way to base 0 numbers. For example, + + = 3 in base 0 becomes + + = in binary. In the same way, 3 = 2 in base 0 becomes = 0 in binary. When you add and subtract binary numbers you will need to be careful when 'carrying' or 'borrowing' as these will take place more often. Key Addition Results for Binary Numbers + 0 = + = = Key Subtraction Results for Binary Numbers 0 = 0 = = 0 9
11 B2.3 Worked Example Calculate, using binary numbers: (a) + 00 (b) (c) + (a) (b) 0 (c) Note how important it is to 'carry' correctly. Worked Example 2 Calculate the binary numbers: (a) 0 (b) 0 (c) 00 0 (a) (b) 0 (c) Exercises. Calculate the binary numbers: (a) + (b) + (c) + (d) + 0 (e) 0 + (f) (g) + 00 (h) (i) (j) (k) 0 + (l) Calculate the binary numbers: (a) 0 (b) 0 0 (c) 0 (d) 00 0 (e) 00 (f) 000 (g) 0 0 (h) 0 0 (i) (j) (k) 0 (l) 0 0
12 B Calculate the binary numbers: (a) + (b) + (c) + (d) + Describe any patterns that you observe in your answers. 4. Calculate the binary numbers: (a) (b) (c) (d) Describe any patterns that you observe in your answers. 5. Solve the following equations, where all numbers, including x, are binary: (a) x + = 0 (b) x 0 = 0 (c) x 0 = 0 (d) x + 0 = 000 (e) x + = 0 (f) x 00 = 0 6. Calculate the binary numbers: (a) 0 (b) 00 (c) 000 (d) 0000 Describe any patterns that you observe in your answers. 7. (a) Convert the binary numbers 0 and 0 to base 0. (b) Add together the two base 0 numbers. (c) Add together the two binary numbers. (d) Convert your answer to base 0 and compare with your answer to (b). 8. (a) Convert the binary numbers 0 and 0 to base 0. (b) Calculate the difference between the two base 0 numbers. (c) Convert your answer to (b) into a binary number. (d) Calculate the difference between the two binary numbers and compare with your answer to (c). 9. Here are 3 binary numbers: Working in binary, (a) add together the two smaller numbers, (b) add together the two larger numbers,
13 B2.3 (c) (d) take the smallest number away from the largest number, add together all three numbers. 0. Calculate the binary numbers: (a) (b) B2.4 Multiplying Binary Numbers Long multiplication can be carried out with binary numbers and is explored in this section. Note that multiplying by numbers like 0, 00 and 000 is very similar to working with base 0 numbers. Worked Example Calculate the binary numbers: (a) 0 00 (b) (c) Check your answers to (a) and (c) by converting each number to base 0. (a) 0 00 = 000 (b) = (c) = Checking: (a) = = 44 and 4 = 44, as expected. (c) =
14 B = 432 and 27 6 = 432, as expected. Note: clearly it is more efficient to keep the numbers in binary when doing the calculations. Worked Example 2 Calculate the binary numbers: (a) 0 (b) 0 0 (c) 0 (d) 0 00 (a) 0 (b) (c) 0 (d) Exercises. Calculate the binary numbers: (a) 0 (b) (c) (d) (e) (f) (g) (h) Check your answers by converting to base 0 numbers. 3
15 B Calculate the binary numbers: (a) (b) 0 (c) 0 0 (d) 0 (e) 0 0 (f) 00 0 (g) 00 0 (h) 00 (i) 00 0 (j) Solve the following equations, where all numbers, including x, are binary: (a) (c) x = 0 (b) x 0 = (d) x = 0 0 x = 0 4. Multiply each of the following binary numbers by itself: (a) (b) (c) What do you notice about your answers to parts (a), (b) and (c)? What will you get if you multiply by itself? 5. Multiply each of the following binary numbers by itself: (a) 0 (b) 00 (c) 000 (d) 0000 What would you expect to get if you multiplied by itself? 6. Calculate the binary numbers: (a) ( ) (b) 0 ( 0) ( ) (d) 0 ( ) (c) Here are 3 binary numbers: Working in binary, (a) multiply the two larger numbers, (b) multiply the two smaller numbers. 8. (a) Multiply the base 0 numbers 45 and 33. (b) Convert your answer to a binary number. (c) Convert 45 and 33 to binary numbers. (d) Multiply the binary numbers obtained in part (c) and compare this answer with your answer to part (b). 4
16 B2.5 Other Bases The ideas that we have considered can be extended to other number bases. The table lists the digits used in some other number bases. Base Digits Used 2 0, 3 0,, 2 4 0,, 2, 3 5 0, 2, 3, 4 The powers of the base number give the place values when you convert to base 0. For example, for base 3, the place values are the powers of 3, i.e., 3, 9, 27, 8, 243, etc. This is shown in the following example, which also shows how the base 3 number 200 is equivalent to the base 0 number 44. Base ( 8) + (2 27) + ( 9) + (0 3) + (0 ) = 44 in base 0 The following example shows a conversion from base 5 to base 0 using the powers of 5 as place values. Base (4 625) + ( 25) + (0 25) + (0 5) + ( ) = 2626 in base 0 Worked Example Convert each of the following numbers to base 0: (a) 42 in base 6. (b) 374 in base 9. (c) 432 in base 5. (a) (4 36) + ( 6) + ( 2 ) = 52 in base 0 (b) (3 8) + (7 9) + (4 ) = 30 in base 0 (c) ( 25) + (4 25) + (3 5) + (2 ) = 242 in base 0 5
17 B2.5 Worked Example 2 Convert each of the following base 0 numbers to the base stated: (a) 472 to base 4, (b) 79 to base 7, (c) 342 to base 3. (a) For base 4 the place values are 256, 64, 6, 4,, and you need to express the number 472 as a linear combination of 256, 64, 6, 4 and, but with no multiplier greater than 3. We begin by writing 472 = ( 256) + 26 The next stage is to write the remaining 26 as a linear combination of 64, 6, 4 and. We use the fact that 26 = (3 64) + 24 and, continuing in this way, 24 = ( 6) = (2 4) + 0 Putting all these stages together, 472 = ( 256) + (3 64) + ( 6) + ( 2 4) + ( 0 ) = 320 in base 4 (b) For base 7 the place values are 49, 7,. 79 = (3 49) + (4 7) + (4 ) = 344 in base 7 (b) For base 3 the place values are 243, 8, 27, 9, 3,. 342 = ( 243) + ( 8) + (0 27) + (2 9) +( 0 3) + (0 ) = 0200 in base 3 Worked Example 3 Carry out each of the following calculations in the base stated: (a) base 5 (b) base 7 (c) base 5 (d) base 3 Check your answer in (a) by changing to base 0 numbers. 6
18 B2.5 (a) Note that 4 + = 0 in base 5. (b) Note that = in base 7. (c) Note that + 4 = 0 in base 5. (d) Note that, in base 3, 2 + = = = Checking in (a): (a) 5 4 ( 5) + (4 ) = (2 5) + ( ) = (4 5) + (0 ) = 20 and 9 + = 20, as expected. Worked Example 4 Carry out each of the following multiplications in the base stated: (a) 4 23 in base 5 (b) 22 2 in base 3 (c) in base 6 Check your answer to (b) by converting to base 0 numbers. 7
19 B2.5 (a) 4 Note that, in base 5, = = (b) 2 2 Note that, in base 3, = (c) 5 2 Note that, in base 6, = = = Checking in (b): (b) ( 9) + (2 3) + (2 ) = ( 3) + (2 ) = ( 8) + (0 27) + (0 9) + (3 ) + ( ) = 85 and 7 5 = 85, as expected. 8
20 B2.5 Exercises. Convert the following numbers from the base stated to base 0: (a) 42 base 5 (b) 333 base 4 (c) 728 base 9 (d) 20 base 3 (e) 47 base 8 (f) 62 base 7 (g) 35 base 6 (h) base 3 2. Convert the following numbers from base 0 to the base stated: (a) 24 to base 3 (b) 6 to base 4 (c) 32 to base 5 (d) 3 to base 6 (e) 34 to base 7 (f) 84 to base 9 (g) 42 to base 3 (h) 67 to base 5 3. Carry out the following additions in the base stated: (a) in base 4 (b) in base 9 (c) in base 8 (d) in base 3 (e) in base 9 (f) in base 6 4. In what number bases could each of the following numbers be written: (a) 23 (b) 2 (c) Carry out each of the following calculations in the base stated: (a) in base 4 (b) in base 5 (c) in base 3 (d) in base 4 (e) in base 7 (f) in base 5 (g) in base 4 (h) in base 5 Check your answers to parts (a), (c) and (e) by converting to base 0 numbers. 6. Carry out each of the following multiplications in the base stated: (a) 3 2 in base 4 (b) 4 3 in base 5 (c) 4 2 in base 6 (d) 3 5 in base 6 (e) 2 2 in base 3 (f) 8 8 in base 9 9
21 B Carry out each of the following multiplications in the base stated: (a) 2 in base 3 (b) 33 2 in base 4 (c) 3 24 in base 5 (d) 42 4 in base 5 (e) 6 24 in base 7 (f) in base 8 (g) in base 5 (h) 20 2 in base 3 Check your answers to parts (a), (c) and (e) by converting to base 0 numbers. 8. In which base was each of the following calculations carried out? (a) = (b) = 3 (c) 8 2 = 7 (d) 4 5 = 32 (e) 3 = 5 (f) 22 4 = 3 9. (a) Change 47 in base 8 into a base 3 number. (b) Change 32 in base 4 into a base 7 number. (c) Change 72 in base 9 into a base 4 number. (d) Change 324 in base 5 into a base 6 number. 0. In which base was each of the following calculations carried out? (a) 7 2 = 2272 (b) 22 2 = 02 (c) = 252 (d) =
MEP Y9 Practice Book A
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More information3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.
SOLUTIONS TO HOMEWORK 2  MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid
More informationUnit 1 Review Part 1 3 combined Handout KEY.notebook. September 26, 2013
Math 10c Unit 1 Factors, Powers and Radicals Key Concepts 1.1 Determine the prime factors of a whole number. 650 3910 1.2 Explain why the numbers 0 and 1 have no prime factors. 0 and 1 have no prime factors
More informationCONTENTS. Please note:
CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationCOMPASS Numerical Skills/PreAlgebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13
COMPASS Numerical Skills/PreAlgebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre
More informationA 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 informationNumber Systems I. CIS0082 Logic and Foundations of Mathematics. David Goodwin. 11:00, Tuesday 18 th October
Number Systems I CIS0082 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 11:00, Tuesday 18 th October 2011 Outline 1 Number systems Numbers Natural numbers Integers Rational
More informationMath 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =
Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could
More informationPreAlgebra Class 3  Fractions I
PreAlgebra Class 3  Fractions I Contents 1 What is a fraction? 1 1.1 Fractions as division............................... 2 2 Representations of fractions 3 2.1 Improper fractions................................
More informationSupplemental Worksheet Problems To Accompany: The PreAlgebra Tutor: Volume 1 Section 1 Real Numbers
Supplemental Worksheet Problems To Accompany: The PreAlgebra Tutor: Volume 1 Please watch Section 1 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm
More informationLesson Plan. N.RN.3: Use properties of rational and irrational numbers.
N.RN.3: Use properties of rational irrational numbers. N.RN.3: Use Properties of Rational Irrational Numbers Use properties of rational irrational numbers. 3. Explain why the sum or product of two rational
More informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationAccuplacer Arithmetic Study Guide
Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole
More information1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
More informationRules of Signs for Decimals
CHAPTER 6 C Rules of Signs for Decimals c GOAL Apply the rules of signs for calculating with decimals. You will need number lines a calculator with a sign change key Learn about the Math Positive and negative
More informationMath Circle Beginners Group October 18, 2015
Math Circle Beginners Group October 18, 2015 Warmup problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd
More informationThis assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More information1 Number System. Introduction. Face Value and Place Value of a Digit. Categories of Numbers
RMAT Success Master 3 1 Number System Introduction In the decimal number system, numbers are expressed by means of symbols 0, 1,, 3, 4, 5, 6, 7, 8, 9, called digits. Here, 0 is called an insignificant
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On wellordering and induction: (a) Prove the induction principle from the wellordering principle. (b) Prove the wellordering
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationRational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have
8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationAPPLICATIONS OF THE ORDER FUNCTION
APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and
More informationOct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: Our standard number system is base, also
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationMathematics Success Grade 8
T92 Mathematics Success Grade 8 [OBJECTIVE] The student will create rational approximations of irrational numbers in order to compare and order them on a number line. [PREREQUISITE SKILLS] rational numbers,
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationInstructions for SA Completion
Instructions for SA Completion 1 Take notes on these Pythagorean Theorem Course Materials then do and check the associated practice questions for an explanation on how to do the Pythagorean Theorem Substantive
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationCalculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1
Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors.
More informationMarch 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationIndiana 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 informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationNow that we have a handle on the integers, we will turn our attention to other types of numbers.
1.2 Rational Numbers Now that we have a handle on the integers, we will turn our attention to other types of numbers. We start with the following definitions. Definition: Rational Number any number that
More informationWarm up. Connect these nine dots with only four straight lines without lifting your pencil from the paper.
Warm up Connect these nine dots with only four straight lines without lifting your pencil from the paper. Sometimes we need to think outside the box! Warm up Solution Warm up Insert the Numbers 1 8 into
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationMath Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
More informationAnchorage School District/Alaska Sr. High Math Performance Standards Algebra
Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,
More informationElementary 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 information2. Perform the division as if the numbers were whole numbers. You may need to add zeros to the back of the dividend to complete the division
Math Section 5. Dividing Decimals 5. Dividing Decimals Review from Section.: Quotients, Dividends, and Divisors. In the expression,, the number is called the dividend, is called the divisor, and is called
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationNUMBER SYSTEMS. 1.1 Introduction
NUMBER SYSTEMS 1.1 Introduction There are several number systems which we normally use, such as decimal, binary, octal, hexadecimal, etc. Amongst them we are most familiar with the decimal number system.
More informationMaths Level Targets. This booklet outlines the maths targets for each sublevel in maths from Level 1 to Level 5.
Maths Level Targets This booklet outlines the maths targets for each sublevel in maths from Level 1 to Level 5. Expected National Curriculum levels for the end of each year group are: Year 1 Year 2 Year
More information1.2 Linear Equations and Rational Equations
Linear Equations and Rational Equations Section Notes Page In this section, you will learn how to solve various linear and rational equations A linear equation will have an variable raised to a power of
More informationExponents, Radicals, and Scientific Notation
General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =
More informationSection 33 Approximating Real Zeros of Polynomials
 Approimating Real Zeros of Polynomials 9 Section  Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros
More informationTI83 Plus Graphing Calculator Keystroke Guide
TI83 Plus Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a keyshaped icon appears next to a brief description of a feature on your graphing calculator. In this
More informationIrrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.
Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that
More informationContents. Introduction and Notes pages 23 (These are important and it s only 2 pages ~ please take the time to read them!)
Page Contents Introduction and Notes pages 23 (These are important and it s only 2 pages ~ please take the time to read them!) Systematic Search for a Change of Sign (Decimal Search) Method Explanation
More informationAlgebra. Selfstudy Textbook for Junior Secondary School. Content. First Term (for Year 8 & 9)
Content Selfstudy Textbook for Junior Secondary School Algebra First Term (for Year & 9) Published by Chiu Chang Math Books & Puzzles Co. Provided by Chiu Chang Mathematics Education Foundation Chapter
More informationI know when I have written a number backwards and can correct it when it is pointed out to me I can arrange numbers in order from 1 to 10
Mathematics Targets Moving from Level W and working towards level 1c I can count from 1 to 10 I know and write all my numbers to 10 I know when I have written a number backwards and can correct it when
More informationMathematics (Project Maths Phase 2)
2013. M230S Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certiﬁcate Examination 2013 Sample Paper Mathematics (Project Maths Phase 2) Paper 2 Higher Level Time: 2 hours, 30 minutes
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More informationInteger 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 informationSet Theory Basic Concepts and Definitions
Set Theory Basic Concepts and Definitions The Importance of Set Theory One striking feature of humans is their inherent need and ability to group objects according to specific criteria. Our prehistoric
More informationAddition Methods. Methods Jottings Expanded Compact Examples 8 + 7 = 15
Addition Methods Methods Jottings Expanded Compact Examples 8 + 7 = 15 48 + 36 = 84 or: Write the numbers in columns. Adding the tens first: 47 + 76 110 13 123 Adding the units first: 47 + 76 13 110 123
More informationSquare Roots. Learning Objectives. PreActivity
Section 1. PreActivity Preparation Square Roots Our number system has two important sets of numbers: rational and irrational. The most common irrational numbers result from taking the square root of nonperfect
More informationChapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
More informationIn 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 informationGREATEST COMMON DIVISOR
DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their
More informationMath 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 informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationYOU CAN COUNT ON NUMBER LINES
Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationThis 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 information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2
More informationM13/5/MATSD/SP1/ENG/TZ2/XX/M MARKSCHEME. May 2013 MATHEMATICAL STUDIES. Standard Level. Paper 1. 25 pages
M13/5/MATSD/SP1/ENG/TZ/XX/M MARKSCHEME May 013 MATHEMATICAL STUDIES Standard Level Paper 1 5 pages M13/5/MATSD/SP1/ENG/TZ/XX/M This markscheme is confidential and for the exclusive use of examiners in
More informationKnowing and Using Number Facts
Knowing and Using Number Facts Use knowledge of place value and Use knowledge of place value and addition and subtraction of twodigit multiplication facts to 10 10 to numbers to derive sums and derive
More informationKey. Introduction. What is a Fraction. Better Math Numeracy Basics Fractions. On screen content. Narration voiceover
Key On screen content Narration voiceover Activity Under the Activities heading of the online program Introduction This topic will cover how to: identify and distinguish between proper fractions, improper
More informationCopy 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 informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More information6 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 informationJust the Factors, Ma am
1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive
More informationUNDERSTANDING ALGEBRA JAMES BRENNAN. Copyright 2002, All Rights Reserved
UNDERSTANDING ALGEBRA JAMES BRENNAN Copyright 00, All Rights Reserved CONTENTS CHAPTER 1: THE NUMBERS OF ARITHMETIC 1 THE REAL NUMBER SYSTEM 1 ADDITION AND SUBTRACTION OF REAL NUMBERS 8 MULTIPLICATION
More informationBinary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
More informationn th roots of complex numbers
n th roots of complex numbers Nathan Pflueger 1 October 014 This note describes how to solve equations of the form z n = c, where c is a complex number. These problems serve to illustrate the use of polar
More informationCommon Core Standards for Fantasy Sports Worksheets. Page 1
Scoring Systems Concept(s) Integers adding and subtracting integers; multiplying integers Fractions adding and subtracting fractions; multiplying fractions with whole numbers Decimals adding and subtracting
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More information2015 Junior Certificate Higher Level Official Sample Paper 1
2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list
More information25 Integers: Addition and Subtraction
25 Integers: Addition and Subtraction Whole numbers and their operations were developed as a direct result of people s need to count. But nowadays many quantitative needs aside from counting require numbers
More informationVocabulary 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 informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationFractions and Decimals
Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first
More informationMajor Work of the Grade
Counting and Cardinality Know number names and the count sequence. Count to tell the number of objects. Compare numbers. Kindergarten Describe and compare measurable attributes. Classify objects and count
More informationGuidance paper  The use of calculators in the teaching and learning of mathematics
Guidance paper  The use of calculators in the teaching and learning of mathematics Background and context In mathematics, the calculator can be an effective teaching and learning resource in the primary
More informationPOLAR COORDINATES DEFINITION OF POLAR COORDINATES
POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationIndices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková
Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead
More informationAppendix F: Mathematical Induction
Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another
More informationMathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)
( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =
More informationBridging Documents for Mathematics
Bridging Documents for Mathematics 5 th /6 th Class, Primary Junior Cycle, PostPrimary Primary PostPrimary Card # Strand(s): Number, Measure Number (Strand 3) 25 Strand: Shape and Space Geometry and
More information