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

Size: px
Start display at page:

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

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 non-recurring 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 4-digit binary number. What could the original base 0 number be? 7. A 4-digit 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

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 information

CHAPTER 3 Numbers and Numeral Systems

CHAPTER 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 information

3 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.

3 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 information

Unit 1 Review Part 1 3 combined Handout KEY.notebook. September 26, 2013

Unit 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 information

CONTENTS. Please note:

CONTENTS. 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 information

26 Integers: Multiplication, Division, and Order

26 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 information

We now explore a third method of proof: proof by contradiction.

We 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 information

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13 COMPASS Numerical Skills/Pre-Algebra 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 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

Number Systems I. CIS008-2 Logic and Foundations of Mathematics. David Goodwin. 11:00, Tuesday 18 th October

Number Systems I. CIS008-2 Logic and Foundations of Mathematics. David Goodwin. 11:00, Tuesday 18 th October Number Systems I CIS008-2 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 information

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 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 information

Pre-Algebra Class 3 - Fractions I

Pre-Algebra Class 3 - Fractions I Pre-Algebra 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 information

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 1 Real Numbers

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 1 Real Numbers Supplemental Worksheet Problems To Accompany: The Pre-Algebra 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 information

Lesson Plan. N.RN.3: Use properties of rational and irrational numbers.

Lesson 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 information

Chapter 4 -- Decimals

Chapter 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 information

Accuplacer Arithmetic Study Guide

Accuplacer 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 information

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.

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. 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 information

Rules of Signs for Decimals

Rules 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 information

Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Math Circle Beginners Group October 18, 2015 Warm-up 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 information

This 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 information

Properties of Real Numbers

Properties 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 information

1 Number System. Introduction. Face Value and Place Value of a Digit. Categories of Numbers

1 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 information

Click on the links below to jump directly to the relevant section

Click 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 information

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

More information

0.8 Rational Expressions and Equations

0.8 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 information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational 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 information

a 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.

a 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 information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS 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 information

Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8

Oct: 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 information

So let us begin our quest to find the holy grail of real analysis.

So 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 information

Homework until Test #2

Homework 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 information

Mathematics Success Grade 8

Mathematics 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 information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

Discrete 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 information

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.1. 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 information

3.1. RATIONAL EXPRESSIONS

3.1. 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 information

MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

MATHEMATICAL 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 information

Instructions for SA Completion

Instructions 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 information

Applications of Fermat s Little Theorem and Congruences

Applications 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 information

Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1

Calculate 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 information

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

March 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 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

Exponents and Radicals

Exponents 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 information

Now that we have a handle on the integers, we will turn our attention to other types of numbers.

Now 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 information

Warm 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. 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 information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: 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 information

Math Common Core Sampler Test

Math 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 information

Anchorage School District/Alaska Sr. High Math Performance Standards Algebra

Anchorage 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 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

2. 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

2. 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 information

5.4 The Quadratic Formula

5.4 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 information

NUMBER SYSTEMS. 1.1 Introduction

NUMBER 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 information

Maths Level Targets. This booklet outlines the maths targets for each sub-level in maths from Level 1 to Level 5.

Maths Level Targets. This booklet outlines the maths targets for each sub-level in maths from Level 1 to Level 5. Maths Level Targets This booklet outlines the maths targets for each sub-level 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 information

1.2 Linear Equations and Rational Equations

1.2 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 information

Exponents, Radicals, and Scientific Notation

Exponents, 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 information

Section 3-3 Approximating Real Zeros of Polynomials

Section 3-3 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 information

TI-83 Plus Graphing Calculator Keystroke Guide

TI-83 Plus Graphing Calculator Keystroke Guide TI-83 Plus Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a key-shaped icon appears next to a brief description of a feature on your graphing calculator. In this

More information

Irrational 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. 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 information

Contents. Introduction and Notes pages 2-3 (These are important and it s only 2 pages ~ please take the time to read them!)

Contents. Introduction and Notes pages 2-3 (These are important and it s only 2 pages ~ please take the time to read them!) Page Contents Introduction and Notes pages 2-3 (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 information

Algebra. Self-study Textbook for Junior Secondary School. Content. First Term (for Year 8 & 9)

Algebra. Self-study Textbook for Junior Secondary School. Content. First Term (for Year 8 & 9) Content Self-study 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 information

I 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

I 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 information

Mathematics (Project Maths Phase 2)

Mathematics (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 information

Florida 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 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 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

Set Theory Basic Concepts and Definitions

Set 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 information

Addition Methods. Methods Jottings Expanded Compact Examples 8 + 7 = 15

Addition 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 information

Square Roots. Learning Objectives. Pre-Activity

Square Roots. Learning Objectives. Pre-Activity Section 1. Pre-Activity 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 non-perfect

More information

Chapter 3. if 2 a i then location: = i. Page 40

Chapter 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 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

GREATEST COMMON DIVISOR

GREATEST 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 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

Solutions of Linear Equations in One Variable

Solutions 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 information

YOU CAN COUNT ON NUMBER LINES

YOU 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 information

Roots and Coefficients of a Quadratic Equation Summary

Roots 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 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

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment 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 information

Knowing and Using Number Facts

Knowing 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 two-digit multiplication facts to 10 10 to numbers to derive sums and derive

More information

Key. Introduction. What is a Fraction. Better Math Numeracy Basics Fractions. On screen content. Narration voice-over

Key. Introduction. What is a Fraction. Better Math Numeracy Basics Fractions. On screen content. Narration voice-over Key On screen content Narration voice-over Activity Under the Activities heading of the online program Introduction This topic will cover how to: identify and distinguish between proper fractions, improper

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

Lesson 9.1 Solving Quadratic Equations

Lesson 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 y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

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

Just the Factors, Ma am

Just 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 information

UNDERSTANDING ALGEBRA JAMES BRENNAN. Copyright 2002, All Rights Reserved

UNDERSTANDING 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 information

Binary Adders: Half Adders and Full Adders

Binary 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 information

n th roots of complex numbers

n 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 information

Common Core Standards for Fantasy Sports Worksheets. Page 1

Common 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

= 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 information

2015 Junior Certificate Higher Level Official Sample Paper 1

2015 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 information

25 Integers: Addition and Subtraction

25 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 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

Solve 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 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 information

Fractions and Decimals

Fractions 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 information

Major Work of the Grade

Major 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 information

Guidance 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 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 information

POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR 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 information

Zeros of a Polynomial Function

Zeros 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 information

Indices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková

Indices 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 information

Appendix F: Mathematical Induction

Appendix 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 information

Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)

Mathematics. (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 information

Bridging Documents for Mathematics

Bridging Documents for Mathematics Bridging Documents for Mathematics 5 th /6 th Class, Primary Junior Cycle, Post-Primary Primary Post-Primary Card # Strand(s): Number, Measure Number (Strand 3) 2-5 Strand: Shape and Space Geometry and

More information