RATIONAL AND IRRATIONAL NUMBERS


 Jocelyn Irene Marsh
 6 months ago
 Views:
Transcription
1 Q.. RATIONAL AND IRRATIONAL NUMBERS Without actual division find which of the following rationals are terminating decimal: (i) 9 5 (ii) 7 (iii) 5 (iv) 7 78 Ans. (i) In 9, the prime factors of denominator 5 are 5, 5. Thus it is terminating decimal. 5 (ii) In 7, the prime factors of denominator are, and. Thus it is not terminating decimal. (iii) In, the prime factors of denominator 5 are 5, 5 and 5. Thus it is 5 terminating decimal. (iv) In 7, the prime factors of denominator 78 are, and. Thus it is not 78 terminating decimal. Q.. Represent each of the following as a decimal number. (i) 4 5 Ans. (i) In (ii) 5 (iii), using long division method: 5 Hence, Math Class IX Question Bank
2 (iii) In Q.. 5, using long division method: Hence, (iv) In 5 55, using long division method: Hence, p q, Express each of the following as a rational number in the form of where q 0. (i) 0.6 (ii) 0.4 (iii) 0.7 (iv) 0.04 Ans. (i) Let x (i) Multiplying both sides of eqn. (i) by 0, we get 0x (ii) Subtracting eqn. (i) from eqn. (ii), we get 0x x x 6 x Hence, required fraction. 9 (ii) Let x (i) Multiplying both sides of eqn. (i) by 00, we get 00x (ii) Subtracting eqn. (i) from eqn. (ii), we get Math Class IX Question Bank
3 00x x x 4 Hence, required fraction 4 99x 4 x p q (iii) Let x (i) Multiplying both sides of eqn. (i) by 0, we get 0x (ii) Multiplying both sides of eqn. (ii) by 00, we get 000x (iii) Subtracting eqn. (ii) from (iii), we get 000x x x 5 990x 5 x 990 p 5 Hence, required fraction. q (iv) Let x (i) Multiplying both sides of eqn. (i) by 0, we get 0x (ii) Multiplying both sides of eqn. (ii) by 000, we get 0000x (iii) Subtracting eqn. (ii) from (iii), we get 0000x x x 0 Hence, required fraction x 0 x Math Class IX Question Bank
4 Q.4. Express each of the following as a vulgar fraction. (i).46 (ii) 4.4 Ans. Let x (i) Multiplying both sides of eqn. (i) by 000, we get 000x (ii) Subtracting eqn. (i) from eqn. (ii), we get 000x Q.5. x x 4 999x 4 x 999 Hence, required vulgar fraction (ii) Let x (i) Multiplying both sides of eqn. (i) by 0, we get 0x (ii) Multiplying both sides of eqn. (ii) by 00, we get 000x (iii) Subtracting eqn. (ii) from eqn. (iii), we get 000x x x x 48 x Hence, required vulgar fraction. 0 Insert one rational number between: (i) and 7 (ii) 8 and Ans. If a and b are two rational numbers, then between these two numbers, one rational number will be ( a + b ). Required rational number between 5 and < < Math Class IX 4 Question Bank
5 (ii) Required rational number between 8 and 8.04 ( ) (6.04) < 8.0 < 8.04 Q.6. Insert two rational numbers between and Ans. and and < + < < < < < < < < < + < < < < < < < < < < Hence, required rational numbers are 9 40 and Q.7. Insert three rational numbers between (i) 4 and 5 (ii) and 5 (iii) 4 and 4.5 (iv) and (v) and Ans. (i) The given numbers are 4 and 5. As, 4 < < < 5 4 < < 5 4 < 4.5 < 5...(i) 9 9 Again, 4 < 4 + < 4 < 4.5 < (ii) Again, 4.5 < < ( ) < < 4.75 < 5...(iii) From eqn. (i), (ii) and (iii), we get 4 < 4.5 < 4.5 < 4.75 < 5. Thus, required rational numbers between 4 and 5 are 4.5, 4.75 and 4.5. Math Class IX 5 Question Bank
6 (ii) The given numbers are and 5 As, < < + < 5 < < < < < < Again, < + < < < < <...(ii) 40 5 Again, < < + < 0 5 < < < <...(iii) 40 5 From eqn. (i), (ii) and (iii), we get < < < < Thus, required rational numbers between and are, and. 40 (iii) The given numbers are 4 and 4.5 As 4 < < ( ) < < 4.5 < (i) 4 < ( ) < < 4.5 < (ii) Again, 4.5 < < ( ) < 4.75 < (iii) From eqn. (i), (ii) and (iii), we have 4 < 4.5 < 4.5 < 4.75 < 4.5 Thus, required rational numbers between 4 and 4.5 are 4.5, 4.5 and Math Class IX 6 Question Bank
7 (iv) The given numbers are As 7 < 7 and i.e., and. 7 7 < + < 7 8 < < < < 7 < <...(i) Again, 7 7 < + < 7 8 < <...(ii) Again, < 0 < + < < < 0 < <...(iii) From eqn. (i), (ii) and (iii), we get < < < <. Thus required rational numbers between and i.e., 7 and are 8, and 0. Q.8. Find the decimal representation of 7 and. Deduce from the decimal 7 representation of, without actual calculation, the decimal representation of, 4, and Math Class IX 7 Question Bank
8 Ans. Decimal representation of 7 using long division method Thus decimal representation of Decimal representation of Decimal representation of Decimal representation of Decimal representation of Decimal representation of Q.9. State, whether the following numbers are rational or irrational: (i) ( + ) (ii) ( )( 5 5 ) Ans. (i) ( + ) Hence, it is an irrational number. (iii) ( ) ( ) ( ) ( ) [Using ( a + b)( a b) a b ] Hence, it is a rational number. Q.0. Given 4 set { 6, 5, 4,,, 0,,,, 8,.0,π, 8.47} From the given set find: (i) Set of rational numbers (ii) Set of irrational numbers (iii) Set of integers (iv) Set of nonnegative integers Ans. The given universal set is 4 { 6, 5, 4,,, 0,,,, 8,.0, π, 8.47} universal Math Class IX 8 Question Bank
9 (i) Set of rational numbers 4 { 6, 5, 4,,, 0,,,,.0, 8.47} (ii) Set of irrational numbers { 8, π} (iii) Set of integers { 6, 4, 0, } (iv) Set of nonnegative integers {0, } Q.. Use division method to show that and 5 are irrational numbers. Ans It is nonterminating and nonrecurring decimals. is an irrational number. It is non terminating and non recurring decimals. 5 is an irrational number. Math Class IX 9 Question Bank
10 Q.4. Show that 5 is not a rational number. p Ans. Let 5 is a rational number and let 5. q Where p and q have no common factor and q 0. Squaring both sides, we get ( ) p p 5 5 p 5q q q p is a multiple of 5 p is also multiple of 5 Let p 5m for some positive integer m....(i) p 5m...(ii) From eqn. (i) and (ii), we get 5q 5m q 5m q is multiple of 5 q is multiple of 5 Thus, p and q both are multiple of 5. This shows that 5 is a common factor of p and q. This contradicts the hypothesis that p and q have no common factor, other than. 5 is not a rational number. Q.. Show that: (i) ( + 7 ) is an irrational number (ii) ( + 5 ) is an irrational number. Ans. (i) Let ( + 7 ) is a rational number. Then square of given number i.e., ( + 7 ) is rational. ( + 7 ) is rational ( ) + ( 7 ) ( 0 + ) is rational But, ( 0 + ) being the sum of a rational and irrational is irrational. This contradiction arises by assuming that ( + 7 ) is rational number. Hence, + 7 is an irrational number. Math Class IX 0 Question Bank
11 (ii) Let ( + 5) is a rational number. Then square of given number i.e., ( + 5 ) is rational. ( + 5 ) is rational ( ) + ( 5 ) ( 8 + 5) rational. But, ( 8 + 5) being the sum of a rational and irrational it is irrational. This contradiction arises by assuming that ( + 5 ) is rational. Hence, ( + 5 ) is irrational number. Q.4. Use method of contradiction to show that Ans. (i) Now, Let is a rational number p (where q 0) q Then, ( ) p q p q p q p is divisible by as p is divisible by. Let p r Then p q is divisible by. 9r (On squaring both sides) q 9r q r is an irrational number....(i) r is also divisible by. q is divisible by....(ii) From (i) and (ii), we get p is divisible by. q Math Class IX Question Bank
12 p and q have as their common factor but p q is a rational number i.e. p and q have no common factor. p q is not rational. So is not rational. Hence, is irrational number. Q.5. Insert three irrational numbers between 0 and. Ans. Three irrational numbers between 0 and can be 0 < < < < Q.6. Rationalise the denominator and simplify. Ans. (i) (ii) (i) (iii) (v) (ii) (iv) (vi) Multiplying numerator and denominator by + 5, we get ( ) ( ) { ( a + b)( a b) a b } ( ) ( 5) 5 + Multiplying numerator and denominator by 5, we get ( ) ( ) ( ) ( ) ( ) ( ) 6( 5 ) 6( 5 ) { ( a + b)( a b) a b } ( ) 5 5 Math Class IX Question Bank
13 (iii) 5 Multiplying numerator and denominator by 5 +, we get ( 5 ) ( ) ( ) ( ) ( ) (iv) (v) (7 + 5) ( 5) (7 5) ( + 5) ( + 5) ( 5) () ( 5) Multiplying numerator and denominator by ( 5 + ), we get ( 5 + ) ( 5 + ) ( 5 + ) ( 5 + ) () ( 5 + ) 5 (5 5) Multiplying numerator and denominator by (7 5) (7 5) (5) , we get Math Class IX Question Bank
14 (vi) ( ) Multiplying numerator and denominator by , we get [( 6 + 5) + ] [( 6 + 5) ][ 6 + 5) + ] ( 6 + 5) ( ) Multiplying numerator and denominator by ( ) , we get Q.7. If a + b, + Ans. a + b + find the value of a and b. Multiplying both sides numerator and denominator of L.H.S. by ( ), we get ( ) ( ) + ( + ) ( ) Math Class IX 4 Question Bank
15 + 4 But a + b, + Comparing both sides a and b so a + b Q.8. If + a + b, find the value of a and b.  Ans. + a + b Multiplying numerator and denominator of L.H.S. by +, we get + ( + ) ( + ) ( ) ( + ) ( + ) () ( ) But + a + b, Comparing both sides, a, b 7 7 Q.9. Simplify: Ans so + 6 a + b (i) + + (ii) (iii) (i) + + By rationalising the denominator of each term, we get () () ( ) ( ) Math Class IX 5 Question Bank
16 (ii) By rationalising the denominator of each term, we get ( 6) ( ) ( 6) ( ) ( 6) (iii) By rationalising the denominator of each term, we get ( ) ( ) (5 ) ( ) ( ) + (5 ) Math Class IX 6 Question Bank
17 Q.0. If x Ans. (i) (ii) and y 5 + ; find : x + y + xy 5 (i) x (ii) y (iii) xy (iv) x + y + xy x ( 5 ) x Squaring both sides, we get x (9 4 5) 8+ 6(5) y y Squaring both sides, we get y ( ) (iii) xy (9 4 5) ( ) 8 80 (iv) x + y + xy Q.. Write down the values of: Ans. (i) (ii) (i) (iii) (i) ( 6 ) (iv) (5 + ) ( 5 + 6) ( ) (5 + ) (5) + ( ) + (5) ( ) [using ( a + b) a + b + ab] (iii) ( 6 ) ( 6) + () 6 [using ( a b) a + b ab] (iv) ( 5 + 6) ( 5) + ( 6) [using ( a + b) a + b + ab] Math Class IX 7 Question Bank
18 Q.. Rationalize the denominator of: (i) (iii) (ii) (iv) Ans. (i) + Multiplying numerator and denominator by, we get ( ) + ( ) ( ) ( ) + ( ) (ii) 7 5 Multiplying numerator and denominator by 7 + 5, we get ( 7 + 5) ( 7) ( 5) ( 7) + ( 5) (6 + 5) (iii) 5 Multiplying numerator and denominator by 5 +, we get ( 5 + ) ( 5) ( ) ( 5) + ( ) Math Class IX 8 Question Bank
19 (4 + 5) (iv) Multiplying numerator and denominator by 5 +, we get ( 5 + ) ( 5) ( ) ( 5) + ( ) ( ) Q.. Find the values of a and b in each of the following: + (i) a + b (iii) a b 7 (ii) a 7 + b (iv) a + b (v) a b + 5 Ans. (i) + a + b (vi) a + b 0 5 Multiplying numerator and denominator of L.H.S. by ( + ), we get ( + ) + 4 () + ( ) + {using ( a + b) a + ab + b } But + a + b. So, a + b Comparing both sides we get: a 7 and b 4 Math Class IX 9 Question Bank
20 7 (ii) a 7 + b 7 + Multiplying numerator and denominator of L.H.S. by 7, we get ( 7 ) ( 7) + () 7 {using ( a b) a ab + b } But a 7 + b. So, 4 7 a 7 + b. 7 + Comparing both sides, we get 4 7 a 7 and b 4 a and b (iii) a b Multiplying numerator and denominator of L.H.S. by +, we get ( ) ( ) + + Also a b. So, + a b Comparing both sides, we get a and b Math Class IX 0 Question Bank
21 (iv) 5 + a + b 5 Multiplying numerator and denominator of L.H.S. by 5 +, we get (5 + ) (5) ( ) (5) + ( ) + 5 {using ( a + b) a + ab + b } Also, 5 + a + b. So, a + b Comparing both sides, we get : 4 0 a and b (v) a b + 5 Multiplying numerator and denominator of L.H.S. by 5, we get : ( ) (5 ) But a b. So, a b Comparing both sides, we get : 7 a and b 9 57 Math Class IX Question Bank
22 (vi) a + b 0 5 Multiplying numerator and denominator of L.H.S. by 5 +, we get ( 5) ( ) Also, a + b 0. So, a + b Comparing both sides, we get : 7 7 a and b 7 7 Math Class IX Question Bank
STRAND B: Number Theory. UNIT B2 Number Classification and Bases: Text * * * * * Contents. Section. B2.1 Number Classification. B2.
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
More informationChapter 1. Real Numbers Operations
www.ck1.org Chapter 1. Real Numbers Operations Review Answers 1 1. (a) 101 (b) 8 (c) 1 1 (d) 1 7 (e) xy z. (a) 10 (b) 14 (c) 5 66 (d) 1 (e) 7x 10 (f) y x (g) 5 (h) (i) 44 x. At 48 square feet per pint
More informationLESSON 1 PRIME NUMBERS AND FACTORISATION
LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers
More informationHFCC Math Lab Intermediate Algebra  7 FINDING THE LOWEST COMMON DENOMINATOR (LCD)
HFCC Math Lab Intermediate Algebra  7 FINDING THE LOWEST COMMON DENOMINATOR (LCD) Adding or subtracting two rational expressions require the rational expressions to have the same denominator. Example
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 informationThe notation above read as the nth root of the mth power of a, is a
Let s Reduce Radicals to Bare Bones! (Simplifying Radical Expressions) By Ana Marie R. Nobleza The notation above read as the nth root of the mth power of a, is a radical expression or simply radical.
More informationFractions to decimals
Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of
More informationSimplifying Radical Expressions
Section 9 2A: Simplifying Radical Expressions Rational Numbers A Rational Number is any number that that expressed as a whole number a fraction a decimal that ends a decimal that repeats 3 2 1.2 1.333
More informationSTRAND A: NUMBER. UNIT A6 Number Systems: Text. Contents. Section. A6.1 Roman Numerals. A6.2 Number Classification* A6.
STRAND A: NUMBER A6 Number Systems Text Contents Section A6.1 Roman Numerals A6. Number Classification* A6.3 Binary Numbers* A6.4 Adding and Subtracting Binary Numbers* A6 Number Systems A6.1 Roman Numerals
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 informationConnect Four Math Games
Connect Four Math Games Connect Four Addition Game (A) place two paper clips on two numbers on the Addend Strip whose sum is that desired square. Once they have chosen the two numbers, they can capture
More informationMATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.
MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.
More informationImproper Fractions and Mixed Numbers
This assignment includes practice problems covering a variety of mathematical concepts. Do NOT use a calculator in this assignment. The assignment will be collected on the first full day of class. All
More informationAlgebra 1: Topic 1 Notes
Algebra 1: Topic 1 Notes Review: Order of Operations Please Parentheses Excuse Exponents My Multiplication Dear Division Aunt Addition Sally Subtraction Table of Contents 1. Order of Operations & Evaluating
More informationAlgebra 1A and 1B Summer Packet
Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
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 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 information6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms
AAU  Business Mathematics I Lecture #6, March 16, 2009 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms 6.1 Rational Inequalities: x + 1 x 3 > 1, x + 1 x 2 3x + 5
More informationScientific Notation and Powers of Ten Calculations
Appendix A Scientific Notation and Powers of Ten Calculations A.1 Scientific Notation Often the quantities used in chemistry problems will be very large or very small numbers. It is much more convenient
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 informationMath Help and Additional Practice Websites
Name: Math Help and Additional Practice Websites http://www.coolmath.com www.aplusmath.com/ http://www.mathplayground.com/games.html http://www.ixl.com/math/grade7 http://www.softschools.com/grades/6th_and_7th.jsp
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 informationHere are some examples of combining elements and the operations used:
MATRIX OPERATIONS Summary of article: What is an operation? Addition of two matrices. Multiplication of a Matrix by a scalar. Subtraction of two matrices: two ways to do it. Combinations of Addition, Subtraction,
More information7 Quadratic Expressions
7 Quadratic Expressions A quadratic expression (Latin quadratus squared ) is an expression involving a squared term, e.g., x + 1, or a product term, e.g., xy x + 1. (A linear expression such as x + 1 is
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 informationOperations on Decimals
Operations on Decimals Addition and subtraction of decimals To add decimals, write the numbers so that the decimal points are on a vertical line. Add as you would with whole numbers. Then write the decimal
More informationSect Exponents: Multiplying and Dividing Common Bases
40 Sect 5.1  Exponents: Multiplying and Dividing Common Bases Concept #1 Review of Exponential Notation In the exponential expression 4 5, 4 is called the base and 5 is called the exponent. This says
More information1.3. Properties of Real Numbers Properties by the Pound. My Notes ACTIVITY
Properties of Real Numbers SUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Think/Pair/Share, Interactive Word Wall The local girls track team is strength training by
More informationSummer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2
Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level
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 informationMaths Area Approximate Learning objectives. Additive Reasoning 3 weeks Addition and subtraction. Number Sense 2 weeks Multiplication and division
Maths Area Approximate Learning objectives weeks Additive Reasoning 3 weeks Addition and subtraction add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar
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 informationSimplifying SquareRoot Radicals Containing Perfect Square Factors
DETAILED SOLUTIONS AND CONCEPTS  OPERATIONS ON IRRATIONAL NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationRadicals  Rational Exponents
8. Radicals  Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. When we simplify
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 information3.3 Addition and Subtraction of Rational Numbers
3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.
More information2 is the BASE 5 is the EXPONENT. Power Repeated Standard Multiplication. To evaluate a power means to find the answer in standard form.
Grade 9 Mathematics Unit : Powers and Exponent Rules Sec.1 What is a Power 5 is the BASE 5 is the EXPONENT The entire 5 is called a POWER. 5 = written as repeated multiplication. 5 = 3 written in standard
More informationFactoring (pp. 1 of 4)
Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common
More informationChapter 15 Radical Expressions and Equations Notes
Chapter 15 Radical Expressions and Equations Notes 15.1 Introduction to Radical Expressions The symbol is called the square root and is defined as follows: a = c only if c = a Sample Problem: Simplify
More informationread, write, order and compare numbers to at least and determine the value of each digit
YEAR 5 National Curriculum attainment targets Pupils should be taught to: Number Number and place value read, write, order and compare numbers to at least 1 000000 and determine the value of each digit
More informationA fraction is a noninteger quantity expressed in terms of a numerator and a denominator.
1 Fractions Adding & Subtracting A fraction is a noninteger quantity expressed in terms of a numerator and a denominator. 1. FRACTION DEFINITIONS 1) Proper fraction: numerator is less than the denominator.
More informationRules for Exponents and the Reasons for Them
Print this page Chapter 6 Rules for Exponents and the Reasons for Them 6.1 INTEGER POWERS AND THE EXPONENT RULES Repeated addition can be expressed as a product. For example, Similarly, repeated multiplication
More informationMath Released Set 2015. Algebra 1 PBA Item #13 Two Real Numbers Defined M44105
Math Released Set 2015 Algebra 1 PBA Item #13 Two Real Numbers Defined M44105 Prompt Rubric Task is worth a total of 3 points. M44105 Rubric Score Description 3 Student response includes the following
More informationYear Five Maths Notes
Year Five Maths Notes NUMBER AND PLACE VALUE I can count forwards in steps of powers of 10 for any given number up to 1,000,000. I can count backwards insteps of powers of 10 for any given number up to
More informationAlgebra I Notes Review Real Numbers and Closure Unit 00a
Big Idea(s): Operations on sets of numbers are performed according to properties or rules. An operation works to change numbers. There are six operations in arithmetic that "work on" numbers: addition,
More informationUSING THE PROPERTIES TO SIMPLIFY EXPRESSIONS
5 (1 5) Chapter 1 Real Numbers and Their Properties 1.8 USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS In this section The properties of the real numbers can be helpful when we are doing computations. In
More informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
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 informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationSolution: There are TWO square roots of 196, a positive number and a negative number. So, since and 14 2
5.7 Introduction to Square Roots The Square of a Number The number x is called the square of the number x. EX) 9 9 9 81, the number 81 is the square of the number 9. 4 4 4 16, the number 16 is the square
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 informationSUBJECTABLE MATHMATICS
SUBJECTABLE MATHMATICS PREFACE (A NOTE TO THE READER) This book, A COMPLETE BOOK ON SUBJECTIVE (CONVEN TIONAL) ARITHMETIC has been specially prepared for candidates appearing for competitive entrance
More informationSIMPLIFYING SQUARE ROOTS
40 (88) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify
More informationProperty: Rule: Example:
Math 1 Unit 2, Lesson 4: Properties of Exponents Property: Rule: Example: Zero as an Exponent: a 0 = 1, this says that anything raised to the zero power is 1. Negative Exponent: Multiplying Powers with
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 informationNumber: Multiplication and Division
MULTIPLICATION & DIVISION FACTS count in steps of 2, 3, and 5 count from 0 in multiples of 4, 8, 50 count in multiples of 6, count forwards or backwards from 0, and in tens from any and 100 7, 9, 25 and
More informationRadicals  Multiply and Divide Radicals
8. Radicals  Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More information3 cups ¾ ½ ¼ 2 cups ¾ ½ ¼. 1 cup ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼
cups cups cup Fractions are a form of division. When I ask what is / I am asking How big will each part be if I break into equal parts? The answer is. This a fraction. A fraction is part of a whole. The
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 informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationSimplifying Radical Expressions
In order to simplifying radical expression, it s important to understand a few essential properties. Product Property of Like Bases a a = a Multiplication of like bases is equal to the base raised to the
More informationGrade 7 Red Math Weekly Schedule December 6 th December 10 th Mr. Murad
Lesson 1&2 6/12/2015 Lesson 3 7/12/2015 Lesson 4 8/12/2015 Lesson 5&6 9/12/2015 Grade 7 Red Math Weekly Schedule December 6 th December 10 th Mr. Murad Date Class Work Home Work 21 find the absolute
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More informationACCUPLACER Arithmetic & Elementary Algebra Study Guide
ACCUPLACER Arithmetic & Elementary Algebra Study Guide Acknowledgments We would like to thank Aims Community College for allowing us to use their ACCUPLACER Study Guides as well as Aims Community College
More informationSelfDirected Course: Transitional Math Module 2: Fractions
Lesson #1: Comparing Fractions Comparing fractions means finding out which fraction is larger or smaller than the other. To compare fractions, use the following inequality and equal signs:  greater than
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationis a real number. Since it is a nonterminating decimal it is irrational.
Name the sets of numbers to which each number belongs. 19. The number 8.13 is a real number. Since 8.13 can be expressed as a ratio where a and b are integers and b is not 0 it is also a rational number.
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationGrade 9 Mathematics Unit #1 Number Sense SubUnit #1 Rational Numbers. with Integers Divide Integers
Page1 Grade 9 Mathematics Unit #1 Number Sense SubUnit #1 Rational Numbers Lesson Topic I Can 1 Ordering & Adding Create a number line to order integers Integers Identify integers Add integers 2 Subtracting
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 informationLesson Plan  Rational Number Operations
Lesson Plan  Rational Number Operations Chapter Resources  Lesson 312 Rational Number Operations  Lesson 312 Rational Number Operations Answers  Lesson 313 Take Rational Numbers to WholeNumber
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 informationExpanding brackets and factorising
Chapter 7 Expanding brackets and factorising This chapter will show you how to expand and simplify expressions with brackets solve equations and inequalities involving brackets factorise by removing a
More informationIntegration Unit 5 Quadratic Toolbox 1: Working with Square Roots. Using your examples above, answer the following:
Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots Name Period Objective 1: Understanding Square roots Defining a SQUARE ROOT: Square roots are like a division problem but both factors must
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationHFCC Math Lab Arithmetic  4. Addition, Subtraction, Multiplication and Division of Mixed Numbers
HFCC Math Lab Arithmetic  Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.
More informationChapter 4 Fractions and Mixed Numbers
Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.
More informationFirst Degree Equations First degree equations contain variable terms to the first power and constants.
Section 4 7: Solving 2nd Degree Equations First Degree Equations First degree equations contain variable terms to the first power and constants. 2x 6 = 14 2x + 3 = 4x 15 First Degree Equations are solved
More informationThis is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).
This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
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 informationDecimal and Fraction Review Sheet
Decimal and Fraction Review Sheet Decimals Addition To add 2 decimals, such as 3.25946 and 3.514253 we write them one over the other with the decimal point lined up like this 3.25946 +3.514253 If one
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationRational Expressions  Complex Fractions
7. Rational Epressions  Comple Fractions Objective: Simplify comple fractions by multiplying each term by the least common denominator. Comple fractions have fractions in either the numerator, or denominator,
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationThe wavelength of infrared light is meters. The digits 3 and 7 are important but all the zeros are just place holders.
Section 6 2A: A common use of positive and negative exponents is writing numbers in scientific notation. In astronomy, the distance between 2 objects can be very large and the numbers often contain many
More informationSimplifying Radical Expressions
9.2 Simplifying Radical Expressions 9.2 OBJECTIVES. Simplify expressions involving numeric radicals 2. Simplify expressions involving algebraic radicals In Section 9., we introduced the radical notation.
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More information12 Properties of Real Numbers
12 Properties of Real Numbers Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Simplify. 1. 5+5 0 2. 1 3. 1.81 4. Find 10% of $61.70. $6.17 5. Find the reciprocal of 4. Objective Identify and use properties
More information6th Grade Vocabulary Words
1. sum the answer when you add Ex: 3 + 9 = 12 12 is the sum 2. difference the answer when you subtract Ex: 179 = 8 difference 8 is the 3. the answer when you multiply Ex: 7 x 8 = 56 56 is the 4. quotient
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 information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationHAROLD CAMPING i ii iii iv v vi vii viii ix x xi xii 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52
More informationChapter 9: Quadratic Functions 9.3 SIMPLIFYING RADICAL EXPRESSIONS
Chapter 9: Quadratic Functions 9.3 SIMPLIFYING RADICAL EXPRESSIONS Vertex formula f(x)=ax 2 +Bx+C standard d form X coordinate of vertex is Use this value in equation to find y coordinate of vertex form
More informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
More informationScholastic Fraction Nation
Scholastic correlated to the ematics: Grades 48 2010 TM & Scholastic Inc. All rights reserved. SCHOLASTIC,, and associated logos are trademarks and/or registered trademarks of Scholastic Inc. Scholastic
More informationText: Scott Foresman CA Mathematics
Grade 6 Mathematics 008009 3 Course Introduction 7 Segment I: Algebra and Decimals 0 Variables and Expressions NS.0 AF. 9 Orders of Operation AF. 4 Powers and Exponents  Comparing and Ordering Decimals
More informationParamedic Program PreAdmission Mathematics Test Study Guide
Paramedic Program PreAdmission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More information