Copyright 2014 Scott Storla Rational Numbers. Copyright 2014 Scott Storla Vocabulary Rational number...

Rational Numbers

Vocabulary

Rational number

Proper fraction

Improper fraction

Mixed number

Prime number

Composite number

Prime factorization

Reciprocal

The Rational Numbers

Irrational NumbersThe real numbers which are not rational.

3 14159265358979. ...

73 14158. ...

1133 1415929. ...

104348

332153 1415926539. ...

Trying to find a rational number that’s equal to pi.

Fractions

Proper Fraction

In a proper fraction the numerator (top) is less than the denominator (bottom).

The value of a proper fraction will always be between 0 (inclusive) and 1 (exclusive).

Improper Fraction

In an improper fraction the numerator (top) is greater than or equal to the denominator (bottom).

The value of an improper fraction is greater than or equal to 1.

Prime Factorization

Prime Number

A natural number,

greater than 1,

which has unique natural number factors 1 and itself.

Ex: 2, 3, 5, 7, 11, 13

Composite Number

A natural number,

greater than 1,

which is not prime.

Ex: 4, 6, 8, 9, 10

Prime Factorization

I have prime factored a composite number when

the number is written as the product of prime factors.

We say 2 2 3 is the prime factorization of 12

since the factors 2 and 3 are prime.

We don't consider 2 6 a prime factorization of 12

because 6 is not prime.

Prime Factorization

To write a natural number as the product of prime factors.

Ex: 12 = 2 x 2 x 3

Factor Rules

Decide if 2, 3, and/or 5 is a factor of

List all positive integers between 51 and 61 inclusive.

List all prime numbers between 51 and 61 inclusive.

List all rational numbers with denominators of 1 between 110 and 120 inclusive.

List all natural numbers between 31 and 40 inclusive.

Building a factor tree for 20

The prime factorization of 20 is 2 x 2 x 5.

Property – The Commutative Property of Multiplication

English: The order of the factors doesn’t affect the product.

Example: 2 4 4 2

Note: Division is not commutative. For instance 4 2 2 4 .

5 2 3 2 2 2 3 5

The Fundamental Theorem of Arithmetic

Every natural number, greater than 1, has a unique prime factorization.

Example: 20 5 4 2 10 2 2 5

Procedure – To Prime Factor a Natural Number

1. Build a factor tree using the factor rules for 2,3,and 5.

2. After step 1 divide uncircled factors by the prime numbers beginning with 7 up to the square root of the number.

3. Write your prime factors in order from smallest to largest.

The prime factorization of 24 is 2 x 2 x 2 x 3.

Find the prime factorization of 24

The prime factorization of 119 is 7 x 17.

Prime Factorization

Reducing Fractions

Property – The Associative Property of Multiplication

English: The grouping of the factors doesn’t affect the product.

Example: 2 3 4 2 3 4

Note: Division is not associative. For instance 8 4 2 8 4 2 .

Property – The Multiplicative Identity

English: 1 is the multiplicative identity. Multiplying an expression by

one results in an equivalent expression.

Example:

Reducing Fractions

A fraction is reduced when the numerator and denominator have no common factors other than 1.

Reducing Fractions

A fraction is reduced when the numerator and denominator have no common factors other than 1.

Procedure – Reducing Fractions

1. Prime factor the numerator and denominator.

2. Reduce common factors.

3. Find the product of the factors in the numerator and the product of the factors in the denominator.

No “Gozinta” method allowed

No “Gozinta” (Goes into) method allowed

Simplify using prime factorization

18 2 3 3

24 2 2 2 3

2 3 2 2

2 2 2 3

2 3 3 5

45 2 2 2 3 5

Reduce using prime factorization

2 3 3 7

2 3 3 13

2 2 2 3 7

3 3 5 7

Reducing Fractions

Multiplying Fractions

using prime factorizationMultiply

51 1 1

5 5 2 32 2 7

15 28 150

Procedure – Multiplying Fractions

1. Combine all the numerators, in prime factored form, in a single numerator.

2. Combine all the denominators, in prime factored form, in a single denominator.

3. Reduce common factors

4. Multiply the remaining factors in the numerator together and the remaining factors in the denominator together.

2 3 5 5

3 5 2 2 7

Multiply using prime factorization

3 5 15 69 25

3 7 2 2 7 2 3 5 18 5 7

21 28 30 2 3 3 5 7 1

1 3 2 3 5 1 7 5

2 7 7 2 3

3 30 35

2 49 6

3 30 55

2 49 6

Dividing Fractions

Reciprocal

The reciprocal of a number is a second number which when multiplied to the first gives a product of 1.

The reciprocal of is because

The reciprocal of is because 7 1

Procedure – Dividing Fractions

1. To divide two fractions multiply the fraction in the numerator by the reciprocal of the fraction in the denominator.

2 3 2 5

6 105 3

Procedure – Dividing Fractions

1. To divide two fractions multiply the fraction in the numerator by the reciprocal of the fraction in the denominator.

2 3 2 55 3

10181524

2 5 1 2 2 2 3

2 3 3 3 5

Divide using prime factorization

355014

2 5 5 2 7

1 2 2 2 2 3 3

15 7 3

3 3 2 7

5 3 73 3 3 5

Dividing Fractions

Copyright 2014 Scott Storla Rational Numbers. Copyright 2014 Scott Storla Vocabulary Rational number...

Documents

Integrating Rational Functions by the Method of Partial Fraction

Adding and Subtracting Rational Numbers. The term, Rational Numbers, refers to any number that can be written as a fraction. This includes fractions that

Rational Expressions Simplifying. Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are

Add and Subtract Rational Expressions€¦ · Simplifying Complex Fractions A complex fraction is a fraction that contains a fraction in its numerator or denominator. A complex fraction

Control Systemslanari/ControlSystems/CS... · Partial fraction expansion (Heaviside) for rational functions ... Partial fraction expansion example: ﬁnd the zero-state output response

11.3 – Rational Functions. Rational functions – Rational functions – algebraic fraction

rcrawfordmath.weebly.com · How do you divide rational expressions? Example 2 Multiply the first fraction by the reciprocal of the second fraction. A) 16x2y 24x2y 8 Ixy2 54x3y3

ACTIVITY Activity/class 7.pdf(ii) Fractions and rational numbers: • Multiplication of fractions • Fraction as an operator • Reciprocal of a fraction ... 4 5 9 Subtract ﬂ from

How do we divide complex numbers? Do Now: Express as an equivalent fraction with a rational denominator

4.6 Rational Equations and Partial Fraction Decomposition

Chapter 12 Final Exam Review. Section 12.4 “Simplify Rational Expressions” A RATIONAL EXPRESSION is an expression that can be written as a ratio (fraction)

Objective - To recognize, graph, and compare rational numbers. Rational Number - any number that can be written as a fraction. including decimals... including

Copyright 2014 Scott Storla Formulas. Copyright 2014 Scott Storla A formula is an equation that uses mathematics to express a fact or rule. Usually the

Weebly · Dividing Rational Functions When dividing rational functions, you multiply the first fraction by the reciprocal of the second fraction, simplifying where you can. SAME-CHANGE-FLIP!

Lecture 5 Rational functions and partial fraction expansion

Partial Fractions. Understand the concept of partial fraction decomposition. Use partial fraction decomposition with linear factors to integrate rational

Rational and Irrational Numbers A rational number is a number that can be expressed as a fraction or ratio (rational). The numerator and the denominator

Rational Functions - Northern Virginia Community College · Rational Functions 1 Introduction A rational function is a fraction with variables in its denominator, and usually in its

Continuing with Operation-Inverse Pairs Copyright 2014 Scott Storla

Quadratic Functions Copyright 2014 Scott Storla