Natural numbers are counting numbers.

Natural Numbers

= {1, 2, 3, 4, 5…}

Whole numbers are natural numbers and zero.

Whole Numbers

= {0, 1, 2, 3, 4, 5…}

N is a subset of W.

Integers are whole numbers and opposites of naturals.

Integers

= {...−3, −2, −1, 0, 1, 2, 3…}

N and W are subsetsof Z.

Rational numbers are integers and all fractions.

Rational Numbers

= { ab a b & b ≠ 0}, ,

Irrational numbers are totally different from rational numbers. The two have nothing in common.

Irrational Numbers

Rationals and irrationals are disjoint sets. In other words, they have no common element.

Irrationals

2, , 5 7, 1.305276...p

Real numbers are rational and irrational.

Real Numbers

= & irrationals

There are an infinite number of rational numbers between each pair of integers. This is called the density of numbers.

Rational Numbers

A rational number is any number that can be written in the form , where a and b are integers and b ≠ 0.

ab

Lowest Terms

A rational fraction is in lowest terms if the GCF of a and b is one.

ab

Rename in lowest terms.Example 1

1218

12 = 2 • 2 • 318 = 2 • 3 • 3GCF = 2 • 3 = 61218

2 x 63 x 6= 2

3=

Rename in lowest terms.2490

2490

2 x 2 x 2 x 3 2 x 3 x 3 x 5=

415=

2 x 2 x 2 x 3 2 x 3 x 3 x 5=

Example 2

Rename in lowest terms.3042

57=

Example

Rename in lowest terms.3,0004,200

57=

Example

Rename in lowest terms.7290

45=

Example

A proper fraction is one whose numerator is less than its denominator.

If the numerator is greater than or equal to the denominator, the fraction is greater than or equal to one and is called an improper fraction.

A mixed number is actually the sum of a whole number and a fraction.

Renaming Improper Fractions as Mixed Numbers

1. Divide the numerator by the denominator.

2. Write the quotient as the whole number.

3. Write the remainder over the divisor as a fraction.

4. If possible, reduce the fraction to lowest terms.

Rename as a mixed number.

19 7

7 192

- 145

5 7= 2

Example 3

Rename as a mixed number.

12 8

8 121

- 84

4 81 1

2= 1

Example 3

Rename the improper fraction as a mixed number.

78 36

1 6= 2

Example

Rename the improper fraction as a mixed number.

93 8− 5

8= −11

Example

Evaluate the expression when y = 38 and z = 2. Write the answer as a mixed number in lowest terms.

y

3z

3 196

- 181

1 3 = 638

3(2)38 6

=

19 3

=19 x 2 3 x 2=

Example 4

Evaluate when x = 2, y = – 3, and z = 5.

3 5= 6 3x – y

z

Example

Evaluate when x = 2, y = – 3, and z = 5.

2 5= 5 – y3x2

z

Example

Evaluate when x = 2, y = – 3, and z = 5.

4 25= −(3x)2

3yz2

Example

Renaming Mixed Numbers as Improper Fractions

1. Multiply the whole number by the denominator.

2. Add the numerator to the product.

3. Write the sum over the denominator.

4. If possible, reduce the fraction to lowest terms.

Rename as an improper fraction in lowest terms.

16 5

=

1 5 3

1 5 3

5(3) + 15=

15 + 15

=

Example 5

Rename as an improper fraction in lowest terms.

31 4

=

6 8 7

6 8 7

8(7) + 68=

56 + 68

=

628= 31 x 2

4 x 2=

Example 5

Rename the mixed number as an improper fraction.

31 11=9

112

Example

Rename the mixed number as an improper fraction.

4 5− 12 64

5= −

Example