number systems

Number Systems Focus on Rational Numbers

The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers.

Introduction

NATURAL NUMBERS 1, 2, 3, 4, 5, . . . The use of three dots at the end of the list is

a common mathematical notation to indicate that the list keeps going forever.

Natural Numbers

At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero.

We call the set of natural numbers plus the number zero the whole numbers.

WHOLE NUMBERS Natural Numbers together with “zero” 0, 1, 2, 3, 4, 5, . . .

Whole Numbers

Hopefully you remember these from grade 8! INTEGERS Whole numbers plus negatives . . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . . Number lines are useful for representing

integers

Integers

Decimals that terminate or repeat are rational numbers.

A rational number is any number that can be written as a fraction, d≠0. Numerator n

d Denominator

Rational Numbers

Terminate means come to an end

Warm UpDivide.

12 24

34

16

1. 36 3 2. 144 6

3. 68 17 4. 345 115

5. 1024 64

Work With a Partner Turn to the person next to you and work together to find a solution

An ice cream parlor has 6 flavors of ice cream. A dish with two scoops can have any two flavors, including the same flavor twice. How many different double-scoop combinations are possible?

21

The goal of simplifying fractions is to make the numerator and the denominator relatively prime.

Relatively prime numbers have no common factors other than 1.

The goal is to have the fraction in its smallest possible form!

The Goal for all Rational Numbers is…

Simplify.

105 GCF = ?

51055

2

1

5

Simplify.

8016 GCF = ?

16801616

5

1

16

Simplify.

2918- GCF = ?

29

18

1

These two numbers are relatively prime!!!

Simplify.

306 GCF = ?

63066

5

1

6

Simplify.

34-17 GCF = ?

1734-1717

2

1

17

–0.8

Write the decimal as a fraction in simplest form.

Tenths

108- GCF = ?

21028-

5

4

2

5.37

Write the decimal as a fraction in simplest form.

Hundredths

100375 GCF = ?

100

375

1

5 and 37 are relatively prime!!

0.622

Thousandeths

1000622 GCF = ?

210002622

500

311

2

Write the decimal as a fraction in simplest form.

8.75

Hundredths

100

758 GCF = ?

25100

25758

4

38

25

Write the decimal as a fraction in simplest form.

11 9

Write the fraction as a decimal.

1.222222…, a repeating decimal

= 1.2

7 20

Write the fraction as a decimal.

0.35, a terminating decimal

= 0.35

Write the fraction as a decimal.

1.363636…, a repeating decimal

15 11 = 1.36

Practice Simplify.

1. 2.

Write each decimal as a fraction in simplest form.

3. 0.27 4. –0.625

5. Write as a decimal

18 42

3 7

15 21

5 7

27 100

–5 8

13 6

2.16

Tommy had 13 hits in 40 at bats for his baseball team. What is his batting average? (Batting average is the number of hits divided by the number of at bats, expressed as a decimal.)

What Does it All Mean?

0.325

Homework Part 1 (More Practice)

1. 0.3

2. 0.2

3. 0.01

4. 5.22

5. -3.25

6. 0.34

7. -1

8. -4.24

9. 44.4

10.12.964

Convert the following decimals into fractions

1. -3

Homework Part 2(more practice)

Convert the fractions into decimals (don’t forget to indicate when a decimal is repeating)

Answers to Homework Part 1

1. 0.3

2. 0.2

3. 0.01

4. 5.22 5

5. -3.25 -3

6. 0.34

7. -1

8. -4.24 -4

9. 44.4 44

10.12.964 12

Answers to Homework Part 2

1. 0.75

2. 0.625

3. 2.25

4. -8.

5. 0.6

6. -3 -3.

7. -1.5

8. 0.

9. 0.8