THE REAL NUMBER SYSTEM

Complex numbers ( C )

Real Numbers (R )

Rational numbes (Q)

Integers (Z)

Whole numbers

Natural numbers(N)

Prime numbers

Even numbers

Composite numbers

Odd Numbers

Zero

Negative integers

Fractions

Irational numbers

Imaginer Numbers

N : 1,2,3,….Z : …,-2,-1,0,1,2,..

Q : 0,,, bZbab

a

Irational: ,3,2

IrasionalQR

Real Numbers

IrrationalNumbers

π

0.101001

√2

Rational Numbers

¼ ⅓

0.6

- ⅔ √0.25

Integers

- 4 -3

-8

Whole Numbers

0

NaturalNumbers

2 5

Concept SummaryExample Natural Whole Integer Rational Irrational Real

0 / / / /

4 / / / / /

-7 / / /

√25 / / / / /

√41 / /

¼ / /0.1212… / /0.0101101

…/ /

Operation of IntegersOperations Properties Form/ Expression

1. Addition 1. Clossure2. Commutativ

e3. Associative4. Identity

element

1. For a,b Є Integers so, a + b Є integers

2. a + b = b + a3. ( a + b ) + c = a + ( b + c)4. a + 0 = 0 + a = a

2. Substraction

Invers of addition

a – b = a + ( -b ), -b is invers of b

3. Multiplication

1. Clossure2. Commutativ

e3. Associative4. Distributive

5. Identity element

1. For a, b Є integers so, a x b Є integers

2. a x b = b x a3. ( a x b ) x c = a x ( b x c )4. a x ( b + c ) = (a x b ) + ( a x c)

a x ( b – c ) = ( a x b ) – ( a x c )5. a x 1 = 1 x a = a

4. Division Invers of multiplication

a : b = a x 1/b

+ - x ¸

plussummore thanincreased bytotalin all

minusdifferenceless thansubtractdecreased by

timesproductmultipliedeachof

dividedquotient

Four Fundamental Operations

Symbol Symbol

√ Radical sign ≤ Less than or equal to

∑ summation ≥ Greater than or equal to

Π phi Є Element of

∞ infinity ∩ intersection

! factorial U union

≠ Not equal to % percent

=>

Equal to

Greater than

°<

Degree

Less than

Positive and Negative Numbers

1.1

You may write:• the temperature of 5 degrees above

zero as 5 ºC, • the temperature of 5 degrees below

zero as -5 ºC.

0

+5

-5

0+5-5

Vertical number line

Horizontal number line

th

erm

om

eter

The numbers 5 and -5 can be drawn on a vertical or a horizontal line as follows.

A set of integers; ..., 3, 2, 1, 0, 1, 2, 3, ... , can be symbolized as Z. The complete number line of integers can be drawn as follows.

0 is neither positive nor negative

Negative integers Positive integers

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Natural numbers ( bilangan Asli ) : 1, 2, 3, 4, …

Whole numbers ( bilangan Cacah ) : 0, 1, 2, 3, 4, …

Integers ( bilangan Bulat ) : …, -2, -1, 0, 1, 2, …

Example 1

a. Determine all integers between 5 and 4.

Solution: 4, 3, 2, 1, 0, 1, 2, 3

b. Write all even integers between 6 and 11.

Solution: The even integers between 6 and 11 are 4, 2, 0, 2, 4, 6, 8, and 10

c. Write all odd integers between 7 and 8.

Solution: The odd integers between 7 and 8 are -5, -3, -1, 1 , 3 , 5, and 7

Comparing and Ordering Integers

What if we use a vertical number line?

Change the sign to <, >, or = on 4 7

Solution:

Example 2

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

The number 4 is located on the right side of 7. Therefore,

4 > 7.

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

decrease increase

Problem 1Arrange the following numbers from the least to the greatest.a. 5, 3, 6, 6, 2, 4, 1b. 9, 5, 6, 12, 17, 8, 14

Problem 1

Solution :

a. -6, -3, -1, 2, 4, 5, 6

b. -14, -12, -5, 6, 8, 9, 17

In a mathematics test, the score for the right answer is 4, for the wrong answer is -1, and 0 for failing to answer.Complete the table below (use a spreadsheet, if possible) and then sort the students based on their scores from the highest to the lowest.

Problem 2

Students’

Names Number of

Correct

Answers

Number of Wrong

Answers

Number of Unanswered

Questions Total Score

Abdullah 5 3 2

Aminah 6 4 0

Galuh Erna 5 2 3

Zainul 8 2 0

Nabila 8 1 1

Zaty 8 0 2

Hamidah 7 1 2

Yusuf 7 3 0

17

20

18

30

31

32

27

25