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MATHS

CHAPTER 4 INTEGERS

Prepared by: Fatmir Kondi

6th grade D

Tirana, on 20 december, 2010



Integers

The set of integers is the set of whole numbers

together with their opposites.

The set 1,2,3, positive integers

The set -1,-2,-3, negative integers

Z= ..-3,-2,-1,0, 1, 2, 3, integers



Addition in Z

The sum of two positive integers is a positive integer.

The sum of two negative integers is a negative integer.

0 1 2 3 4 5 6 7 8 9

+4 +5

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

-3-4

4 + 5 = 9

-3 + -4 =-7



Properties of addition in Z

1. Closure propertyWhen we add any two integers,we always obtain an integer.For example, 4 and -6 are integers, and (+4) + (-6) = (-2) is alsointeger.

If a, b Z, then a+b Z. Therefore, Z is closed under addition.

2. Commutative propertyWhen we add two integers, if we change the places of theintegers then the sum doesnt change.

If a, b Z, then a+b = b+a. Therefore, Z is commutative underaddition.



3. Associative propertyIf we change the places of the brackets when we add threeintegers, then the sum doesnt change.

If a, b, c Z , then (a+b) + c = a +(b+c). Therefore, Z is associative

under addition

4. Identity elementWhen we add zero to any integer in any order, the sum is

always the same integer. For example 5+0 = 0+5 =5.

If a Z, then a + 0 = 0 + a = a. Therefore, 0 is the additiveidentity in Z.

Properties of addition in Z



Subtraction in Z

0 1 2 3 4 5 6 7 8 9

+8

- (+4)

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

-3-4

8 4 = 4

-3 - 4 = -7

+4

-7



Properties of subtraction in Z

1. Closure propertyThe difference of any two integers is also an integer. For

example, 5 and (-3) are integers, and 5 (-3) = 5 + 3 = 8 is also

an integer.

If a, b Z, then a b Z. Therefore, Z is closed undersubtraction.

2. Commutative property

If we change the order of the numbers in a subtractionoperation then the result is not the same.

For example, 5,3, Z, and 5 3 = 2 but 3 5 = -2. Therefore,

5 3 3 5, and so subtraction is not commutative in Z.



3. Associative propertyIf we change the order of two subtraction operations, then the

result is not the same.

For example, 4,3,5 Z, and (4 - 3)- 5 = 1 - 5 = -4, but 4 -(3 - 5) =

4 - (-2)= 6

Therefore, (4-3) -5 4-(3-5), and so subtraction is not

associative in Z.

4. Identity elementThere is no identity element for subtraction in Z.

Properties of subtraction in Z



Multiplication in Z

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

- 2 -2 x +3 = -6

-6

- 2 - 2

Rules:

The product of two like signs is positive.

The product of two opposite signs is negative.



Properties of multiplication in Z1.

Closure propertyWhen we multiply two integers, we always obtain an integers

If a, b Z, then a x b Z. Therefore, Z is closed under multiplication.

2. Commutative propertyTwo integers can be multiply in any order.

For example, (-3), (+5) Z and

(-3) x (+5) = -15 Therefore, (-3) x (+5) = (+5) x (-3)(+5) x (-3) = -15

If a, b Z, then a x b = b x a. Therefore, multiplication is commutative in Z.



3. Associative propertyWhen we multiply three integers, if we change the order of multiplication

then the result does not change.

If a,b,c Z,then a x (b x c) = (a x b) x c. Therefore multiplication is associativein Z.

4. Identity elementIf we multiply an integer by one, the product is always the same integer.

If a Z, then a x 1 = 1 x a. Therefore, 1 is the multiplicative identity in Z.

Properties of multiplication in Z



Powers of ten

Powers of ten are important in our number systems.List of powers of ten for positive and negative integers:

Every odd power of -10 is negative

Every even power of -10 is positive

10° = 1

10¹ = 10

10² =100

10³ =1000

10 =10000

10=100000

(-10°) = 1

(-10¹) = -10

(-10²) = 100

(-10³) = -1000

(-10 )= 10000

(-10)= -100000



Division in Z

� Rule:1. The quotient of two like signs is positive and the

quotient of two different signs is negative.

� Rule:

1. Zero divided by any non-zero integers is zero.

2. Any non-zero integer divided by zero is undefined.



Properties of division in Z

1. Closure PropertyThe quotient of some integers is not an integer.

Therefore, Z is not closed under division.

2. Commutative propertyDivision is not commutative in Z.

3. Associative Property

Division in not assocuative in Z



4. Identity ElementThere is no identity element for division in Z: 1/a a/1

5. Null element

There is no null element for division in Z: 0/a = 0, but a/0 0

6. Distributive Property

Division is not distributive over addition in Z.

Therefore, 16 (8 + 8) (16/8) + (16/8)Division in not distributive over subtraction in Z.

Therefore, 16 /(8 - 4) (16/8) (16/4).

Properties of division in Z



THANK YOU !

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