Integers

INTEGERS

UNDERSTANDING INTEGERS

• Integers form a bigger collection of numbers which contains whole numbers and negative numbers.

• The numbers _ _ _, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 _ _ _ etc are integers.

• 1, 2, 3, 4, 5 _ _ _ are Positive integers.• _ _ _-5, -4 , -3, -2, -1 are Negative integers.• Integer ‘0’ is neither a positive nor negative integer.• Integer ‘0’ is less than a positive integer and greater

than negative integer.

NUMBER LINE

ADDING INTEGERS ON NUMBER LINE• On a number line when we• add a positive integer, we move to the right.• E.g.: -4+2=-2

• add a negative integer, we move to left.• E.g.: 6+(-4)=2

SUBTRACTING INTEGERS ON NUMBER LINE• On a number line when we • Subtract a positive integer, We move to the left• E.g.: (-4)-2=-6

• Subtract a negative integer, We move to the right• E.g.: 1-(-2)=3

SIGNS

SIGN

E.g.: 6+3=9

E.g.: -9-2= -11 greater value E.g.: +2-4= -2 -2+4=+2

ADDITIVE INVERSE

INTEGER ADDITIVE INVERSE 10 -10 -10 10 76 -76 -76 76 0 0

PROPERTIES OF ADDITION

AND SUBTRACTION OF

INTEGERS

CLOSURE PROPERTY

• ADDITION: Integers are closed under addition. In general

for any two integers a and b, a+b is an integer. E.g.: -2+4=2• SUBTRACTION: Integers are closed under subtraction. If a and

b are two integers then a-b is also an integer. E.g.: -6-2=-8

COMMUTATIVE PROPERTY• ADDITION: This property tells us that the sum of two

integers remains the same even if the order of integers is changed. If a and b are two integers, then a+b = b+a

E.g.: -2+3 =3+(-2)• SUBTRACTION: The subtraction of two integers is not

commutative. If a and b are two integers ,then a-b = b-a

E.g.: 4-(-6) = -6-4

ASSOCIATIVE PROPERTY

• ADDITION: This property tells us that that we can group integers in a

sum in any way we want and still get the same answer. Addition is associative for integers. In general, a+(b+c) = (a+b)+c

E.g.: 2+(3+4) = (2+3)+4 =9• SUBTRACTION: The subtraction of integers is not associative. In general,

a-(b-c) = (a-b)-c E.g.: 3-(5-7) = (3-5)-7 5 = -9

MULTIPLICATION OF INTEGERS• Multiplication of two positive integers: If a and b are two positive integers then their product is also a

positive integer i.e.: a x b = ab• Multiplication of a Positive and a Negative Integer:

While multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign(-) before the product. We thus get a negative integer. In general, a x (-b) = -(a x b)

• Multiplication of two negative integers: Product of two negative integers is a positive integers. We

multiply two negative integers as whole numbers and put the positive sign before the product. In general,

-a x -b = a x b

IF THE NUMBER OF NEGATIVE INTEGERS IN A

PRODUCT IS EVEN, THEN THE PRODUCT IS POSITIVE BUT IF

THE NUMBER OF THE NEGATIVE INTEGERS IN THE PRODUCT IS ODD THEN THE

PRODUCT IS NEGATIVE.

PROPERTIES OF MULTIPLICATION OF INTEGERS

• Closure under Multiplication: The product of two integers is an integer.

Integers are closed under multiplication. In general, a x b is an integer.

e.g.: -2 x 2 = -4• Commutativity of Multiplication: The product of two integers remain the same

even if the order is changed. Multiplication is commutative for integers. In general, a x b =b x a

e.g.: 2 x (-3) = -3 x 2

• Associativity of multiplication: The product of three integers remains the same, irrespective of

their arrangements. In general, if a, b and c are three integers, then a x (b x c) = (a x b)

x c e.g.: -2 x (3 x 4) = (-2 x 3) x 4 = -24• Multiplication by zero: The product of any integer and zero is always. In general, a x 0 = 0 x a =0 e.g.: -2 x 0 =0• Multiplicative identity: The product of any integer and 1 is the integer itself. In general, a

x 1 = 1 x a = a e.g.: -5 x 1= -5

DISTRIBUTIVE PROPERTY

• Distributivity of multiplication over addition: If a, b and c are three integers, then a x (b+c) = a x b + a x c e.g.: -2 x (4+5) = -2 x 4 + -2 x 5• Distributivity of multiplication over subtraction: If a, b and c are three integers, then a x (b-c) = a x b - a x c e.g.: -9 x (3-2) = -9 x 3 – (-9) x 2

DIVISION OF INTEGERS• Division of two Positive Integers: If a and b are two positive integers then their quotient is also a positive integer. e.g.: 4 ÷ 2 = 2• Division of a positive and a negative integer: When we divide a positive integer and a negative integer, we divide them as whole numbers

and then put a minus sign (-) before the quotient. We, thus, get a negative integer. In general, a÷ (-b) = (-a) ÷ b where b = 0

• Division of two negative integers: When we divide two negative integers, we first divide them as two whole numbers and then

put a positive sign (+). We, thus, get a positive integer. In general, (-a) ÷ (-b) = a÷b where b = 0

PROPERTIES OF DIVISION OF INTEGERS • Integers are not closed under division. In other words if

a and b are two integers, then a ÷ b may or may not be an integer.

• Division of integers is not commutative. In other words, if a and b are two integers, then a ÷ b = b ÷ a.

• Division by 0 is meaningless operation. In other words for any integer a, a ÷ 0 is not defined whereas 0 ÷ a = 0 for a = 0.

• Any integer divided by 1 give the same integer. If a is an integer, then a ÷ 1 = a.

• For any integer a, division by -1 does not give the same integer. In general, a ÷(-1) = -a but -a ÷ (-1) = a

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Made by : Samyak JainClass: VII D