INTEGERS:
INTEGERSUNDERSTANDING 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:
NUMBER LINEADDING INTEGERS ON 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)=2SUBTRACTING INTEGERS ON NUMBER LINE:
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)=3SIGNS:
SIGNS SIGN E.g.: 6+3=9 E.g.: -9-2= -11 greater value E.g.: +2-4= -2 -2+4=+2ADDITIVE INVERSE :
ADDITIVE INVERSE INTEGER ADDITIVE INVERSE 10 -10 -10 10 76 -76 -76 76 0 0PowerPoint Presentation:
Properties of addition and subtraction of integersCLOSURE PROPERTY:
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=-8COMMUTATIVE PROPERTY:
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) = -6-4ASSOCIATIVE PROPERTY:
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 = -9MULTIPLICATION OF INTEGERS:
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 bPowerPoint Presentation:
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 :
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 2PowerPoint Presentation:
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= -5DISTRIBUTIVE PROPERTY:
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 2DIVISION OF INTEGERS:
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 = 0PROPERTIES OF DIVISION OF INTEGERS :
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) = aPowerPoint Presentation:
THANK YOU Made by : Samyak Jain Class: VII D