INTEGERS INTEGERS FROM A BIGGER COLLECTION OF NUMBERS WHICH CONTAINS WHOLE NUMBER AND NEGATIVE NUMBERS A NUMBER LINE REPRESENTING INTEGERS ON A NUMBER LINE WHEN WE1. Add a positive integer, we move to the right.2. Add a negative integer, we move to the left.3. Subtract a positive integer, we, move to the left.4. Subtract a negative unteger, we move to theright.
PROPERTIES OFINTEGERSFOR ANY TWO INTEGERS A AND Ba-b=a + additive inverse of b=a + (-b)a-b(-b) = a + additive inverse of (-b) = a+bClosure under additionSum of two whole numbers is again a whole number. Forexample, 17 + 24 = 41 which is again a whole number.His property is known as the closure property for addition ofthe whole numbers. Integers are closed under addition ingeneral. For any two integers a and b, a + b is an integer.Integers are closed under subtraction. Thus, if a and b aretwo integers then a-b is also an intger.
Commutative propertyAddition is commutative for integers.In general, for any two integers a and b, we can say a + b = b+aSubtraction is not commutative for integers.Because 5 –(-3)=5+3=8, and (-3)-5=-3-5=-8. Associative property Addition is associative for integers. In general for any intgers a,b and c, we can say a+(b+c)=(a+b)+c
Additive identity In general, for any integer a a+0=a=0+a 1. (-8)+0= - 8 2. 0+(-8)= - 8MULTIPLICATION OF A POSITIVE AND ANEGATIVE INTEGERFOR ANY TWO POSITIVE INTEGERS A AND B WECAN SAY a X (-b) = (-a) X b = - (a X b ) (-33) X 5 = 33 X (-5) = -165
MULTILICATION OF TWO NEGATIVE INTEGERSWE MULTIPLY THE TWO NEGATIVE INTEGERS ASWHOLE NUMBERS AND PUT THE POSITIVE SIGNBEFORE THE PRODUCT.THUS, WE HAVE ( -10 ) x ( -12 ) = 120SIMILARLY ( -15) x ( - 6 ) = 90IN GENERAL, FOR ANY TWO POSITIVE INTEGERSA AND B, ( -a ) x ( -b ) = a x b
Product of three or more Negative Integers If the number of negative integers in a product is even, then the product is a positive integer; if the number of negative integers in a product is odd. Then the product is a negative integer.a. ( -4 ) x ( -3 ) = 12b. (-4 ) x ( -3 ) x ( -2 )= [ ( -4 ) x ( -3 ) ] x ( -2 ) = 12 x ( -2 )= -24
CLOSURE UNDER MULTIPLICATIONIntegers are closed under multiplication.In general, a x b is an integer, for all integers a and b.Statements Inferences(-20 ) x ( -5 ) = 100 Product is an integer MULTIPLICATION BY ZERO IN GENERAL, FOR ANY INTEGER A. a x 0 = 0 x a =0 ( -3 ) x 0 = 0 0 x ( -4 ) = 0
MULTIPLICATIVE IDENTITY IN GENERAL, FOR ANY INTEGER A WE HAVE. ax1=1xa=a ( -3 ) x 1 = -3 1x5=5ASSOCIATIVITY FOR MULTIPLICATION Product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers. In general, for any three integers a, b and c ( a x b) x c = a x (b x c ) [ (-3) x (-2)] x 5 = 6 x 5 = 30 (-3) x [(-2) x 5] = (-3) x (-10) = 30
DISTRIBUTIVE PROPERTYIN GENERAL, FOR ANY INTEGERS A, B AND C, a x ( b + c ) =a x b + a x cA ( -2 ) x (3+5) = - 2x8= -16 and [(-2) x 3] +[ (-2) x5]= (-6)+ (-10) =-16IN GENERAL, FOR ANY INTEGERS A, B AND C, ax(b-c)=a x b–a x c4x(3-8)=4x(-5)= -204x3 - 4x8=12 -32= -204x(3-8)=4x3 - 4x8(1) (-18) x (-10) x 9 (-18) x (-10) x 9 = [(-18)x(-10)]x9 = 180x9=1620(2) (-20) x (-2) x (-5) x7 (-20) x (-2) x (-5) x 7 = -20x (-2 x -5) x7 = [-20x10] x7 = -1400