INTEGERSIntegers form a bigger collection of numberswhich contains whole numbers andnegative numbers. For example-…-4, -3, -2,-1, 0, 1, 2, 3….
TYPES OF INTEGERSPOSITIVEINTEGERS- 1, 2, 3,4…i.e. the natural numbersare called positive integers.NEGATIVEINTEGERS- -1, -2, -3,-4… are called negativeintegers.On a number line points tothe right of zero arepositive integers and thepoints to the left of zeroare negative.
ZEROZero is neithernegative norpositive. It isgreater thanevery negativeinteger andsmaller thanevery positiveinteger.
MATHEMATICAL APPLICATIONS OF INTEGERSThe sum of two integers is also an integer.Example: -5 +7=+2The difference of two integers is also an integer.Example: +8-(-4)=8+4=12The product of two positive integers is alwayspositive. Example:+8 x +4 =32The product of two negative integer is alwayspositive. Example: -3 x -4 =+12The product of a positive integer with a negativeinteger is negative. example: +3 x -2=-6.The product of zero and an integer is alwayszero. Example: 0x 2=0
ADDITION AND SUBTRACTION
MATHEMATICAL APPLICATIONS OF INTEGERS (contd.)If the dividend and divisorare of the same sign thenthe quotient is a positiveinteger. Example: -4/-2=+2If dividend and divisor areintegers of opposite signthen the quotient is anegative integer.Example:4/-2=-2Any integer divided by 1gives the same integer.Zero divided by anyinteger is equal to zero.Division by zero is notpossible.
MULTIPLICATION AND DIVISION
COMMUTATIVE PROPERTYThe whole number can be added inany order. example: 5+(-6)=(-6)+5.Addition and multiplication iscommutative for integers.Whole numbers cannot be subtractedin any order. Example:5-6 is notequal to 6-5. Subtraction anddivision is not commutative forintegers.
ASSOCIATIVE PROPERTYFor all integers a, b and c,(a+b)+c=a+(b+c) i.e. addition isassociative for integers.For any three integers a, b and c,(a*b)*c=a*(b*c) i.e. for integers themultiplication is associative.
DISTRIBUTIVE PROPERTYFor any three integers a, b and c,a*(b+c)=a*b+a*c i.e. integers showdistributive property under additionand multiplication.
ADDITIVE IDENTITYWhen we add zero to any wholenumber, we get the same wholenumber.Zero is an additive identity for wholenumbers.
MULTIPLICATIVE IDENTITYOne is themultiplicativeidentity forwhole numbersas well as forintegers.Thusa*1=1*a=a.
ADDITIVE INVERSEThe additive inverse of any integer a is –aand the additive inverse of –a is a.The number zero is its own additiveinverse. INTEGERS ADDITIVE INVERSE 10 -10 -10 10
IMPORTANT POINTS Integers are closed for addition and subtraction both i.e. if a and b are integers then a+b and a-b are also integers. For any two integers a and b, a*b is an integer i.e. integers are closed under multiplication. For any integer a, we have:i. a/0 is not definedii. a/1 is equal to a Absolute value of an integer is its numerical value regardless of its sign.