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Integers

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  • 1. • INTRODUCTION THE first numbers -to be discovered were NATURAL NUMBERS i.e. 1,2,3,4,…If we include zero to the collection of natural numbers we get a new collection known as WHOLE NUMBERS i.e. 0,1,2,3,4,... There are also numbers known as negative numbers .If we put the whole numbers and the negative numbers together we get a new collection of numbers which will look like 0,1,2,3,4,…,-1,-2,-3,-4,- 5…In this collection 1,2,3,…said to be positive integers and -1,-2,- 3,…are said to be negative integers but 0 is neither of them.
  • 2. ADDITIVE INVERSE OFINTEGERS• For every integer a there exists its opposite –a such that :-• a +{-a } = 0 = {-a } + a• Integers a and –a are called OPPOSITES or NEGATIVE or ADDITIVE INVERSE of each other.EXAMPLE:- {-1 } + {1} =0 and 1+{-1}=0,:.,{-1 }+{1}= 0 ={1}+{-1 }NOTE:- THE ADDITIVE INVERSE OF 0 IS 0.
  • 3. ABSOLUTE VALUE OFINTEGERS The absolute value of an integer is the numerical value of the integer regardless of its sign. The absolute value of an integer is denoted by |a|.
  • 4. OPERATIONS ONINTEGERS The four basic operations namely:- ADDITION SUBTRACTION MULTIPLICATION DIVISION Can easily be done on integers.
  • 5. ADDITION OFINTEGERS{properties} PROPERTY 1- Closure PROPERTY 3- Associative property of addition law of addition; If a,b,c The sum of two integers are any three integers is always an integer. then {a+b}+c=a+{b+c} PROPERTY 2- PROPERTY 4- If a is any Commutative property of integer then a+0=a and addition 0+a=a If a and b are any two PROPERTY 5- The sum of integers then a+b=b+a. an integer and its. opposite is 0.
  • 6. ADDITION OFINTEGERS{RULES}• RULE 1- If two positive positive and a negative integers or two negative integer, we find the integers are added, we difference between their add their values absolute values regardless regardless of their signs of their signs and give the and give their common sign of the greater integer sign to the sum. to it.• EXAMPLE:- +1+4=+5 EXAMPLE:- -3+2=-1• RULE 2- To add a -
  • 7. SUBTRACTION OFINTEGERS{Properties and Rules} PROPERTY 1- Closure property If a and b are integers then{a-b}is also an integer. PROPERTY 2- If a is any integer then {a-0}=a. PROPERTY 3- If a , b and c are integers and a>b then{a-c}>{b-c}. RULE 1- To subtract one integer from another, we take the additive inverse of the integer to be subtracted and add it to the other integer. Thus, if a and b are two integers then a-b=a+{-b}. EXAMPLE:- 2-7=+5
  • 8. MULTIPLICATION OFINTEGERS{properties} PROPERTY 1- Closure property of multiplication ;The product of two integers is always an integer. PROPERTY 2- Commutative law for multiplication ; For any two integers a and b ,we have a*b=b*a. PROPERTY 3- Associative law for multiplication ; If a ,b and c are any three integers then {a*b}*c=a*{b*c}. PROPERTY 4- Distributive law ; If a, b and c are any three integers then a*{b + c}=a*b + a*c. PROPERTY 5- For any integer a we have a*1=a.1 is known as the multiplicative identity for integers. PROPERTY 6- For any integer a we have a*0=0.
  • 9. MULTIPLICATION OFINTEGERS{Rules} RULE 1- To find the product of two integers with unlike signs ,we find the product of their values regardless of their signs and give a minus sign to the answer. Example- -5*7=-35 RULE 2- To find the product of two integers with the same sign ,we find the product of their values regardless of their signs and give a plus sign to the answer. Example- -18*{-10}=+180.
  • 10. DIVISION OFINTEGERS{properties} PROPERTY 1-If a and b are integers then {a/b} is not necessarily an integer. PROPERTY 2- If a is an integer and a is not equal to 0 then {a/a}=1 PROPERTY 3- if a is an integer then {a/1 }= a PROPERTY 4- if a is a non –zero number then {0/a }=0 but {a/0}is not meaningful.
  • 11. Division of integers{rules}RULE 1 - for dividing one integer by another , the two having unlike signs we divide their values regardless of their sign and give a minus {-} sign to the quotient EXAMPLE {–36}/4={ -9 }RULE 2- for dividing one integer by another , the two having like signs , we divide their values regardless of their signs and give a plus{+} sign to the quotient. EXAMPLE{-26}/{-2}=+13