Multiplication Of Integers


 
 
Concept Explanation
 

Multiplication Of Integers

Multiplication of Integers:  The rules that govern on how to multiply and divide integers are very similar. Rules on How to Multiply Integers:

  • Step 1: Multiply their absolute values.
  • Step 2: Determine the sign of the final answer (in this case it is called the product because we are multiplying) using the following conditions.
  • Condition 1: If the signs of the two numbers are the same, the product is always a positive number.
  • This is an illustration showing that when you multiply two numbers with the same sign, the answer is always positive. That is, positive times positive is positive and negative times negative is negative. In math symbols, we have (+)*(+)=+ and (-)*(-)=+.

  • Condition 2: If the signs of the two numbers are different, the product is always a negative number.
  • This diagram shows that when you multiply two numbers with different signs, the answer is always negative. That is, positive times negative is negative and negative times positive is negative. In math symbols, we have (+)*(-)=- and (-)*(+)=-.

    Example: Multiply the integers (+4) and (-7)

    Solution: (+4) x (-7) = -28Multiply or find the product of the absolute values (4 x 7 = 28) . Now, determine the sign of the final answer. The rule states that if the signs of the two integers are different then the final answer will be negative. Hence, -28.

    Example: The opposite of   ( - 2)   x   7  x   ( - 1 )  is  __________

    Solution: Multiply the absolute values (2 x 7 x 1 = 14). The product of two negative integers is positive. The opposite of 14 is (-14).

    Properties of Integers under Multiplication:

  • Closure Property: The product of two integers is always an integer. Like for any two integers m and n, m x n is an integer. Integers are closed under multiplication.
  • Example: Verify Closure Property of (+30) by (-5).

    Solution: Multiply +30 x (-5) = -150. Hence, +30, -5 and -150 are integers.

  • Commutative Property: Multiplication of integers is commutative. For integers a and b: a x b = b x a.  for any integers a and b. for example: 3*4=4*3=12.
  • Example: Verify Commutative Property of (+10) and (-5).

    Solution: Let a = +10 and b = -5. LHS = a x b = (+10) x (-5) = -50   and  RHS = b x a = (-5) x (+10) = -50. Hence Verified.

  • Associative Property: The multiplication of integers is associative, i.e., for any three integers a, b, c, we have:   a × ( b × c) = (a × b) × c.

  •  

    Example: Verify Associative Property of (+9) ,  (-5)  and  (-2)

    Solution: Let a = +9 ,  b = -5  and c = -2. LHS = a × ( b × c) = +9 x (-5 x (-2)) = +9 x (+10) = +90

    and  RHS = (a × b) × c = (+9 x  (-5)) x (-2) = (-45) x (-2) = +90. Hence Verified.

  • Multiplicative Identity Property: For every integer a, we have: a × 1 = a = 1 × a. The integer 1 is called the multiplicative identity for integers. For any integer, we have a × 0 = 0 = 0 × a.

  • Example: Simplify:     a)  (-18) x 1    b)  (-7) x 0

    Solution:  a)  (-18) x 1 = (-18)     b)  (-7) x 0  =  0

    Sample Questions
    (More Questions for each concept available in Login)
    Question : 1

    Multiplication of a negative integer for even times gives a  _________  result.

    Right Option : C
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    Explanation
    Question : 2

    If the expression [( - 43) x 109] can also be written as [a + ( - 387 ), what is the value of 'a' ?

    Right Option : B
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    Explanation
    Question : 3

    68 x  _______  = - 68

    Right Option : C
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    Explanation
     
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