Basic Arithmetic Operations Involving Integers

 
1.2  Basic Arithmetic Operations Involving Integers
 
Addition of integers:
 
  • Positive integers is represented by moving towards the right.
  • Negative integers is represented by moving towards the left.
 

 
Subtraction of integers:
 
  • Positive integers is represented by moving towards the left.
  • Negative integers is represented by moving towards the right.
 

 
  Example  
     
 

Solve:

(i)

\(\begin{aligned}6-(-7)&=6+7 \\\\&=13. \end{aligned}\)

(ii)

\(\begin{aligned}-21+(3)&=-21+3 \\\\&=-18. \end{aligned}\)

 
 
Multiplication of integers:
 
Operation Sign of the product
\((+)\times(+)\) \(+\)
\((+)\times(-)\) \(-\)
\((-)\times(+)\) \(-\)
\((-)\times(-)\) \(+\)
 
Division of integers:
 
Operation Sign of the quotient
\((+)\div(+)\) \(+\)
\((+)\div(-)\) \(-\)
\((-)\div(+)\) \(-\)
\((-)\div(-)\) \(+\)
 

In general,

  • The product or quotient of two integers with the same signs is a positive integer.
  • The product or quotient of two integers with different signs is a negative integer.
 
  Example  
     
 

Calculate:

(i)

\(\begin{aligned}9\times(-11)&=-(9\times11) \\\\&=-99. \end{aligned}\)

(ii)

\(\begin{aligned}-48\div(-8)&=+(48\div8) \\\\&=6 \end{aligned}\)

 
 
Combined basic arithmetic operations of integer:
 

 
  Example  
     
 

Solve:

(i)

\(\begin{aligned}&\space49\div(-8+1)\\\\&=49\div(-7) \\\\&=-7. \end{aligned}\)

(ii)

\(\begin{aligned}&\space\dfrac{22+(-4)}{-7-2} \\\\&= \dfrac{22-4}{-9}\\\\ &=\dfrac{18}{-9}\\ \\&=-2. \end{aligned}\)

 
 
Laws of arithmetic operations:
 

Commutative Law

\(\begin{aligned} a+b&=b+a \\\\a\times b&=b\times a \end{aligned}\)

 

Associative Law

\(\begin{aligned} (a+b)+c&=a+(b+c) \\\\(a\times b)\times c&=a\times(b\times c) \end{aligned}\)

 

Distributive Law

\(\begin{aligned} a\times(b+c)&=a\times b+a\times c \\\\a\times(b-c)&=a\times b-a\times c \end{aligned}\)

 

Identity Law

\(\begin{aligned} a+0&=a \\\\a\times 0&=0 \\\\a\times 1&=a \\\\a+(-a)&=0 \\\\a\times\dfrac{1}{a}&=1 \end{aligned}\)

Basic Arithmetic Operations Involving Integers

 
1.2  Basic Arithmetic Operations Involving Integers
 
Addition of integers:
 
  • Positive integers is represented by moving towards the right.
  • Negative integers is represented by moving towards the left.
 

 
Subtraction of integers:
 
  • Positive integers is represented by moving towards the left.
  • Negative integers is represented by moving towards the right.
 

 
  Example  
     
 

Solve:

(i)

\(\begin{aligned}6-(-7)&=6+7 \\\\&=13. \end{aligned}\)

(ii)

\(\begin{aligned}-21+(3)&=-21+3 \\\\&=-18. \end{aligned}\)

 
 
Multiplication of integers:
 
Operation Sign of the product
\((+)\times(+)\) \(+\)
\((+)\times(-)\) \(-\)
\((-)\times(+)\) \(-\)
\((-)\times(-)\) \(+\)
 
Division of integers:
 
Operation Sign of the quotient
\((+)\div(+)\) \(+\)
\((+)\div(-)\) \(-\)
\((-)\div(+)\) \(-\)
\((-)\div(-)\) \(+\)
 

In general,

  • The product or quotient of two integers with the same signs is a positive integer.
  • The product or quotient of two integers with different signs is a negative integer.
 
  Example  
     
 

Calculate:

(i)

\(\begin{aligned}9\times(-11)&=-(9\times11) \\\\&=-99. \end{aligned}\)

(ii)

\(\begin{aligned}-48\div(-8)&=+(48\div8) \\\\&=6 \end{aligned}\)

 
 
Combined basic arithmetic operations of integer:
 

 
  Example  
     
 

Solve:

(i)

\(\begin{aligned}&\space49\div(-8+1)\\\\&=49\div(-7) \\\\&=-7. \end{aligned}\)

(ii)

\(\begin{aligned}&\space\dfrac{22+(-4)}{-7-2} \\\\&= \dfrac{22-4}{-9}\\\\ &=\dfrac{18}{-9}\\ \\&=-2. \end{aligned}\)

 
 
Laws of arithmetic operations:
 

Commutative Law

\(\begin{aligned} a+b&=b+a \\\\a\times b&=b\times a \end{aligned}\)

 

Associative Law

\(\begin{aligned} (a+b)+c&=a+(b+c) \\\\(a\times b)\times c&=a\times(b\times c) \end{aligned}\)

 

Distributive Law

\(\begin{aligned} a\times(b+c)&=a\times b+a\times c \\\\a\times(b-c)&=a\times b-a\times c \end{aligned}\)

 

Identity Law

\(\begin{aligned} a+0&=a \\\\a\times 0&=0 \\\\a\times 1&=a \\\\a+(-a)&=0 \\\\a\times\dfrac{1}{a}&=1 \end{aligned}\)