What are Integers?
The set of all the positive and negative numbers including zero are called integers. They don’t have any fractional or decimal parts, and are complete numbers. For example \( 2,~9,~-4,~0,~134 \) are integers but \( \frac{1}{2},~-\frac{3}{4},~2.5,~0.12,~-8.325 \) do not belong to the set of integers.
Properties of Integers
- Commutative property
- Associative property
- Additive inverse property
- Addition property of zero
- Multiplication property of zero.
- Distributive property
- Closure Property
Commutative Property
The commutative property states that the order of terms does not matter while the operation is performed. This property is valid for addition and multiplication of integers. This implies that addition or multiplication of any two integers provides the same result even when their order is reversed.
For example, there are two integers \( 2 \) and \( -3 \) then
\( 2 + (-3) = -1 \) and also \( -3 + 2 = -1 \)
\( 2 \times (-3) = -6 \) and also \( -3 \times 2 = -6 \)
In general, If there are two integers \( x \) and \( y \) then can say that,
- \( x + y = y + x \) (commutative property of addition)
- \( x \times y = y \times x \) (commutative property of multiplication)
But \( 2 – (-3) = 5 \) and \( -3-2= -5,~5 \neq -5 \)
\( 2 \div (-3)= -\frac{2}{3} \) and \( -3 \div 2 = -\frac{3}{2},~-\frac{2}{3} \neq -\frac{3}{2} \)
Since, \( x – y \neq y – x \) and \( x \div y \neq y \div x \)
Therefore, we can say that commutative property for subtraction and division is not valid.
Associative Property
The associative property of integers states that, addition or multiplication of more than two integers is always the same irrespective of their order of addition or multiplication.
For example, there are three integers \( 2,~5 \) and \( -4 \).
\( 2+5+(-4)=(2+5)+(-4)=7+(-4)=3 \)
Also,
\( 2+5+(-4)=2+(5+(-4))=2+1=3 \)
\( 2 \times 5 \times (-4) = (2 \times 5) \times (-4) = 10 \times (-4) = -40 \)
\( 2 \times 5 \times (-4) = 2 \times (5 \times (-4)) = 2 \times (-20) = -40 \)
In general, If there are three integers \( x \), \( y \) and \( z \) then
- \( (x+y)+z=x+(y+z) \) (Associative property of addition)
- \( (x \times y) \times z = x \times (y \times z) \) (Associative property of multiplication)
But, \( 2-5-(-4)=(2-5)-(-4)= -3+4=1 \)
\( ~~~~~~~~2-5-(-4)=2-(5-(-4))=2-9= -7, ~1 \neq -7 \)
\( ~~~~~~~~2 \div 5 \div (-4) = (2 \div 5) \div (-4) = \frac{2}{5} \times (-\frac{1}{4})= -\frac{1}{10} \)
\( ~~~~~~~~2 \div 5 \div (-4) = 2 \div (5 \div (-4)) = \frac{2}{1} \times (-\frac{4}{5})= -\frac{8}{5}, ~-\frac{1}{10} \neq -\frac{8}{5} \)
Since \( (x-y)-z \neq x-(y-z) \) and \( (x \div y)\div -z \neq x \div (y \div z) \)
Therefore, associative property for subtraction and division is not valid.
Additive Inverse Property
Additive inverse property states that, the sum of an integer and its additive inverse (or opposite) is equal to zero(0).
For example-
\( 8+(-8)=0 \)
\( -12+12=0 \)
In general, if \( x \) is an integer then the additive inverse of \( x \) is \( -x \).
Addition Property of Zero
The addition property of zero states that the sum of any number and \( 0 \) is equal to that number.
For example-
\( 9+0=9 \)
\( -5+0= -5 \)
In general, if \( x \) is an integer then \( x+0=x \).
Multiplication Property of Zero
It states that, the product of any number and \( 0 \) is equal to \( 0 \).
For example–
\( 7 \times 0 = 0 \)
\( -9 \times 0 = 0 \)
In general, if \( x \) is a number then \( x \times 0 = 0 \).
Distributive Property
Distributive property of integers states that the product of numbers can be distributed over addition and subtraction.
For example–
\( x \times (y + z) = x \times y + x \times z \) (Distributive property over addition)
\( x \times (y – z) = x \times y – x \times z \) (Distributive property over subtraction)
Closure Property
The closure property under addition and subtraction states that, the addition or subtraction of any two integers will always be an integer.
If x and y are any two integers, then \( x + y \) and \( x – y \) will also be an integer.
For example-
\( 5-3 = 5 + (-3) = -2 \)
\( -5 + 8 = 3 \)
The results, \( -2 \) and \( 3 \) are integers.
The closure property under multiplication states that, the product of any two integers will give an integer.
If \( x \) and \( y \) are two integers, then \( x \times y\) will also be an integer.
For example-
\( 7 \times 8 = 56; ~(-6) \times (3) = -18 \), which are integers.
The division of integers doesn’t follow the closure property.
The quotient of any two integers x and y, may or may not be an integer value.
For example–
\( (-1) \div (-2) = \frac{1}{2} \), is not an integer.
Solved Examples
Example 1: Use the distributive property to find the product \( 5\times 22 \).
Solution:
\( 5\times 22 = 5 \times (20+2) \) Write \( 22 \) as the sum of \( 20 \) and \( 2 \)
\( ~~~~~~~~~~~= 5\times 20 + 5 \times 2 \) Distributive property over sum
\( ~~~~~~~~~~~= 100 + 10 \) Find the product
\( ~~~~~~~~~~~= 110 \) Add
So, \( 5\times 22 = 110 \).
Example 2: Use the distributive property to find the product \( \frac{1}{2} \times 3\frac{2}{5} \).
Solution:
\( \frac{1}{2} \times 3\frac{2}{5} = \frac{1}{2} \times (3 + \frac{2}{5}) \) Write \( 3\frac{2}{5} \) as sum of \( 3 \) and \( \frac{2}{5} \)
\( ~~~~~~~~~~~~=\frac{1}{2} \times 3 + \frac{1}{2} \times \frac{2}{5} \) Distributive property over sum
\( ~~~~~~~~~~~~= \frac{3}{2} + \frac{2}{10} \) Find the product
\( ~~~~~~~~~~~~= \frac{17}{10} \) Add
\( ~~~~~~~~~~~~= 1\frac{7}{10} \) Convert to mixed fraction
So, \( \frac{1}{2} \times 3\frac{2}{5} = 1\frac{7}{10} \).
Example 3: Simplify the expression \( 5(3+y+7) \) using properties of integers.
Solution:
\( 5(3+y+7) \) Write the expression
\( = 5 \times 3 + 5 \times y + 5 \times 7 \) Distributive property over sum
\( = 15 + 5y + 35 \) Find the product
\( = 5y + 15 + 35 \) Commutative property of addition
\( = 5y + 50 \) Add \( 15 \) and \( 35 \)
So, \( 5 (3 + y + 7) = 5y + 50 \).
Example 4: Simplify the expression \( 9+(5x+7) \) using properties of integers.
Solution:
\( 9+(5x+7) \) Write the expression
\( = (9+7)+5x \) Associative property over addition
\( = 16+5x \) Add \( 9 \) and \( 7 \)
So, \( 9+(5x+7)= 16+5x \).
Example 5: Find the sum of \( 25+32+40+(-25) \).
Solution:
\( 25+32+40+(-25) \) Write the sum
\( = 32+40+25+(-25) \) Commutative property of addition
\( = 32+40+(25+(-25)) \) Associative property of addition
\( = 32+40+0 \) Additive inverse property
\( = 72+0 \) Add \( 32 \) and \( 40 \)
\( = 72 \) Addition property of \( 0 \).
Since \( -10+10=0 \), the additive inverse of \( -10 \) is \( 10 \).
No, commutative property is only valid for addition and multiplication.
If \( x,y \) and \( z \) are integers, then \( x \times (y-z)=x \times y-x \times z \).
- Addition
- Subtraction
- Multiplication
- Division
The addition property of zero states that the sum of any number and zero is equal to that number.