# Properties Of Integers

Mathematical equations have their own manipulative principles. These principles or properties help us to solve such equations. The properties of integers are the basic principle of the mathematical system and it will be used throughout the life. Hence, it’s very essential to understand how to apply each of them to solve math problems. Basically, there are three properties which outline the backbone of mathematics. They are:

• Associative property
• Commutative property
• Distributive property

All properties and identities for addition and multiplication of whole numbers are applicable to integers also. Integers include the set of positive numbers, zero and negative numbers which can be represented with the letter $$Z$$.

$$Z$$ = {$$……….-5,-4,-3, -2, -1, 0, 1, 2, 3, 4, 5………$$}

Some of the Properties of Integers are given below:

Property 1: Closure property

Among the various properties of integers, closure property under addition and subtraction states that the sum or difference of any two integers will always be an integer i.e. if x and y are any two integers, $$x ~+~ y$$ and $$x~-~y$$ will also be an integer.

Ex: $$3~ –~ 4$$ = $$3~ + ~(-4)$$ = $$-1$$;

$$(–5)~ +~ 8$$ = $$3$$,

The results are integers.

Closure property under multiplication states that the product of any two integers will be an integer i.e. if $$x$$ and $$y$$ are any two integers, $$xy$$ will also be an integer.

Ex: $$6~ ×~ 9$$ = $$54~; ~(– 5)~ ×~ (3)$$ = $$-15$$, which are integers.

Division of integers doesn’t follow the closure property, i.e. the quotient of any two integers x and y, may or may not be an integer.

Ex : $$(-3)~ ÷~ (-6)$$ = $$\frac{1}{2}$$, is not an integer.

Property 2: Commutative property

Commutative property of addition and multiplication states that the order of terms doesn’t matter, result will be same. Whether it is addition or multiplication, swapping of terms will not change the sum or product. Suppose, $$x$$ and $$y$$ are any two integers, then

$$⇒~ x~ + ~y$$  =  $$y ~+ ~x$$

$$⇒~x~ ×~ y$$ = $$y ~× ~x$$

Ex: $$4 ~+ ~(-6)$$ = $$-2$$ = $$(-6) ~+ ~4;$$

$$10~ ×~ (- 3)$$ = $$- 30$$ = $$(-3) ~ × ~10$$

But, subtraction ($$x ~-~ y~ ≠~ y ~-~ x$$) and division ($$x ~÷~ y~ ≠~ y~ ÷~ x$$) are not commutative for integers and whole numbers.

Ex: $$4~ -~ (-6)~=~10~ ; ~(-6)~–~ 4~=~ -10$$

$$~⇒ ~4~ -~ (-6)~≠~(-6) ~–~ 4$$

Ex: $$10~÷~2~=~5~;~2~÷~10$$ = $$\frac{1}{5}$$

$$⇒~ 10 ~÷ ~2 ~≠~ 2 ~÷~10$$

Property 3: Associative property

Associative property of addition and multiplication states that the way of grouping of numbers doesn’t matter; the result will be same. One can group numbers in any way but the answer will remain same. Parenthesis can be done irrespective of the order of terms. Let x, y and z be any three integers, then

$$⇒~x~ +~ (y ~+ ~z)$$ = $$(x ~+ ~y)~ +z$$

$$⇒~x~ ×~ (y~ × ~z)$$ = $$(x ~× ~y) ~×~ z$$

Ex: $$1~+~(2 (-3))$$ = $$0$$ = $$(1~ + ~2)~+~(-3);$$

$$1~× ~(2~× ~(-3))$$ =$$-6$$ = $$(1 ~×~ 2)~×~(-3)$$

Subtraction of integers is not associative in nature i.e. $$x~-~(y ~-~ z)~≠~(x ~- ~y)~-~z$$.

Ex: $$1 ~-~ (2~ -~ (-3))$$ = $$-4;~(1~ – ~2)~ – ~(-3)$$ = $$-21~ – ~(2~ – ~(-3)) ~≠~(1 ~- ~2) ~-~ (-3)$$

Property 4: Distributive property

Distributive property explains the distributing ability of an operation over another mathematical operation within a bracket. It can be either distributive property of multiplication over addition ordistributive property of multiplication over subtraction. Here integers are added or subtracted first and then multiplied or multiply first with each number within the bracket and then added or subtracted.This can be represented for any integers x, y and z as:

$$⇒~x~ × ~(y ~+ ~z)$$ = $$x~×~y~ + ~x~×~ z$$

$$⇒~x ~×~ (y~- ~z)$$ = $$x~×~ y~-~ x~×~ z$$

Ex: $$-5 ~(2~ + ~1)$$ = $$-15$$ = $$(-5~× ~ 2)~ + ~(-5 ~×~ 1)$$

Property 5: Identity Property

Among the various properties of integers, additive identity property states that when any integer is added to zero it will give the same number. Zero is called additive identity. For any integer $$x$$,

$$x~ +~ 0$$ = $$x$$ = $$0 ~+~ x$$

Multiplicative identity property for integers says that whenever a number is multiplied by the number 1 it will give the integer itself as the product. Therefore, the integer 1 is called the multiplicative identity for a number. For any integer $$x$$,

$$x~ ×~ 1$$ = $$x$$ = $$1 ~×~ x$$

If any integer multiplied by 0, product will be zero:

$$x~ × ~0$$ = $$0$$ =$$0~ × ~x$$

If any integer multiplied by -1, product will be opposite of the number:

$$x ~×~ (-1)$$ = $$-x$$ = $$(-1) ~× ~x$$<

 Property Addition Multiplication Commutative x+ y = y+ x x × y = y × x Associative x + (y + z) = (x + y) +z x × (y × z) = (x × y) × z Distributive x × (y + z) = x× y + x× z Identity x + 0 = x =0 + x x * 1 = x = 1 * x

To solve more problems on properties of integers, download BYJU’S – The Learning App from Google Play Store and watch interactive videos. Also, take free tests to practice for exams.

##### Properties of Integers

After studying about the multiplication of integers, we will now learn about various properties of integers. Let us try to understand them one by one.

##### Properties of Integers: Closure Property

We will understand this property using various examples.

We know that,

(-5) x (-7) = 35, which is an integer.

8 x (-9) = -72, which is an integer.

(-6) x 5 = -30, which is also an integer.

Based on the observation of the above examples, we can say that integers are closed under multiplication. In other words, the product of multiplication of two integers is always an integer.

##### Properties of Integers: Commutative Property

Consider the following examples:

4 x (-6) = (-6) x 4 = -24

(-2) x (-3) = (-3) x (-2) = 6

4 x 9 = 9 x 4 = 36

What do we observe? We can say that multiplication is commutative for integers, i.e. multiplication of two integers is irrespective of its order.

##### Properties of Integers: Multiplication by 0

We have studied that multiplication of any whole number by 0 yields the product as 0. Now visualize the given patterns-

(-5) x 0 = 0

7 x 0 = 0

0 x (-9) = 0

Thus, we can say that the multiplication of any integer by 0 will give the product as 0.

##### Properties of Integers: Multiplicative Identity

Any number ‘a’ when multiplied with an integer gives the product as the same integer then ‘a’is called the multiplicative identity of that integer.

When you multiply any integer with 1, the product is the same integer. For e.g.,

1 x (-9) = -9, (-7) x 1 = -7

But what happens if we multiply an integer with (-1). Check the following examples-

9 x (-1) = -9

(-5) x (-1) = 5

We see that when we multiply an integer with (-1) we get the additive inverse of the same integer.

##### Properties of Integers: Associative Property

Associative property of multiplication of integers states that the product of three integers does not depend upon the grouping of integers.

This property will become clear using the given example:

(-2) x (-6) x (-3) = [(-2) x (-6)] x (-3) = (-2) x [(-6) x (-3)] = -36

##### Properties of Integers: Distributive Property

Consider the following example,

(-3) x (4+6) = (-3) x 10 = -30

Also, (-3) x (4+6) = [(-3) x 4] + [(-3) x 6] = -12 + (-18) = -30

Thus, distributivity of multiplication over addition is true for integers.