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.