Closure Property
2026-02-28 01:38 Diff
https://brightchamps.com/en-us/math/numbers/closure-property

According to the closure property, real numbers are closed under addition, subtraction, multiplication, and division (except division by zero, which is undefined).

Closure property for integers

The set of integers is represented as:

Z = {…, –4, –3, –2, –1, 0, 1, 2, 3, 4, …}

The closure property applies to the arithmetic operations such as addition, subtraction, and multiplication of integers, but not to division.

Closure property under addition

According to this property, the sum of any two integers always results in another integer. That is, for the integers a and b, their sum (a + b) is also an integer.

For instance:

\((–7) + 9 = 2.\)

\(5 + 12 = 17.\)

Closure property under subtraction

The difference between two integers will always result in an integer. That is, for the integers a and b, (a – b) will also be an integer.

For instance:

\(12 – 7 = 5.\)

\((– 8) – (– 2) = – 6\)

Property of closure under multiplication

When two integers are multiplied, their product will always be an integer. For the integers a and b, their product (a × b)  will also be an integer.

For instance:

\(4 × (–6) = –24\)

\((–9) × (–5) = 45\)

Property of closure under division

Division does not always yield an integer. So, integers are not closed under division. 

For instance:

\((–15) ÷ 3 = – 5\) (an integer) 

\((–8) ÷ (–20) = 0.4\) (not an integer)