Integers Properties Quiz

Integers

Integers are a set of numbers that includes whole numbers and their negatives. They are generated from the set of counting numbers 1, 2, 3, and so on, along with their negative counterparts. The set of integers is infinite and is denoted as {...,-4,-3,-2,-1,0,1,2,3,4,...}. Integers are used in various mathematical operations such as addition, subtraction, multiplication, and division.

Properties of Addition and Subtraction of Integers

Closure under Addition and Subtraction

For every integer a and b, a+b and a–b are also integers.

Commutativity Property for Addition

For every integer a and b, a+b=b+a.

Associativity Property for Addition

For every integer a, b, and c, (a+b)+c=a+(b+c).

Additive Identity and Additive Inverse

For every integer a, a+0=0+a=a, where 0 is the additive identity. Similarly, for every integer a, a+(–a)=0, where –a is the additive inverse of a.

Properties of Multiplication of Integers

Closure under Multiplication

For every integer a and b, a×b is an integer.

Commutative Property of Multiplication

For every integer a and b, a×b=b×a.

Multiplication by Zero

For every integer a, a×0=0×a=0.

Multiplicative Identity

For every integer a, a×1=1×a=a, where 1 is the multiplicative identity for integers.

Associative Property of Multiplication

For every integer a, b, and c, (a×b)×c=a×(b×c).

Introduction to Zero

Zero is neither a positive nor a negative integer. It is the neutral element for addition and multiplication of integers.

Properties of Division of Integers

Division by 1

For every integer a, a÷1=a.

Division by 0

For every integer a, a÷0 is undefined.

Division by a Non-Zero Integer

For every integer a, a÷(–a)=–1, where –a is the additive inverse of a.