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Questions and Answers
Every integer can be expressed using whole numbers and their negatives.
Every integer can be expressed using whole numbers and their negatives.
True
The set of integers is finite and denoted as {...,-2, -1, 0, 1, 2,...}.
The set of integers is finite and denoted as {...,-2, -1, 0, 1, 2,...}.
False
Multiplication of any two integers will result in another integer.
Multiplication of any two integers will result in another integer.
True
Zero is the multiplicative identity for integers.
Zero is the multiplicative identity for integers.
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Zero is considered a positive integer in the set of integers.
Zero is considered a positive integer in the set of integers.
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For every integer a, a+0=a.
For every integer a, a+0=a.
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Study Notes
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.
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Description
Test your knowledge on the properties of integers, including addition, subtraction, multiplication, division, and the role of zero. Explore concepts such as closure, commutativity, associativity, additive identity, and more.
