Integers begin with the concept of digits. There are ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
The position of the digits is crucial in our number system. Take the number 11537, for example. This can be represented as follows in a place value table.
Positive, zero, and negative numbers constitute a group of numbers known as the group of integers.
Integers are like whole numbers, but they can also be negative, such as –4, –3, –2, –1, 0, 1, 2, 3, 4…
- Positive Integer: All whole numbers greater than zero are positive integers, such as 1, 2, 3, 4, 5, and so on. It has a positive sign.
- Negative Integer: Negative integers are numbers that are less than zero, such as –1, –2, –3, –4, –5, and so on. Its sign is said to be negative.
- Zero Integer: Zero is an integer that has no positive or negative value.
Integer Number Line
The number zero is neither positive nor negative.
Natural numbers, zero, and negative numbers are all examples of integers. There are no fractional components in them. There are various aspects of integers that make it simple for us to perform integer-based calculations. In this lesson, we will study the properties of integers.
Type of Integers
Consecutive numbers are integers that come after each other in a row, with each number being one more than the one before it, for example, 91, 92, 93, 94, etc…
Consecutive integers can be expressed more broadly as n, n +1, n + 2, n + 3…, where n is an integer.
Odd, Even Integers
Odd integers cannot be divided evenly by two, such as –3, –11, –87, 1, 15, 47… An odd integer must always conclude in one of the following numbers: 1, 3, 5, 7, or 9.
Even integers can be split equally by two, such as –2, –34, 0, 2, 90… An even integer is always a number that ends in 0, 2, 4, 6, or 8.
Zero is regarded as an even integer.
Examine the digit in the one’s location to determine whether an integer is even or odd. That single number indicates whether the full integer is odd or even.
For example, the numeral 9,395 is an odd integer since it terminates in the number 5, also an odd integer. Similarly, 902 is an even integer because it terminates in a two.
Properties of Integer
The following are the Different Properties of Integers.
The commutative property of integers is similar to the associative property; the main difference is that we use only two numbers in this property.
According to the commutative property of integers under addition and multiplication, the result of adding and multiplying two integers is always the same regardless of their order.
This property does not apply to subtraction or division operations. Consider the commutative property of integers in the contexts of addition, subtraction, multiplication, and division.
The associative property of numbers under addition and multiplication asserts that the outcome of adding and multiplying more than two integers is always the same, regardless of how the integers are grouped.
The associative property of integers does not apply to subtraction and division of integers since the order of the numbers is critical and cannot be modified in the case of subtraction and division.
To make calculations easier, the distributive property of integers asserts that the multiplication operation can be dispersed over addition and subtraction.
The closure property of integers states that adding, subtracting, and multiplying two integers always yields an integer.
The closure property of integers does not apply to integer division since the division of two integers does not always result in an integer. For instance, we know that 3 and 4 are integers, but 3/4 = 0.75 is not. As a result, the closure property does not apply to integer division. It holds for integer addition, subtraction, and multiplication.
The identity property of integer addition asserts that every number added to 0 produces the same number. For example, if ‘a’ is an integer, then means that a + 0 = 0 + a = a. Consider the negative integer -6 as an example. We get -6 if we add 0 to -6. The outcome remains unchanged.
As a result, we can argue that 0 is the additive identity or the identity element of integer addition.
Look at the graphic below for an overview of the application of various integer properties on four basic arithmetic operations on integers.
|Communicative Property||a + b = b + a||a – b ≠ b – a||a × b = b × a||a ÷ b ≠ b ÷ a|
|Associative Property||a + (b + c) = (a + b) + c||(a – b) – c ≠ a – (b – c)||a × (b × c) = a × (b × c)||(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)|
|Distributive Property||a × (b + c) = a × b + a × c||a × (b − c) = a × b − a × c||Not Applicable||Not Applicable|
|Closure Property||a + b ∈ Z||a – b ∈ Z||a × b ∈ Z||a ÷ b ∉ Z|
|Identity Property||a + 0 = a =0 + a||a – 0 = a ≠ 0 – a||a × 1 = a = 1 × a||a ÷ 1 = a ≠ 1 ÷ a|
Solved Examples of properties of Integers
1) In a test, (+8) marks are awarded for each correct answer and (-2) marks are deducted for each incorrect answer. (i) Anmol answered all questions and received 50 marks despite getting 10 correct answers. (ii) Vijay also answered all questions and received (-22) marks despite giving 3 correct answers.
How many erroneous responses had they given?
One right answer = 8 marks.
As a result, the marks allowed for 10 correct responses = 8 x 10 = 80.
Anmol has a score of 50.
Marks awarded for inappropriate responses = 50 – 80 = –30.
Marks are deducted for each incorrect response = –2.
As a result, the number of wrong responses = (–30) / (–2) = 15.
(ii) Marks awarded for 3 correct answers = 8 x 3 = 24.
Vijay’s score is –22.
Marks awarded for wrong responses = –22 – 24= –46.
Marks are deducted for each incorrect response = –2.
As a result, the number of erroneous responses = (–46) / (–2) = 23.
2) Solve: (−12) × (−3) × 4× (−6) =?
(−12) × (−3) × 4 × (−6) = (36)× 4 × (−6)
=144 × (−6)
3) Fill in the blank:
−112 × _______ =+112
−112 × (−1) = +112.
1) Is there any whole number that cannot be an integer?
Whole numbers begin with zero and end with infinity. For example, 0, 1, 2, 3, 4, and so on. In contrast, integers are numbers that can be both negative and positive. For example, -5, -4, -3, -2, -1, 0, 1, 2, 3, and so on. As a result, every whole number is an integer, but not every integer is a whole number.
2) Which of the following equations demonstrates the distributive property of multiplication?
The distributive property of multiplication is demonstrated by the equation a (b + c) = a b + a c. Because of one or more variables, the terms within the parenthesis cannot be simplified here.
3) What Does “Commutative Property of Addition” Mean?
Natural numbers can be added in any sequence, and the result will be the same, thanks to the commutative feature of addition. The formula for this property is a + b = b + a, which holds for any a, b ∈ N. For example, 1 + 2 or 2 + 1, both will give the same answer.
Therefore, we hope you can now clearly understand the properties of integers. Vedantu is here to aid you in learning the principles of integers, which is one of the basic chapters in mathematics. Our panel of teachers is here to answer your questions and help you achieve good exam results.