What are Integers?
Integers are numbers that have no decimal or fractional part. Integers can belong to the group of numbers that are both negative and positive sets of numbers along with 0. The symbol used to represent integers is z. Here are the following examples of integers:
- Positive integers: These integers are positive and greater than 0.
- For example, 3, 4, 5, …
- Negative integers: These integers are negative and lesser than 0.
- For example, – 5, – 6, – 7, …
0 is neither positive nor negative and a whole number, but it’s an integer as well.
What are Mathematical Operational Rules for Integers?
The arithmetic operations for integers have the following rules:
- Sum of two positive integers will give an integer as result.
- Sum of two negative integers will give an integer as result.
- Product of two positive integers gives an integer.
- Product of two negative integers gives an integer.
- The addition of positive and negative integers of the same value will result in 0.
What are the Properties of Integers?
Here are the properties of integers:
Commutative Property
The commutative property of integers states that when the positions of the operands are switched in an arithmetic operation, the result remains the same.
For example,
2 + 5 = 5 + 2,
2 \( \times\) 5 = 5 \( \times\) 2
Associative Property
The associative property of integers states that when the grouping of two integers is changed, the result remains unaffected.
For example:
3 + (5 + 2) = (3 + 5) + 2 = 10,
2 \( \times\) (3 \( \times\) 5) = (2 \( \times\) 3) \( \times\) 5 = 30
Distributive Property
The distributive property of integers states that when there is an expression of the form a × (b + c), the operand a gets distributed among the other operands, b, and c, and is written as: a × (b + c) = a × b + a × c
Solved Examples on Integers
Example 1: Amy’s football team gained 16 yards after losing 5 yards in the previous game. What was Amy’s team’s total gain or loss for the two matches?
Solution:
According to the summary, on the first day of the game, Amy’s football team lost 5 yards. On the second day, they won 16 yards. Here, both the numbers are integers as they don’t have any decimals or fractional parts. Now, in order to find total gain or loss, we will subtract.
16 – 5 = 11 yards.
Since the number is a positive integer, this means it was a gain in total for Amy’s football team.
Example 2: Solve the following product of integers:
(a) (- 4) \( \times\) 8
(b) (+ 5) \( \times\) (+ 6)
(c) 6 \( \times\) (- 2)
solution:
Part (a): When two integers with different signs are multiplied, the result is negative.
Thus, (- 4) \( \times\) 8 = – 32 is the answer.
Part (b): When two integers with the same sign are multiplied, the result is positive. Thus, (+ 5) \( \times\) (+ 6) = + 30 is the answer.
Part (c): This is similar to the (a) sum. So, 6 \( \times\) (- 2) = – 12 will be the answer.
Example 3: Which property is used for the following: 8 \( \times\) 9 = 9 \( \times\)8?
Solution:
The commutative property states that when the operands’ positions are switched, the result remains the same. Let us use that property for this sum to identify the ideal property.
Hence, 8 \( \times\) 9 = 9 \( \times\) 8 = 72 is the answer.
Thus the property used in this expression is commutative property.
Yes, integers can be negative.
There are three types of integers: zero, positive and negative integers.
Zero is neither a negative integer nor a positive integer. In other words we can say that it is neutral in nature.
Fractions or decimal values are not integers. Integers only consists of positive and negative numbers such as 1, 2, -3, -60, 100 and so on.
Integers like whole numbers and natural numbers can be represented on a number line.