What Are Integers?
The set of numbers that include negative numbers, positive numbers as well as zero are known as integers. As shown in the image, we can observe negative numbers and whole numbers as part of the set of integers.
In terms of a set, integers are represented by the letter, ‘Z’ and is written as, Z={…-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8…}
Additionally, all the four basic math operations, addition, subtraction, multiplication and division can be performed on the set of integers.
What Is a Number Line?
While in lower grades there was an introduction to a horizontal line with numbers placed at an equal distance from each other. This is how numbers are represented on a number line.
This image shows us a number line that extends to the left and right of the zero. To the left of the zero we find negative integers and to the right of the zero we can see that the number line contains positive whole or natural numbers.
With the help of the number line we can add, subtract, multiply as well as divide integers.
Representing integers on the number line also helps us in identifying opposites. For example 34 and -34 are opposites as 34 is a positive integer and -34 is a negative integer. The signs of these two numbers are different, but they are located at an equal distance either to the left or right of the zero on a number line.
Graphing Integers on the Coordinate Plane
Another interesting way that integers can be represented is by making use of the coordinate plane or what is also known as a Cartesian plane. A horizontal line that resembles the number line that was just explained and a vertical line intersect at a point on the coordinate plane.
Notice that there are two lines with numbers at equal intervals. The horizontal line is called the x-axis and the vertical line is called the y-axis. Both these lines intersect at a point called the origin.
There are also negative and positive numbers to the left, right, top and bottom of the origin, which marks the point where zero is written on the coordinate plane.
Such a method of expressing integers is used in algebra to find solutions and to ‘plot’ points on the coordinate plane.
Rapid Recall
- Positive numbers are written on the right of the zero on a number line as they are greater in value than zero and the negative numbers.
- Negative numbers are written on the left of the zero on a number line as they are lesser in value when compared to zero or the positive integers.
- Zero is always placed in the middle of the number line as it is neither positive nor negative.
Solved Examples
Example 1:
Represent these situations as a positive or negative integer:
- You travel to the ground floor from the tenth floor
- John wins five rounds of a tennis match
- A truck travels 50 miles downhill
- A leaf falls 200 inches
- A mountaineer summits a mountain at 7000 feet
Solution
Representing the situations as integers:
- You travel to the ground floor from the tenth floor: -10
- John wins five rounds of a tennis match: +5 or 5
- A truck travels 50 miles downhill: -50
- A leaf falls 200 inches: -200
A mountaineer summits a mountain at 7000 feet: + 7000 or 7000
Example 2:
While walking to buy lettuce for his mother Jim starts from his house, walks 20 minutes to reach the farmer’s market. He realizes he has forgotten a list of things to buy for his science experiment at school. So he walks back home from the farmer’s market. He takes the list and walks for thirty minutes to the hardware store and then to the local supermarket for another twenty minutes to buy a snack and a cola. Based on these events represent the situations as integers:
- The time taken for Jim to walk from his house to the supermarket
- The time taken for Jim to get back home from the farmer’s market
- The time taken for Jim to walk back home from the hardware store
Solution:
- The time taken for Jim to walk from his house to the supermarket = + 50 or 50 minutes
- The time taken for Jim to get back home from the farmer’s market = – 20 minutes
- The time taken for Jim to walk back home from the hardware store = – 30 minutes
Note: The integers are represented such that returning home is a negative and walking towards a location is considered positive.
Fractions and decimals are not considered to be integers as the set of integers only consist of positive or negative whole numbers and zero.
Rational numbers that can also be represented on a number line, are numbers that are expressed as a ratio or a fraction of two integers, in the form of ‘p over q’, and the denominator, ‘q’ cannot be zero.
Yes, fractions can also be represented on a number line usually between whole numbers.