How to Arrange Integers from Least to Greatest and Vice Versa
Integers are numbers that are not fractions but whole numbers. Integers are of two types-negative and positive integers. The numbers are classified as positive and negative integers only according to their placement on the number line.
If the numbers are placed on the left side of 0 on the number line, the integers are called negative integers. If the integers are placed on the right side of 0 on a number line, they are called positive integers. Apart from this difference, the negative integers always have a negative sign in front of the number, while the positive integers don’t have one. Let us know how to put these integers in order.
Ordering Integers
Ordering integers means one has to arrange the integers in a particular sequence. To order the integers, one has to put them on a number line. The most basic rule you should remember is that the integer on the left on a number line is always smaller.
A Number Line(self-made)
Have a look at the picture of the number line in the above image.
You can see that $- 1$ is placed on the left side of 0. This means that 0 is greater than $- 1$; hence, $0 > - 1$. Have a look on the right side of 0. Since 1 is placed before 2, you can write that 2 is greater than 1, $2 > 1$. Therefore, wherever you go from left to right on the numberline, the numbers keep increasing. This rule applies to all integers. Now you may understand the meaning of the temperatures in a thermometer with a minus sign.
A Thermometer
Things to Remember While Ordering Integers
Positive integers are always greater than negative integers.
Zero is greater than every negative integer but smaller than every positive integer.
The greater the number, the lesser is the value of its negative integer. For example, 5 is greater than 0, but $- 5$ is much less than 0.
Solved Questions on the Ordering of Integers
1. Arrange the given integers from greater to lesser. $“9, 2, 0, -5, -7, -1, -2”$.
Solution: The question says that you have to order the integers from greater to lesser. Negative integers are always less than positive integers, so you must write the largest integer first. Hence, the order is $9, 2, 0, - 1, - 2, - 5, - 7$.
2. Arrange the given integers in lesser to greater. $"2,-4,5,8,-10,-3,3"$.
Solution: The question says that the order of the integers must be from lesser to greater. As the negative integers are always less than the positive integers, the negative integers will be written first while ordering the integers. Hence, the order of the given integers is $- 10, - 4, - 3, 2, 3, 5, 8$.
3. Arrange in ascending order. $"20, -10, 0, 12, -13"$.
Solution: As the question says that the given integers must be in ascending order, one has to arrange them in a lesser to greater order. The negative integers are always less than the positive integers. Hence, the order of the given integers is $- 13, - 10,0,12,20$.
4. Arrange in descending order. $"17,16,-13,-14,12"$.
Solution: According to the question, one has to solve the given integers in descending order or from greater to lesser. Since the positive integers are greater than the negative integers, one has to write the greater positive integers first. Hence, the order of the integers is $"17,16,12,-13,-14"$.
Conclusion
In this article, you have learnt about positive and negative integers, the representation of integers on a number line, and ordering integers from greater to lesser. You have also learnt that the numbers on the right are always greater than the ones on the left on a number line. According to this simple rule, you can arrange the numbers in either ascending order or descending order as per the questions.
FAQs on Ordering of Integers: Step-by-Step Guide
1. What is the fundamental rule for ordering integers on a number line?
The fundamental rule for ordering integers is based on their position on the number line. Any integer to the right of another integer is always greater. Consequently, any integer to the left is always smaller. This means all positive integers are greater than 0 and all negative integers, while any negative integer is less than 0 and all positive integers. For example, -3 is greater than -9 because -3 is to the right of -9 on the number line.
2. What is the main difference between comparing and ordering integers?
The main difference lies in the number of integers involved. Comparing integers is the process of evaluating just two integers to see which is greater or smaller (e.g., determining that -5 < 2). In contrast, ordering integers involves arranging a set of three or more integers in a specific sequence, either from smallest to largest (ascending order) or largest to smallest (descending order).
3. Can you provide a step-by-step example of ordering a set of mixed integers?
Certainly. To order the set {4, -7, 0, 2, -1} in ascending (smallest to largest) order, follow these steps:
Step 1: Identify all negative integers. Here, they are -7 and -1. The negative integer furthest from 0 is the smallest, so -7 comes first.
Step 2: Arrange the negative integers. After -7, the next smallest is -1.
Step 3: Place zero. Zero is greater than all negative integers, so it comes next.
Step 4: Arrange the positive integers (2 and 4) in increasing order.
The final ordered set is -7, -1, 0, 2, 4.
4. Does the negative sign (-) on an integer always mean something is being subtracted?
No, and this is a common point of confusion. A negative sign in front of an integer, like -10, indicates its value and position on the number line—specifically, that it is 10 units to the left of zero. It only represents the operation of subtraction when it is placed between two numbers, such as in the mathematical expression 15 - 10.
5. How does the concept of 'absolute value' help in ordering negative integers?
The absolute value of an integer is its distance from zero, which is always positive. When ordering negative integers, the rule is inverted: the integer with the larger absolute value is actually the smaller number. For example, the absolute value of -15 is 15, and the absolute value of -8 is 8. Since 15 > 8, the original integers are ordered as -15 < -8, because -15 is further from zero in the negative direction.
6. In what real-world situations is ordering integers an important skill?
Ordering integers is a crucial skill used in many practical, real-world scenarios. Examples include:
Finance: Ranking financial transactions, where a debt of ₹5000 (-5000) is a smaller financial position than a debt of ₹2000 (-2000).
Geography: Comparing elevations of different geographical locations relative to sea level (0). A point at -100m is lower than a point at -50m.
Temperature: Arranging temperatures from coldest to warmest, which often involves ordering a mix of negative and positive values (e.g., -11°C, -2°C, 0°C, 5°C).
