Comparing Negative Numbers: How to Determine Which is Greater Between -0.7 and -1.5

When it comes to negative numbers, determining which one is greater can sometimes be a bit confusing. In this article, we will explore how to compare two negative numbers, specifically -0.7 and -1.5, and explain the logic behind it.

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are used to represent quantities that are opposite in direction or quality. For example, if a bank account shows a balance of -50 dollars, it means the account has a 50 dollar debt.

Using a Number Line

The simplest and most visual way to compare negative numbers is by using a number line. In a number line, numbers are arranged in order from left to right, with negative numbers to the left of zero and positive numbers to the right. Negative numbers are generally represented with a minus sign.

Number Line Representation

Let's compare -0.7 and -1.5 using a number line:

Number Line

2 1 0 -0.7 -1.5 -1.8 -2.3

In this illustration, -0.7 is closer to zero than -1.5. When comparing two numbers on a number line, the number that is closer to zero is considered greater. Therefore, -0.7 is greater than -1.5.

This can be observed by looking at the positions of -0.7 and -1.5 on the number line. -0.7 is to the right of -1.5, indicating that it is greater.

Using Absolute Values

Another method to determine which negative number is greater is by using the concept of absolute values. The absolute value of a number is its distance from zero, regardless of direction. Mathematically, the absolute value of a number is represented by placing the number between two vertical bars: |number|.

Example 1: Subtracting the Numbers

To find out which number is greater, we can subtract -1.5 from -0.7:

-0.7 - (-1.5) -0.7 1.5 0.8

Since the result is a positive number (0.8), it indicates that -0.7 is greater than -1.5.

Example 2: Subtracting the Positive Values

Alternatively, we can subtract the positive values:

-0.7 - 1.5 -2.2

To compare the original numbers, we can also subtract the absolute values:

|1.5| - |0.7| 1.5 - 0.7 0.8

A positive result (0.8) again indicates that -0.7 is greater than -1.5.

Mathematical Logic

Understanding why -0.7 is greater than -1.5 can be further explained by considering the order of magnitude. When dealing with negative numbers, the closer a number is to zero, the greater it is. This is because subtracting a larger negative number from zero results in a smaller negative number being added to the result.

For instance, -0.7 is 0.7 units away from zero, while -1.5 is 1.5 units away from zero. Since 0.7 is smaller than 1.5, -0.7 is greater than -1.5.

Summary

When comparing negative numbers like -0.7 and -1.5, the key is to look at their positions on a number line. The number that is closer to zero is considered greater. Alternatively, using absolute values and simple subtraction can also help determine which negative number is greater.

By remembering these mathematical principles, you can confidently compare negative numbers and solve problems involving negative numbers with ease.