Subtraction of Rational Numbers

Subtracting rational numbers involves adding the additive inverse of the second number to the first. It requires changing the signs of the terms of the number being subtracted and then performing the arithmetic operation. There are three components of a rational number:

Numerators: These are the integers that represent parts of a whole.

Denominators: These are non-zero integers that specify the number of equal parts the whole is divided into.

Operators: For subtraction, the operator is the minus (-) symbol.

When subtracting rational numbers, students should follow these rules:

Common Denominator: Ensure both rational numbers have the same denominator before subtraction.

Subtract the numerators: After having a common denominator, subtract the numerators.

Simplify the result: Simplify the fraction to its lowest terms after subtraction.

The following are the methods of subtraction of rational numbers:

Method 1: Horizontal Method

To apply the horizontal method for subtraction of rational numbers, use the following steps.

Step 1: Write both rational numbers in the same line using a minus sign in between.

Step 2: If needed, rewrite the rational numbers to have a common denominator.

Step 3: Subtract the numerators and place the result over the common denominator. Simplify if possible.

Let’s apply these steps to an example: Question: Subtract 3/4 from 5/2

Step 1: Write both numbers in the same line,

Step 2: Make denominators the same: 3/4 = 6/8, 5/2 = 20/8

Step 3: Subtract the numerators: 20/8 - 6/8 = 14/8 = 7/4 Answer: 7/4

Method 2: Fraction Bar Method

When subtracting rational numbers using the fraction bar method, we place one fraction over the other.

Ensure like denominators, then subtract the numerators and simplify the result. For example, Subtract 2/3 from 7/6

Solution: Arrange the like denominators 7/6 ← Minuend (from which we subtract) - 2/3 ← Subtrahend (what we subtract, rewritten as 4/6) ------- 7/6 - 4/6 = 3/6 = 1/2

Therefore, upon subtracting, we get 1/2

In rational numbers, subtraction has some characteristic properties. These properties are listed below:

• Subtraction is not commutative In subtraction, changing the order of the numbers changes the result, i.e., A - B ≠ B - A

• Subtraction is not associative Unlike addition, we cannot regroup in subtraction. When three or more numbers are involved, changing the grouping changes the result. (A − B) − C ≠ A − (B − C)

• Subtraction is the addition of the opposite sign Subtracting a number is the same as adding its opposite, so to make calculations easier, you can convert subtraction into addition by changing the signs of the second number. A − B = A + (−B)

• Subtracting zero from a number leaves the number as is Subtracting zero from any number results in the same rational number: A - 0 = A

Tips and tricks are useful for students to efficiently deal with the subtraction of rational numbers. Some helpful tips are listed below:

Tip 1: Always ensure the denominators are the same before subtracting the numerators.

Tip 2: Simplify the fraction to its lowest terms after subtraction for clarity.

Tip 3: Beginners can benefit from visual aids like fraction bars or number lines to avoid common mistakes.

Subtraction of rational numbers can be challenging due to the need for common denominators, often leading to common mistakes. However, being aware of these errors can help students avoid them.