Addition, Subtraction, and Rational Numbers

Background

We now know how to make rational numbers. Let's learn how to add and subtract them:

Question

Evaluate:

1.   
2.  
3. 0.25 + 0.5 

Answer

1. 
2.  
3. 0.25 + 0.5 = 0.75 

Analysis

Question 1

When we're looking at adding and subtracting fractions, we can think of the operation in a couple of different ways.

One way is to think of the number line. The denominator, the 2, tells us the number of jumps we need to make to get from 0 to 1. If we're making the same number of jumps in our two fractions, then we can add up the number of jumps we've made (that's the numerator). So we can say:



Another way to view this is to think of a pizza. We take a pizza and cut it into the number of pieces in the denominator (i.e. 2). The numerator tells us the number of slices we have. And so if I have 1 out of 2 slices of pizza (that's the first fraction) and I add to it another slice of the same size, I now have an entire pizza to myself!



Question 2

When we're looking at two fractions with different denominators, we have to first make the denominators equal. Why? Well... we can answer that a couple of ways.

If we're thinking of the denominator as the size of a jump from on a number line from 0 to 1, if we're going to add the jumps together, we need to have the jumps be the same size!

If we're thinking of the denominator as the size of a piece of pizza, the two different denominators means we had two different sizes of pizza slices. We can't simply say we ate two pieces of pizza, because one was bigger and one was smaller.

When we're adding fractions, we want to add up the numerators and to do that we need the denominators to be the same. And we can do that by multiplying by clever forms of the number 1.

With our current problem, we have:



We have denominators of 2 and 3 and we'd like them to be equal. We'll find a common denominator (sometimes abbreviated as CD) and ideally the lowest common denominator (LCD). Which can get a bit confusing because this is the same as the lowest common multiple (LCM) of the denominators.

For 2 and 3, we can pretty easily see that the LCD is 6. However, let's run through a prime factorization to see it:

2 = 2 x 1
3 = 3 x 1

For the LCD, we look at each prime number and grab the biggest group of each one. For our question, there is a single 2 and a single 3. We multiply them together to get 6. So we want our denominators to each be 6. We get there by multiplying with clever forms of 1:









and so



Question 3

When we add decimals, we can put the numbers one under the next and line up the decimals (we can fill in any spots needed with 0s). In our case, we have:

0.50
0.25

and now we add down:

0.50
0.25
0.75

Vocabulary used:

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Where might you have come from?

Fact-orials Index

Numbers:

• Rational Numbers

Operations:

• Addition
• Subtraction

Operations with different kinds of numbers:

• Addition, Subtraction, and Integers
• Addition, Subtraction, and Integers - Practice Problems
• Multiplication, Division, and Rational Numbers

Associated Operations:

• Prime Factorization
• Multiples

Where might we go?

Operations with different kinds of numbers:

• Addition, Subtraction, and Rational Numbers - Practice Problems
• Multiplication, Division, and Rational Numbers - Practice Problems

Relations:

• Equality and Rational Numbers