Addition, Subtraction, and Integers

Background

We've covered how to add and subtract positive numbers. With the inclusion of negative numbers, we now have to learn how to add and subtract them.

Question

Evaluate the following:

• 3 + 4
• 3 + 2
• 3 - 4
• 3 - 2
• 3 + (-4)
• 3 + (-2)
• 3 - (-4)
• 3 - (-2)
• -3 + 4
• -3 + 2
• -3 - 4
• -3 - 2
• -3 + (-4) 
• -3 + (-2)
• -3 - (-4)
• -3 - (-2) 

Answer

• 3 + 4 = 7
• 3 + 2 = 5
• 3 - 4 = -1
• 3 - 2 = 1
• 3 + (-4) = -1
• 3 + (-2) = 1
• 3 - (-4) = 7
• 3 - (-2) = 5
• -3 + 4 = 1
• -3 + 2 = -1
• -3 - 4 = -7
• -3 - 2 = -5
• -3 + (-4) = -7
• -3 + (-2) = -5
• -3 - (-4) = 1
• -3 - (-2) = -1 

Analysis

One of the trickiest concepts students run into when learning about integers is how to add and subtract (especially since practice problems will force students to make convoluted calculations). I hope this section makes adding and subtracting integers easier.

To briefly recap, integers are whole numbers that sit on either side of the number 0. We have positive integers (2, 6, 103, etc) and negative integers (-2, -6, -103, etc). So let's add and subtract some and talk about what's going on.

To make things a bit more tangible, I'm going to use a temperature analogy for this. So imagine that you are in a room and we start off at a certain number of degrees (Fahrenheit, Celsius, or whatever units you want to use) and let's say that Hot is Positive and Cold is Negative.

We start from our initial point on the number line (that first number in the equation) and then we either add or subtract heat or cold. Let's think about this for a second - if I add heat (add a positive number), the temperature of the room will go up.

If I take away heat (subtract a positive number), the temperature of the room will go down.

Ok - now what happens if we add cold (add a negative number)? The temperature of the room will go down.

And now for the hardest one of all - if I take away cold (subtract a negative number), the temperature of the room will go up.

We can now work our questions:

Positive integer plus a positive integer (it's hot and we add more heat)

3 + 4 = 7
3 + 2 = 5



Positive integer minus a positive integer (it's hot and we take away heat)

3 - 4 = -1
3 - 2 = 1



Positive integer plus a negative integer (it's hot and we add cold)

3 + (-4) = -1
3 + (-2) = 1



Positive integer minus a negative integer (it's hot and we take away cold)

3 - (-4) = 7
3 - (-2) = 5



And we can start with a negative number and it'll work the same way:

Negative integer plus a positive integer (it's cold and we add heat)

-3 + 4 = 1
-3 + 2 = -1



Negative integer minus a positive integer (it's cold and we take away heat)

-3 - 4 = -7
-3 - 2 = -5



Negative integer plus a negative integer (it's cold and we add more cold)

-3 + (-4) = -7
-3 + (-2) = -5



Negative integer minus a negative integer (it's cold and we take away cold)

-3 - (-4) = 1
-3 - (-2) = -1



You might be thinking at this point that there is no difference between 3 - 4 = -1 and 3 + (-4) = -1, that there's no difference between subtraction and adding a negative number... and you'd be absolutely right! There is no difference! And the same goes for adding a positive number and subtracting a negative number, 3 + 4 = 7 and 3 - (-4) = 7 - they're exactly the same as well!

They are treated exactly the same. So why do we have to learn this? Because the way formulas and other expressions get written, sometimes things will look like 3 - 4 and other times they'll look like 3 + (-4), and it's important that you be able to recognize them.

Vocabulary used:

• Integers - the set of numbers that start at 0 and increase by 1 (for positive integers) and decrease by 1 (for negative integers), so (...-3, -2. -1, 0, 1, 2, 3,...)
• Whole Numbers - the set of numbers that start at 0 and increase by 1 (0, 1, 2, 3,...)

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Numbers:

• Integers
• Addition, Subtraction, and Whole Numbers

Where might we go?

Operations with different kinds of numbers:

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