Basic Arithmetic: Rules & Concepts

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  • 0:00 What Is Arithmetic?
  • 0:24 Addition
  • 1:43 Subtraction
  • 2:16 Multiplication
  • 3:09 Division
  • 4:31 Lesson Summary
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Lesson Transcript
Shalonda Whitmore
Expert Contributor
Robert Ferdinand

Robert Ferdinand has taught university-level mathematics, statistics and computer science from freshmen to senior level. Robert has a PhD in Applied Mathematics.

This lesson provides basic rules and concepts of arithmetic, the area of math that involves addition, subtraction, multiplication, and division. This lesson shares details about each operation and how they relate to one another in arithmetic.

What Is Arithmetic?

As fancy as the word sounds, arithmetic actually describes one of the most basic areas of mathematics. When you think about doing math, what usually pops into your head first? It's probably some form of adding or subtracting, maybe even division and multiplication. Well, guess what? All four of those mathematical functions make up arithmetic!


When performing addition, you take at least two numbers and add them together. You started with 1 apple and then you added 1 more? Oh, now you have 2 apples. You're standing alone and 3 friends join you? Oh, now there are 4 of you. By simply putting two numbers together, you are performing arithmetic.

There are certain rules, or properties, that go along with addition. The first is the commutative property. Basically, this tells us that 2 + 1 is the same as 1 + 2. The order you follow when adding numbers doesn't matter. Either way, the answer is 3.

If you add a third number, the grouping of the numbers to be added does not matter. In other words, (6 + 1) + 2 = (2 + 6) + 1 = (1 + 2) + 6 and so on. That is the associative property. You can think of it this way: it doesn't matter which pair(s) of numbers are associated with each other, the sum remains the same.

Another important property is the identity property. If you add 0 to any number, the number will remain the same. If someone gives you nothing, you're still you, right? It works the same with numbers:

  • 6 + 0 = 6
  • 3 + 0 = 3

As you can imagine, addition is a pretty big part of arithmetic.


Let's say you want to try subtraction instead, taking items away from a group. This is the inverse, or opposite, of addition, and it's a part of arithmetic, too. You see 2 people walk away from your group of 4? There are now 2 people left. Lost 1 of those 2 apples? Now, you're left with 1, and you've just completed more arithmetic.

It doesn't stop there, though. As previously mentioned, arithmetic also includes multiplication and division, two operations that many people find at least a little more difficult than addition and subtraction.


In multiplication, you're basically counting groups instead of individual numbers. What exactly does that mean? Let's say you need to figure out 3 multiplied by 5, or 3 times 5. This is like having 3 groups with 5 people in each group. If you counted these people one by one, you would find that you have 15 total. Multiplication allows us to count differently, adding the number 5 to itself 3 times, as follows:

5 + 5 + 5 = 15

Looking at it this way, you see that 3 fives gives you 15. Interestingly, 5 times 3 is the same as 3 times 5, like this:

3 + 3 + 3 + 3 + 3 = 15

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Additional Activities

Practice Questions

1. Using the numbers 0, 2, 7 and 9, illustrate the identity property, the commutative property and the associative property of addition.

2. Is subtraction commutative? Please explain why or why not.

3. Is division associative? Explain why or why not.


1. We start with the identity property. Using any of the numbers 2, 7 or 9 (say we pick 7), to illustrate the identity property as:

0 + 7 = 7 + 0 = 7

Choosing any two numbers from the given numbers (we pick 2 and 9), the commutative property of addition is illustrated as follows:

2 + 9 = 9 + 2 = 11

To describe the associative property of addition, we pick three numbers 2, 7 and 9 to get:

2 + (7 + 9) = (2 + 7) + 9 = 18

2. No subtraction is NOT commutative. As a counterexample, we pick two numbers 12 and 15.


12 - 15 = -3


15 - 12 = 3

Since 12 - 15 does NOT equal 15 - 12, we have evidence that subtraction is NOT commutative.

3. Division is NOT associative.

To show a counterexample, we pick three numbers 100, 50, 5.


100/(50/5) = 100/10 = 10


(100/50)/5 = 2/5 = 0.4

Since 10 does NOT equal 0.4, we have evidence that division is NOT associative.

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