The Laws of Algebra: Understanding the Commutative and Associative Laws

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The Basic Laws

Although many people find algebra intimidating, in truth it really is not that complex. Once you understand the fundamental laws, along with the concept of a variable, everything else follows. Here we’ll discuss those lows and how to use them.

Over time, mathematicians have discovered that numbers always behave in a certain way when exposed to the basic arithmetic operations. Understand these behaviors, and you understand the nature of algebra itself.

The Commutative Law of Addition

The commutative law of addition states that the sum of two numbers will always be the same, whatever order they’re written in. For example, 3 + 5 is the same as 5 + 3. This makes intuitive sense; if I have one collection of items and another collection of items, I have the same number of items altogether regardless of which collection I look at first. Alternatively, if Bob owes me $5 and Jim owes me $7, then if they both pay me what they owe me I now have $12, regardless of who paid me first.

The Associative Law of Addition

The associative law of addition is very similar to the commutative law; it says that the sum of three or more numbers is the same regardless of how they are grouped. That is, (2 + 3) + 5 is the same as (2 + 3) + 5. If I have three piles of socks, and I combine two of the piles into a larger pile before adding in the rest, I have the same number of socks no matter which piles I combine first.

As a result of this law and the previous one, we can add up numbers in whatever order is the most convenient. For example, suppose I have a long list of single-digit integers to add up. I can put them in order before I start adding them up, then use multiplication. If my addition problem is 2+3+4+5+4+3+2+3+4+5, I can reorder this to be 2+2+3+3+3+4+4+4+5+5 - two each of 2 and 5, three each of 3 and 4, and I know the total is 4+9+12+10=35.

The Commutative Law of Multiplication

The commutative law of multiplication states that the product (which is what we get when we multiply) of two numbers is always the same, regardless of what order we multiply them in. In other words, 2 x 5 is the same as 5 x 2.

This can make mental arithmetic easier. If you need to think of the answer to 7 x 6, but you remember that 6 x 7 is 42, then you already have the answer to your problem. Getting paid $6 seven times is essentially the same thing as getting paid $7 six times; either way, you end up with $42. Knowing the laws of algebra gives us one less thing to remember!

The Associative Law of Multiplication

Like the similar law for addition, the associative law of multiplication says that the product of three or more numbers is the same regardless of what order we put them in. That is, 2 x (3 x 4) is the same as (2 x 3) x 4.

Combining this law with the previous one means that if we have a long multiplication problem, we can group the numbers however we like to make solving the problem easier. (However, note that this is only true if the problem contains only multiplication!) For example, if we want to find the answer to 2 x 2 x 4 x 5 x 5 x 5, we can rearrange and group it as (2 x 5) x (2 x 5) x (4 x 5) = 10 x 10 x 20 = 2000.

The Distributive Law

The distributive law says that if you multiply one number by the sum of two or more other numbers, the result is the same as if you had multiplied the first number by each of the other numbers individually.

For example, 5 x (2 + 3 + 5) = 5 x 10 = 50; this is the same as 5 x 2 + 5 x 3 + 5 x 5 = 10 + 15 + 25 = 50.

Using the Laws

Understanding the laws of algebra allows us to solve math problems more easily. If you have trouble remembering which is which, think of a group of people choosing to associate with with other; since a group is more than two, the associative laws apply when more than two numbers are involved. Remember that the commutative and associative laws apply only when all operations within the group being rearranged are of the same type, either multiplication or addition; if you try to apply both to the same problem, watch out for your order of operations!

Here’s a sample problem. Can you rearrange the following to solve it more easily?

5 x 2 + 5 x 3 + 4 x 4 + 5 x 5 + 4 x 5 + 4 x 6 =

5 x (2 + 3 + 5) + 4 x (4 + 5 + 6) =

5 x (5 + 5) + 4 x (4 + 6 + 5) =

5 x 10 + 4 x (10 + 5) =

50 + 4 x 15 = 50 + 60 = 110

This post is part of the series: Mathematical Fundamentals

What are the basic rules that you need to know to make sense of arithmetic and algebra?

  1. Math Guide: Understanding the Order of Operations
  2. Learning the Laws of Algebra
  3. Ratios: Definition & Sample Problems