You Gotta Have Rules

This is math, so you are getting used to the idea of rules. Math is about using logic, rules, and organization so that you can get the same answer every time you do a problem. If the rules changed, 1 + 1 would not always equal 2. That would get too confusing.

If you have studied science, you know about laws. There are the law of gravity and the laws of motion in physics. Math also has laws. They set up rules that always allow you to do certain things. You have already used three of those laws in these addition pages. Do you remember how we are able to rearrange numbers or group numbers when we add? We can do that because there are laws that say it is okay and the math will continue to work properly.

Rearrange the Numbers

When you rearrange numbers, you are using the Commutative Law of Addition. The word commutation may seem big, but it just means rearrange. If you want to, you can call it the Rearrangement Law of Addition. The law lets us move all of the addends around in any addition problem.

For example:
1 + 95 + 1,345 + 2 + 15 + 7 = ?
• This one is a little weird to look at.
1 + 2 + 7 + 15 + 95 + 1,345 = ?
• When it is rearranged, it's a little easier to imagine adding the numbers.

You can't just rearrange anything. You still need to pay attention to parentheses and other operations. But for addition, reorder the values all you want. If you were a mathematician, you would use letters called variables to write out the idea. In math, we use letters to represent any number we want. You'll learn more about variables in algebra. For now, know that "a" and "b" can be any number you imagine. The official description of the commutative law is...

a + b = b + a

The order doesn't matter when looking at the addition function.

Grouping the Numbers

Now you know that you can rearrange numbers in addition. You can also group them. You have already been doing that in many of your problems.

For example:
1 + 5 + 9 + 6 + 5 + 4 = ?
• Rearrange the values and group them...
(1+9) + (5+5) + (6+4) = ?
• It's a lot easier to see that the answer is 30 when they are grouped.

Mathematicians saw that grouping was helpful, so they made a law called the Associative Law of Addition. The law looks at the way numbers can associate with each other in addition. You might have a group of two and a group of three. The law lets you break up the groups and move things around. For example...

(1+2+75) + (3+4+25) = (1+2+3+4) + (75+25) = (1+2) + (3+4) + (75+25)

See how we just shifted those parentheses around? We set up new groups of addends. You also need to notice that we can group any number of addends. Whatever makes it easier for you to solve the problem. However, this law only works for adding numbers. You need to pay attention to parentheses and other operations such as subtraction or division.

For example:
(75-1+2) x (3-4+25)
• You can't rearrange this or group them in other ways.
• You need to pay attention to the other symbols in the problem.
• Addition is special. Not every operation in math is associative.

The official way to describe the Associative Law of Addition using variables would look like this...
(a+b) + c = a + (b+c) = a + b + c

And Then There was Zero

You already know about adding zero to a number: 1+0=1. Zero is the value for nothing when it stands alone. You are able to add zero to any number and get the same number in the sum. You can do this because of another law in math. The "adding zero" law is called the Identity Law of Addition. Any number added to zero is equal to itself. It is officially called the Additive Identity.

For example:
1 + 0 = 1
536 + 0 = 536
7,851,498,523 + 0 = 7,851,498,523

Or, as a mathematician might say it with a variable...

a + 0 = 0 + a = a


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