# Associative Property Calculator

By Maciej Kowalski, PhD candidate
Last updated: Apr 12, 2021

Welcome to Omni's associative property calculator, where we'll come to understand, befriend, and eventually love the associative property of addition and multiplication. In essence, it's an arithmetic rule that lets us choose which part of a long formula we do first. We often combine the associative property in math with the commutative and distributive one to make our lives so much easier in the process. Don't worry: we will explain it all slowly, in detail, and provide some nice associative property examples in the end.

But what does the associative property mean exactly?

## Associative property definition – what is associative property?

What is this associative property all about? Informally, it says that when you have some long expression, you can do the calculations in the back before those in the front. Formally (i.e., symbolically), it's as follows.

 💡 The associative property of addition says that: `(a + b) + c = a + (b + c)`.Analogously, the associative property of multiplication states that: `(a * b) * c = a * (b * c)`.

So what does the associative property mean? If you have a series of additions or multiplications, you can either start with the first ones and go one by one in the usual sense, or, alternatively, begin with those further down the line, and only then take care of the front ones.

Also, observe how we said, "a series of additions or multiplications" while the associative property definition only mentions three numbers. That is because we can extend the whole reasoning to as many terms as we like, as long as we keep to one arithmetic operation. For instance, the associative property of addition for five numbers allows quite a few choices for the order:

`a + b + c + d + e = (a + b) + (c + d) + e` `= a + (b + c) + (d + e)` `= (a + b) + c + (d + e)` `= a + ((b + c) + (d + e))` `= ...`

Of course, we can write similar formulas for the associative property of multiplication.

## When can we use the associative property in math?

The above definition is one thing, and translating it into practice is another. Below, we've prepared a list for you with all the important information about the associative property in math.

1. The rule applies only to addition and multiplication. In other words, subtraction and division are not associative.
2. The associative property applies to all real (or even complex) expressions. To be precise, the symbols in the definition above can refer to integers (positive or negative), fractions, decimals, square roots, or even functions.
3. If you change subtraction into addition, you can then use the associative property. Symbolically, this means that changing `a - b - c` into `a + (-b) + (-c)` allows you to apply the associative property of addition.
4. Similarly, if you change division into multiplication, you can use the rule. Again, symbolically, this translates to writing `a / b` as `a * (1/b)` so that the associative property of multiplication applies.
5. The associative property appears in many areas of mathematics. It applies to other, more complicated operations done not only on numbers but objects such as vectors or matrices. In some sense, it describes well-structured spaces, and weird things happen when it fails. Fortunately, we don't have to care too much about it: the associative properties of addition and multiplication are all we need for now (and most probably the rest of your life)!

Alright, that seems like enough formulas for today, don't you think? Let's now use the knowledge and go through a few associative property examples!

## Associative property of addition and multiplication: examples

In total, we give four associative property examples below divided into two groups: two on the associative property of addition and two on the associative property of multiplication. In each pair, the first is a straightforward case using the formula from the above section (also used by the associative property calculator), while the second is a longer, trickier expression.

• `13 + (7 + 19) = (13 + 7) + 19 = 20 + 19 = 39`

Note how easier it got to obtain the result: `13` and `7` sum up to a nice, round `20`. From there, it's fairly simple to add the remaining `19` and get the answer.

• `3 - 1.2 + 7.5 + 11.7 = 3 + (-1.2) + 7.5 + 11.7`

`= 3 + ((-1.2) + 7.5) + 11.7`

`= 3 + (6.3 + 11.7)`

`= 3 + 18`

`= 21`

Observe how we began by changing subtraction into addition so that we can use the associative property. Furthermore, we applied it so that the pesky decimals vanish, and all we had left were integers.

The associative property of multiplication:

• `(4 * (-2)) * 5 = 4 * ((-2) * 5) = 4 * (-10) = -40`

Note how we were careful to keep the sign in `-2` when swapping brackets. Moreover, just like with addition above, we managed to make our lives easier: we got a nice `-10` which is simple to multiply by.

• `(-4) * 0.9 * 2 * 15 = (-4) * 0.9 * (2 * 15)`

`= (-4) * (0.9 * 30)`

`= (-4) * 27`

`= -108`

As before, we used the associated property in such a way to kill the decimal dot almost effortlessly. From there, it was a walk in the park.

Hmm, let's see if we covered everything.

• What is the associative property in math? ✓
• Possible extensions ✓
• Associative property examples ✓
• Using the associative property calculator ✘

Oh, it seems like we have one last thing to do!

## Using the associative property calculator

For simplicity, let's have the instructions neatly in a numbered list.

1. At the top of our tool, choose the operation you're interested in: addition or multiplication.
2. Once you select the correct option, the associative property calculator will show a symbolic expression of the corresponding rule with `a`, `b`, and `c` (the symbols used underneath).
3. Input your three numbers under `a`, `b`, and `c` according to the formula.
4. The moment you give the third value, the associative property calculator will spit out the answer below.
5. Enjoy the calculator, the result, and the knowledge you acquired here.

That's all for today, folks. Remember that the associative property in math is just one of the few basic rules in arithmetic, so make sure to check out other Omni tools in this category!

## FAQ

### What is the difference between associative property and commutative property?

The associative property says that you can calculate any two adjoining expressions, while the commutative property states that you can move the expressions as you please.

For instance, by associativity, you have `(a + b) + c = a + (b + c)`, so instead of adding `b` to `a` and then `c` to the result, you can add `c` to `b` first, and only then add `a` to the result.

On the other hand, commutativity states that `a + b + c = a + c + b`, so instead of adding `b` to `a` and then `c` to the result, you can add `c` to `a` first and, lastly, `a` to all that. Note how associativity didn't allow this order.

### Are associative properties true for all integers?

Yes. Even better: they're true for all real numbers, so fractions, decimals, square roots, etc. However, you need to be careful with negative numbers since they cannot be separated from their sign by, for example, a bracket.

### How do I use associative property?

To use the associative property, you need to:

1. Make sure you have only addition or only multiplication.
2. Specify which two adjoining numbers you want to do first.
3. Calculate the two numbers' operation (don't change the rest).
4. Repeat as many times as you need.
5. Enjoy finding the answer in a clever way.

### Does the associative property work for subtraction?

No. However, subtracting a number is the same as adding the opposite of that number, i.e., `a - b = a + (-b)`. And since the associative property works for negative numbers as well, you can use it after the change. For instance, we have:

`a - b - c = a + (-b) + (-c) = (a + (-b)) + (-c) = a + ((-b) + (-c))`.

### Can you use the associative property with subtraction and division?

No. However, you can use a little trick: change subtraction into adding the opposite of the number and change division into multiplying by the inverse. In other words, we can always write `a - b = a + (-b)` and `a / b = a * (1/b)`. From there, you can use the associative property with `-b` and `1/b` instead of `b`, respectively.

Maciej Kowalski, PhD candidate
Use the associative property of
a + (b + c) = (a + b) + c
Input
a
b
c
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