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Laws of arithmetic math

Many computational laws of mathematics are presented here. We show you this:

  • A Explanation what laws of calculation there are and how to apply them.
  • Lots Examples as an introduction to this topic.
  • tasks / Exercises so that you can practice this yourself.
  • Videos to the various laws of calculation.
  • A Question and answer area to this topic.

If you are already looking for a certain law of arithmetic, you can also look directly at the law or the rule: commutative law, distrubutive law, associative law as well as dot before line, bracketed calculation and calculation sequence.

Explanation of the laws of calculation

Let us start with the three arithmetic laws, the commutative law, the distributive law and the associative law.

Commutative law:

The commutative law (also known as the law of exchange) says that it doesn't matter in which order you add or multiply two numbers. Examples of this:

  • 5 + 3 = 8
  • 3 + 5 = 8
  • In general: a + b = b + a
  • 3 · 5 = 15
  • 5 · 3 = 15
  • In general: a · b = b · a

This law does not apply to subtraction or division. You can find more about this arithmetic law under commutative law.

Associative law:

The associative law - also called the law of connection - extends from two to three numbers. It says that it doesn't matter in which order three numbers are added or multiplied. Here are some examples too:

  • (2 + 4) + 6 = 12
  • 2 + (4 + 6) = 12
  • 2 + 4 + 6 = 12
  • (2 · 4) · 6 = 48
  • 2 · (4 · 6) = 48
  • 2 · 4 · 6 = 48

Generally:

  • (a + b) + c = a + (b + c)
  • (a + b) + c = a + b + c
  • (a b) c = a (b c)
  • (a * b) * c = a * b * c

If you want to learn more about this arithmetic law, please take a look at the associative law.

Distributive law:

We still have the distributive law, which is also called the distribution law. This helps to multiply brackets or to form brackets. Let's take a look at some examples (explanation below):

  • a * (b + c) = a * b + a * c
  • 2 · (3 + 4) = 2 · 3 + 2 · 4
  • 14 = 14
  • (a + b) c = a c + b c
  • (2 + 3) · 4 = 2 · 4 + 3 · 4
  • 20 = 20

You can multiply brackets by multiplying the number (or variable) in front of the bracket by everything inside the bracket. More about this law under distributive law.

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Examples of laws of calculation

In addition to the three mathematical laws above, there are also rules of mathematics for the order of the calculations. In elementary school you learn the law of arithmetic from left to right. In other words, we have additions (plus arithmetic) and subtractions (minus arithmetic). It looks like this:

Or:

  • 5 - 2 + 3 - 1
  • = 3 + 3 - 1
  • = 6 - 1
  • = 5

If multiplications or divisions appear during the calculation, then these are calculated first. This is called point before line.

Another example:

  • 6 + 12 : 2 - 5
  • = 6 + 6 - 5
  • = 12 - 5
  • = 7

More about this calculation rule under point before line.

Bracketed calculation:

What happens if an addition or subtraction should be calculated first? In this case you can put a bracket around what should be calculated first. First the bracket is calculated, then point before line applies.

Example:

  • 3 · 5 + (2 + 4)
  • = 3 · 5 + 6
  • = 15 + 6
  • = 21

Another example:

  • 6 : 2 · (8 - 4)
  • = 6 : 2 · 4
  • = 3 · 4
  • = 12

You can find more about calculating with brackets under the bracket calculation.

Videos Laws of Calculation

Would you like to watch individual content as a video? Then follow these links:

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Questions with answers about the laws of calculation

This section deals with typical questions with answers about the laws of calculation.

Q: When are the laws on arithmetic in school covered?

A: Arithmetic from left to right is already covered in 1st grade or 2nd grade. The rule point before line then mostly follows in the 3rd grade or 4th grade. The calculation in brackets follows, i.e. either in the 4th grade or at the latest in the 5th grade. The arithmetic laws of the commutative law, the distributive law and the associative law follow either at the end of the 4th grade in elementary school or at the latest in the 5th grade.