# Why do the laws of mathematics exist

### Laws of arithmetic for adding

You already know 2 calculation rules that always apply:

- calculate from left to right
- Calculate brackets first

There are also 2 special laws for adding:

- the law of commutation or commutation
- the law of connection or associative law

These 2 arithmetic laws can help you with arithmetic.

### Interchangeability law

Investigate what happens when you flip the numbers in an addition problem.

**Example:**

$$46 + 78 = 124$$

$$78 + 46 = 124$$

So $$ 46 + 78 $$ is the same as $$ 78 + 46 $$.

The law of commutation or commutation states:

When adding, you can swap the summands. The result remains the same.

$$83 + 92 = 92 + 83$$

Or in general:

$$ a + b = b + a $$

$$ a $$ and $$ b $$ are arbitrary numbers.

The technical terms at a glance:

Summand $$ + $$ Summand $$ = $$ sum

Minuend $$ - $$ Subtrahend $$ = $$ difference

### Be careful with the subtraction

Examine the interchanging in subtraction.

**Example:**

$$100-50+45 = 95$$

$$100-45+50 = 105$$

So $ 100-50 + 45 $ **Not** the same as $$ 100-45 + 50 $$.

Mathematically: $$ 100-50 + 45! = 100-45 + 50 $$

When subtracting, you can minuend and subtrahend **Not** swap. The swapping gives different results.

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

### Mental arithmetic with interchangeability law

Sometimes you can calculate more easily if you swap the summands.

See if adding round numbers comes out. You add these numbers up first. **Example 1:** $$+$$$$87+$$

$$=$$ $$+$$$$+$$$$87$$

└──┬──┘

$$+$$$$87$$

$$=$$$$177$$

If you first swap $$ 34 $$ and $$ 56 $$ here, it will make the calculation easier for you. **Example 2:** $$+$$ $$+$$ $$+$$

$$=$$ $$+$$ $$+$$ $$+$$

└──┬──┘ └───┬───┘

$$+$$

$$=300$$

This does not work for all tasks. But always first see if you can calculate skillfully.

### Connection Act

The law of connection has to do with brackets. Try what happens in an addition problem when you put parentheses.

Task: $$ 54 + 23 + 77 $$

1st possibility: from left to right

$$54+23+77$$

└──┬──┘

$$=$$$$77$$$$+$$$$77$$

$$=$$$$154$$

2nd possibility: put brackets

$$54+(23+77)$$

└──┬──┘

$$=$$$$54+$$$$100$$

$$=$$$$154$$

3rd possibility: put brackets somewhere else

$$(54+23)+77$$

└──┬──┘

$$=$$$$77$$$$+$$$$77$$

$$=$$$$154$$

You may or may not use parentheses. It always comes out $$ 154 $$.

The connecting or associative law states:

When adding, you can put or leave out brackets as you like. The result remains the same.

$$26 + 73 + 37 = ( 26 + 73) + 37$$

$$26 + 73 + 37 = 26 + ( 73 + 37 )$$

Or in general:

$$ a + b + c = (a + b) + c = a + (b + c) $$

$$ a $$, $$ b $$ and $$ c $$ are arbitrary numbers.

It is up to you how you can calculate faster, more confidently and better.

### Be careful with the subtraction

Investigate the use of brackets in subtraction.

**Example:**

$$123-73-27$$

└──┬──┘

$$=$$$$50$$$$-$$$$27$$

$$=$$$$23$$

$$123-(73-27)$$

└──┬──┘

$$=$$$$123-$$$$46$$

$$=$$$$77$$

So $$ 123-73-27 $$ is not the same as $$ 123- (73-27) $$.

Mathematically: $$ 123-73-27! = 123- (73-27) $$.

When subtracting, you can **Not** put any brackets. The use of brackets leads to different results.

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

### Mental arithmetic with connection law

Sometimes you can do math more easily if you put brackets or think in your head.

Look at which summands result in round numbers. You add them up first. That is, you put brackets.

**Example 1:**

$$34+12+18$$

$$=$$$$34+(12+18)$$

└──┬──┘

$$=$$$$34+$$$$30$$

$$=$$$$64$$

If you add $$ 12 $$ and $$ 18 $$ here first, you get the round number $$ 30 $$.

**Example 2:**

$$52+58+18+123+77$$

$$=$$$$(52+58)+18+(123+77)$$

└──┬──┘└───┬───┘

$$=$$$$110$$$$+$$ $$18$$$$+$$$$200$$

$$=$$$$328$$

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