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### Less than zero

In mathematics you can calculate with numbers that are **less than zero**.

##### Explanation

They are called negative numbers. We don't know exactly what that is. Only for positive numbers we can imagine what they mean. Mathematicians don't care about it. But note: strange things can happen. We agree to write negative numbers with a minus sign (–), so

means that there are 2 things less than zero, whatever that maybe. We never write +3 if we mean three real things, but that is allowed. We start making an addition

Here we use parentheses, because otherwise you get confused. That went well, that is why we now continue with subtraction

Do we still know what we are doing? We have 6 things and subtract 4 things that are not there from it and find 10. So basically we have added 4. Let's look carefully at this, and write everything in detail

Now we add negative numbers

and once more we subtract

The parenthesis do not really help, so we omit them

When you have a sequence of – and + it gets negative. And the reverse, first + and then – works the same. If you have – and – it becomes positive. That is quite strange. In a multiplication it is not different, so

now we multiply two negative numbers, where we will use parentheses for clarity

and the result is positive, as the same rules apply. By a divisiom it is the same, so

Because we understand it we make an extra calculation

and find that it is not so difficult at all. So we continue with exponentiation, because that must also be in accordance with the rules, so

and here the parentheses really are impotant. You can not just do anything with negative numbers. You will notice when extracting a root that

because you can't just solve it. For this, you have to work with complex numbers. To be precise

You must always apply the rules, and those are

+ + = +

+ – = –

– + = –

– – = +