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1. What are negative numbers?


THAT IS, numbers with the opposite enlarged direction (180 degrees) of positive numbers.
Here, I want to use a number line.
I will talk another explain about a number line in 11. What is a number line?
Please look it.
(I will make that article.)

On 0 (zero) of a number line, we set a mirror.
The mirror world of positive numbers is the negative number's world.

If you depart from 0, a distance represented by positive numbers from 0
will become big in the positive number world.
But in the negative number world,
farther and farther you depart from 0 (zero),
the distance from 0 (zero) is become big,
but the number itself means smaller amount of number.
Negative numbers shows the opposite meanings of positive numbers
and added - (minus) makes number's direction change (180 degrees).

We have images that additions make larger answers than an added number.
But this situations happen only when we add a positive number and a positive number.

If we add negative number to some number,
the sum is smaller than an added number.
It is different from situations when both numbers are positive.
Let's think additions in a figure.

negative number added

When we add 2 to 3, a point on 3 moves in the "right" direction by two.
Then the answer is 5.
When we add (-2) to 3, a point on 3 moves in the "right" direction by two,
because negative (-, we also call it minus) means the opposite direction of positive numbers.
So, the number of the answer become small by 2,
then the answer is 1.

In the end, we subtract a positive number in that situation
(negative numbers addition situation).
We did opposite meaning operations on negative numbers calculations.

Let's think about subtract situations of negative numbers.
If we subtract a positive number from some number,
a point will move in the "left" direction.
3-2 is 1.
But if we subtract a negative number from some number,
a point will move in the "right" direction,
because the direction of movement changes to the opposite direction.
So, 3-(-2) turns 5, because a point on 3 moves in the "right" direction by 2.
Then, the number of the answer becomes big.

A project of
settling into the opposite direction world
enlarged by logic
is one of aims of our studies of negative numbers

In the future, in math, we encounter many worlds enlarged by logic.
The first project in math studies, is negative numbers,
which we meet first in junior high school studies.
During we use negative numbers,
our brain will get comfortable to them.
Then we will feel real feelings in that world in our brains, I think.
(At first, we have better concentrate in doing correct operations in logic.)

And let's think what we will do,
when we add to or subtract from a negative number.
(If I write this answer, I will write too much, I think.
I do not want to rob your thinking abilities and your willing for thinking even for just a bit.
So I will not write answers.
Let's solve them by yourself.)


Next is about multiplication.
When we multiply a negative number to some number,
that number's (enlarged) direction become opposite.
A positive number turns to a negative number,
and a negative number turns to a positive number.
Let's think also about multiplication in a figure.
What happens, when we multiply (-2) to 1?

negative number multiplied

1 is a number, which moves a distance 1 from 0 (zero) in the right direction.
As our tries,
when we multiply a positive number, 2,
a multiplied number become twice bigger.
So the answer of 1x2 is 2,
and it keeps positive condition.
When we multiply a negative number, (-2),
a multiplied number becomes twice bigger in the opposite direction. So the answer of 1x(-2) becomes (-2),
which is made twice larger in the opposite direction of 1 (a positive number),
that is, the answer departs in the negative direction.

Then, let's multiply a negative number to a negative number.
What happens when we multiply (-2) to (-1)?
(I will not write this answer, either.
Let's think it by yourself.)

Let's think about division situations.
I recommend you to think about situations of 2/(-2) and (-2)/(-2), for example.
I give you a hint that divisions by positive number make divided number's direction opposite
and divided number's distance from 0 (zero) changes like divided by a positive number.


If I worte all of how to think,
I give you targets for your straight memorizing.
So I do not write all.
I strongly recommend you to think the rest of discussions of this article by yourself,
or you will not get your thinking powers.

Let's try other examples by yourself.





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