3. Multiplying with Negative Numbers
- Why is the product of two numbers with the negative sign positive?
- To begin with, please look at the Figure 1.
The war record of this year of thier top team of Team Dragons is not good. There are few spectators of the game which exists once per week by the cause, and the deficit has come out by a unit of 5,000 dollars for every game. The present management situation of the club team in yesterday's game is 12,000 dollars in the red.
Figure 1
Dragon "Since it has lost in the top team recently. At last, the management of Dragons became a deficit on the other day."
Snake "Everybody should watch the game, in order to cheer it up more. I surely watch it weekly."
Jason "But our top team is weak. And it is boring if it loses a game."
Selica "By the way, till when was Dragons in the black?"
Novy "Selica, this answer is easy for me. It is why... since the team is 12,000 dollers in the red now, so it is 7,000 one month ago, it is 2,000 two months ago... then it is three months ago."
Selica "Yes. Although I understand it. However, if it is so, how is it calculated? It's {-120 + (-50) · (-3) = -120 + (-150) = -270}. Look! It is wrong."
Dragon "Selica, The first expression is correct. But it's {-120 + (-50) · (-3) = -120 + 150 = +30}."
Selica "What? Is it {(-50) · (-3) = +150}? Is a negative number times a negative number equals positive number? But, probably the answer is +30 and correct, isn't it?"
Dragon "Selica, It's a good question. Why is the product of two numbers with the negative sign positive number? Everybody, let's consider the reason together."
- The reason of {(-1) (-1) = +1}
- Why is the product of two numbers with the negative sign positive number?
Before considering the reason, Let's review a multiplication with unlike signs like {(+A) · (-B)} and {(-A) · (+B)}.
{(-A) · (+B)} is the same solution as {(+A) · (+B)}.
First of all, please consider by yourself.
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Did you remember? The foundation of multiplication are repetitions of addition.
{(+A) · (+B)} is
A + A + ... + A + A
Isn't it?
That is, {(-A) · (+B)} becomes
(-A) + (-A) + ... + (-A) + (-A)
= (-A) · (+B).
Next, let's consider A times -B.
Since this formula cannot be considered by addition like previously, A times -B is changed into the form of -A times B.
A solution is following.
A · -B = C.
Then,
(A · -B) / A = C / A
- B = C / A
- B · A = (C / A) · A
- B · A = C
After all, it becomes,
A · -B = -B · A.
That is, A times -B and -B times A equal -(A · B).
Then, consider - A times -B at last.
If it is {-1 · -1 = +1}, it turns out simply that it is {-A · -B = +(A · B)}.
Proof #1
I prove from the expression {-1 · (2 - 1) = -1}.
Since -1 times (2 - 1) is -1 times 1, it becomes -1, doesn't it?
-1 · (2 - 1) = -1
-1 · 2 + (-1 · -1) = -1
-2 + (-1 · -1) = -1
-1 · -1 = -1 + 2
-1 · -1 = +1
What a surprise! It became {-1 · -1 = +1}.
Proof #2
Next, I will solve for the difference of the two equal negative values.
(-1) - (-1) = 0
(-1) + (-1) · (-1) = 0
-1 · -1 = +1
Since they are the difference of the equal values, right-hand side of an equality is infallible zero, isn't it?
As I thought, it is {-1 · -1 = +1}.
Why is it {-1 · -1 = +1}? There is no kind of trick whatever behind anything I do. In familiar to us, there are few phenomena of multiplying with negative numbers, more overwhelmingly than the other ones. First of all, since even a sign called a negative could not be found a long time ago, after the sign of a negative was invented, we came to think a thing called the multiplication of negative numbers. So if you imagine a few phenomena of multiplying with negative numbers a lot, your sense of incongruity will fade.
One-Point Advice (1) About A.D. 876, they said that the sign of zero was used in India. Moreover, about 1150, similarly in India, they said that Bhaskara explained a negative in a meaning called a debt. They said that it is beginning that Leonardo who was Italian (Pisa) told that method (a system of numerical notation) and Arabian mathematics to Europe.
- Imagine multiplying with negative numbers
- Let's imagine many phenomena which are multiplying with negative numbers.
Please prepare a sheet of paper and a pen. And let's consider why it becomes so, drawing a figure.
Image #1
Let's solve for the area of the rectangle which is {x - 2} height and {x - 3} width.
(x - 2) · (x - 3)
= x - 2x - 3x + (-2 · -3)
If the expression is only {x - 2x - 3x}, it subtracts too much only the area which is overlapped -2x and -3x. Therefore, it is possible that we consider this expression is compensating the part of {(-2 · -3) = +6} from the area subtracted too much.
Image #2
Let's draw a number line and consider {A - B}. By using a number line, we can imagine negatives easily.
{A - B} is to slove for the difference between A and B. A difference expresses the distance on a number line. Please calculate the difference which is substituted various numbers for A and B, and consider why it becomes so.
Image #3
Let's consider profit and debt.
Let's compare and imagine the following expressions. For example, if it is , then you imagine like that "I got 3 times of profit of 200 dollars!".
POSITIVE · POSITIVE = I got profit = PROFIT
NEGATIVE · POSITIVE = I got debt = LOSS
POSITIVE · NEGATIVE = I lost profit = LOSS
NEGATIVE · NEGATIVE = I lost debt = PROFIT
Image #4
Suppose you are watching a video of soccer, a certain player is running 8m/s and you suspend it. And if you fast-forward or rewind a tape, what does the player become? Suppose the stopped position is set to zero (a base), the attack direction is as POSITIVE and the defense direction is as NEGATIVE based on that, let's imagine that where is the player and how far is it from the base position. For example, if it is , you imagine like that. The player is in attack, and if I fast-forward a tape for 2 seconds, he will be the position where is 16m rather than at the moment at the attack direction.
POSITIVE · POSITIVE = in attack · f-forward = ATTACK DIR.
NEGATIVE · POSITIVE = in defense · f-forward = DEFENSE DIR.
POSITIVE · NEGATIVE = in attack · rewind = DEFENSE DIR.
NEGATIVE · NEGATIVE = in defense · rewind = ATTACK DIR.
* f-forward(fast-forwarding), DIR.(DIRECTION)
- The role of negatives
- Now, let's consider a little more abstractly about negatives.
First of all, what is the role of negatives in a number?
Please consider negatives on the number line which can be seen visually.
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Do you understand anything?
Then, let's look at the Figure 2.
It is possible that negatives shows an opposite direction on the base of zero to the number of original.
Figure 2
Please compare the following expression.
1 = 1
1 · (-1) = (-1) = -1
1 · (-1) · (-1) = (-1) = +1
1 · (-1) · (-1) · (-1) = (-1) = -1
1 · (-1) · (-1) · (-1) · (-1) = (-1) = +1
Furthermore, the following thing can be said if it is {n > 0}.
(-1) = -1 (The nagative sign is odd pieces)
(-1) = +1 (The nagative sign is even pieces)
You fear nothing just because much negatives came out. What is necessary is just to know that {(-1) · (-1) = +1} is important.
Let's play the game and consider the above-mentioned by amount-of-money calculation of coin.
Here is two coins of 10 cents and throw them. If the surface comes out then you get +10 cents. But the back, you got -10 cents. Now, when you threw the coins, both of them showed on the back. How much is the amount of money?
on the back is the surface after all.
Let me write the above to expression,
-10 · (-1) · 2
= -10 · (-1) · 2
= +20
At most, this is only things. There is no troublesome calculation.
The important thing is you can recognize what is positive and what is negative.
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Selica "I see. The situation of three weeks ago, that case is 5,000 dollars in the red weekly, is {-50 · (-3) = +150}. So it can be thought that it was 15,000 dollars profits rather than the present, wasn't it?"
Dragon "Exactly! Since the amount is 12,000 dollars in the red at present, {-120 + 150 = +30}, so it was 3,000 dollars in the black three weeks ago."
Selica "Right."
Dragon "For us, although the multiplication of negatives is very unfamiliar, it comes out in large numbers in mathematical class. Then, in order to supplement this gap, it is necessary to imagine in search of the multiplication of negatives."
Novy "But, it is because the ancient people invented negatives in its origin. Oh my goodness! I wish negatives had never been invented."
Snake "Although my mathematics' record is always , but Novy always gets . HAHAHA!"
Novy "Boohoo. He was hard on me again. Coach Dragoooooooooon!"
©2004 EYG System