# B Geometric meaning of signs in multiplications

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1. Aug 26, 2018

### hatem240600

What does 1/-1 (one divided by minus one) mean?
What does -1 X -1 (minus one multiplied by minus one) mean?
What are the best graphic representations of multiplication and division?

2. Aug 26, 2018

### Staff: Mentor

Welcome to PF!

What is the context of these questions?

Are you taking a course in Algebra?

3. Aug 26, 2018

### DaveC426913

Came across this cute little example, showing concepts of negatives and positives:

The enemy (-) of (*) my enemy (-) is my friend (+).
The friend (+) of (*) my enemy (-) is my enemy (-).
The enemy (-) of (*) my friend (+) is my enemy (-).
The friend (+) of (*) my friend (+) is my friend (+).

4. Aug 26, 2018

### DaveC426913

For starters, division is defined as the inverse of multiplication.

So, 4 x 2 = 8 has the inverse operation of 8 / 2 = 4.
And 4 x -2 = -8 has the inverse: -8 / -2 = 4.

Not the best example, but...

I dove to 60 feet below sea level (-60) in 10 intervals of 6 foot descents (10 x -6 = -60).

How far from my final depth was I when I was two descents away (i.e. 2 less than)?
-6 x -2 = 12
I was 12 feet from my final depth.

Often, debts and assets.

5. Aug 26, 2018

### verty

OP, you just have to learn it. -1*-1 = +1. The minuses cancel. 1 / (-1) = -1 because there is only 1 minus. $${1 \over -1} = {-1 \over 1} = -{1 \over 1} = -1$$

6. Aug 26, 2018

### Staff: Mentor

Actually you should write

$1/-1 = (1/-1)*(-1/-1) = (1*-1)/(-1*-1) = -1/1 = -1$

7. Aug 26, 2018

### Staff: Mentor

The best explanation I found was the following:

$1 \cdot 1 = 1$ is the area of a square with side length $1$. Now think of an orientation of this area. Then the signs lead to

A purely algebraic answer would probably lead a bit far as a few definitions and concepts will be needed.

8. Sep 8, 2018

### Best Pokemon

9. Oct 11, 2018

### hatem240600

Thank you all guys. Sorry for the late response. I really appreciate your help. I am just trying to "put a face" on a concept that seems vague for me.

10. Oct 11, 2018

### hatem240600

Just trying to understand!

11. Oct 11, 2018

### hatem240600

Thank you! These are really good examples!

12. Oct 11, 2018

### hatem240600

Please share. I think I can handle some algebra.

13. Oct 11, 2018

### hatem240600

14. Oct 11, 2018

### Staff: Mentor

$\{\,-1,+1\,\}$ are the units in the ring of integers, which is the basic domain our number system with characteristic zero is built upon. The integers are the basis. In fact, it are the natural numbers and the integers are already the first step, in which we constructed a group from the monoid $(\mathbb{N},+)$ in order to reverse additions. As it turns out, $\mathbb{Z}$ is also a ring, i.e. allows a distributive multiplication. Now the units of any ring form a multiplicative group, in this case of order $2$. There is only one neutral element in a group, which we usually write as $1$ in the multiplicative case ($0$ in the additive case). This requires to have $(-1)\cdot 1 = -1$ from which all other formulas follow:

$(1 / (-1))= 1\cdot (-1)^{-1}=(-1)^{-1}= -1$ for otherwise we would have
$-1\stackrel{(*)}{=}1\cdot (-1)\stackrel{(**)}{=}(-1)^{-1}\cdot (-1)\stackrel{(***)}{=}1$ which cannot be in the case of characteristic zero.
$(*)$ definition of unique $1$
$(**)$ assumption $(-1)^{-1}= 1$
$(***)$ definition of unique inverse

Next we have $(-1)\cdot (-1)\stackrel{(a)}{=}(-1)\cdot (-1)^{-1}\stackrel{(b)}{=}1$
$(a)$ by the previous formula
$(b)$ by the definition of the inverse

The proofs that units of a ring form a multiplicative group, and that a group has a unique neutral element and unique inverse elements depend a bit on how you define a group (laws vs. solvability of equations).

15. Oct 16, 2018

### wahidovic123

When We Multiply:
Example
× two positives make a positive: 3 × 2 = 6

× two negatives make a positive: (−3) × (−2) = 6

× a negative and a positive
make a negative: (−3) × 2 = −6

× a positive and a negative
make a negative: 3 × (−2) = −6
Yes indeed, two negatives make a positive, and we will explain why, with examples!

Signs

"+" is the positive sign, "−" is the negative sign.

When a number has no sign it usually means that it is positive.

Example: 5 is really +5

16. Oct 30, 2018

### Grinkle

I don't think the OP is asking about the mechanics of computing the result of a multiplication.

A multiplication of two positive numbers leads to the direct and intuitively obvious rectangle. One can put the rectangle on its side and immediately see that X * Y is the same as Y * X. I thought this was very cool.

Negative numbers are a bit harder to grasp geometrically. One can talk about holes, I suppose, but a hole needs depth, and a rectangle has no depth. Erase a rectangle out of a larger area, I guess?

17. Oct 30, 2018

### Staff: Mentor

The clue lies in the insight, that volume is an orientated quantity!
This explains $(-1)\cdot (-1) = 1$ up to $v \wedge w = - w \wedge v$.

Strangely this fact isn't questioned in case the volume is a length. Measured to the left differs from measured to the right and nobody asks why. But this observation doesn't change with dimension. That's all.

18. Oct 30, 2018

### symbolipoint

Number Line; the real number line, using zero as a reference point, negative numbers pointing to the left and positive numbers pointing to the right. Start is to learn addition and subtraction on the number line, as done in first grade, and as done again when beginning to learn algebra. From this, Multiplication is reviewed and is also given the reminder that multiplication is a shortened way of representing Addition.

The posts which came before this one were also helpful. Those are discussing the rules.