Are there any brainiacs out there who can 'splain to me (in English!) why a negative number divided by a negative number ='s a positive number?

Asked by Val123 (12651) March 25th, 2010

THIS IS NOT A HOMEWORK QUESTION!! .... Well, actually, it kind of is. I’m relearning algebra (which I aced way back when) and I’m my own teacher. :O

I learn better when I understand the reasoning behind things, and I don’t understand how -50 / -2 = 25 and not -25. It just seems counter intuitive…

I am now a teacher in an adult high school degree completion program, not to be confused with a GED program. The students go through all of the actual Junior and Senior High School curriculum, from Math 800 to Calculus, from Health to World History, so I’m one busy Flutherer re-learning all of this stuff, and teaching myself. Can’t teach it ‘lessn you know it! I LOVE MY JOB!!!!

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It has to do with the identity.

So:
-50 / -2 = (50 * -1) / (2 * -1)
(50 * -1) / (2 * -1) = (50/2) * (-1/-1)
:. 25 * 1 = 25.

That last bit about the -1 / -1 = 1 is this:

For any real number x:
x / x = 1

So 50 / 50 = 1
12.382379827398723872397823 / 12.382379827398723872397823 = 1
-1 / -1 = 1

That help?

grumpyfish (6618)

I hate how the average educator explains math. The average person is not supposed to be a calculator. Think of it like this:

You got -3. How many -1’s goes into -3? Don’t think of the negative numbers as things that you’re holding. There’s more energy involved in the equation than what’s represented by the numbers alone. It’s in the division, where the positive power comes from.

It’s not the actual math that’s counter-intuitive, it’s how these robot-brained freaks teach that’s counter-intuitive. No offense to you robot-brained freaks.

zophu (5686)

(raises hand)

@grumpyfish
Teacher, while you’re at it, can you also explain to me why x ^ 0 = 1 for every possible value of x and his mum?

Fyrius (14525)

When you see -50 / -2.
It’s actually asking “how many times does -2 go into -50” and the anwser is 25 times.

Silence04 (4668)

@silence04

that’s the best explanation

zophu (5686)

All of these years and Silence finally got through to me the answer to this vexing question…thanks!

TheLoneMonk (2892)

While @zophu and @silence04 give good layman answers, they then open up the question of why -50/2 equals -25?

I think @grumpyfish‘s proof handles that case, but doesn’t really explain why either.

Anyone able to provide a complete layman’s answer?

robmandu (21242)

why -50/2 equals -25

Works the same.

How many -25s go into -50?

( If you asked me yesterday I would not have had a short answer, @zophu made it plain for me)

jaytkay (25763)

@Silence04 Perfect!

@Fyrius Haven’t a clue! =)

grumpyfish (6618)

What do these English sentences mean?

1) “She is a great woman and an expert of lawn mowers and not a resident of Germany.”
2) “She is not a nonsmoker.”

We can’t simplify the first sentence. In terms of math (logic) we are looking at an addition, something like a + b + (-c) or a + b – c

The second one means “She is a smoker.” In terms of math (logic) we are looking at a multiplication, something like – (-s) = s, so you see minus times minus is plus.

Now a division is almost like a multiplication, for example m/n is the same as m times 1/n. Handling negatives is easy too and the rule is -m is the same as -1 times m. Which also means that -m/-n is the same as -1 times m divided by -1 times n.

When 2 numbers as numerator and denominator are the same they cancel each other out. Simply strike both -1. Makes sense?

mattbrowne (31537)

The original topic seems to be well in hand, so I’ll answer @Fyrius’ question.

Any exponent (x^n) means that we’re multiplying x with itself n number of times.

This is best illustrated with an example:

2^4 = 2 * 2 *2 *2 = 16

So let’s look at a simple sequence:

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
etc.

What are some defining characteristics of that sequence?

Well, on the left side of each equation, the power goes up by 1. On the right side, for each successive term, we’re multiplying by 2.

What if we go in reverse order?

That means to go from 2^5 = 32 to 2^4 = 16, we’re dividing the right side by two. So let’s take our sequence in reverse direction.

2^5 = 32
2^4 = 16
2^3 = 8
2^2 = 4
2^1 = 2
2^0 = 1

And you can keep going and see that the same principle holds for exponentiating by a negative number.

This is by no means proof, but you wanted a layman’s explanation. I hope it helps. :)

elenuial (1403)

@elenuial
Well, I’ll be buggered.
That makes perfect sense. Thank you.

Fyrius (14525)

@elenuial That’s actually how I remember it, but I never remember the proof of it.

Ah, it’s really simple, and explained here: http://en.wikipedia.org/wiki/Exponent#Exponents_one_and_zero

grumpyfish (6618)

@zophu Thanks for the explanation, but I’m teaching myself via an online tutorial service. The same one the kids take.

Wow you guys. Thanks! (@mattbrowne…..I’m not an expert on lawn mowers! But thanks anyway!)

Val123 (12651)