Is signed division intuitive?

4/2 = 2 makes sense because there are two groups of 2 in Four.
4/-2 does not make sense to me.  I want to understand it intuitively.  How should I think of this problem?  How many groups of negative two are there in four?  

Thanks.
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naseeamAsked:
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ozoCommented:
Are any of these steps non-intuitive?
-2 * -2
= -1 * 2 * -1 * 2
= -1 * -1 * 2 * 2
= -1*-1 * 4
= -(1*-1) * 4
= -(-1) * 4
= (0 - -1) * 4
= 1 * 4
= 4

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Guy Hengel [angelIII / a3]Billing EngineerCommented:
> How many groups of negative two are there in four?  
yes, exactly.
you have to take away two times (-2) from 0 the value -2  to get as result 4.



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naseeamAuthor Commented:
Sorry, I don't understand the response.  Are you saying something like,
0 - -2 = 2,    2 - -2 = 4  .  What does this have to do with how many groups of negative two
in Four?

Thanks.
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Guy Hengel [angelIII / a3]Billing EngineerCommented:
>What does this have to do with how many groups of negative two
how many means  n times x
that means the same as
+ x + x + x + ... + x  , n times

now setting x => -2
-2 + -2 + -2  ....  + -2   = 4  , searching for n will be impossible, as the + goes into the wrong direction
so we have to use - , which then corresponds to
-n times x

-(-2) - (-2) = 4, so we have to subtract (take away) 2 groups of -2 to get to 4
or also
- ( -2 + -2 ) = 4, so n = 2, but has to be taken negatively.


In practice, you don't care the sign to find the absolute number, but only divide 4 by 2, and then apply the rules about the signs:

-  x - => +
- x + => -
= + x + => +

this also applies for the division:

- / - => +
- / + => -
+ / - => -
+/ + => +
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ozoCommented:
Is signed multiplication intuitive to you?
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moorhouselondonCommented:
4/(-2) is of course equivalent to -4/2.

Draw an X axis on a piece of paper.  Now draw a Y axis in the middle of the X axis you have drawn.  At the highest X point put 4, and at the lowest point put -4.  Now divide the positive X axis into 2 equal portions, and label this mid-point 2.  Now do the same on the negative side of the axis, with the mid-point labelled -2, you have divided -4 into 2 portions.  
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ozoCommented:
Is the concept of a negative number intuitive to you?
http://mathforum.org/library/drmath/view/51925.html
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ozoCommented:
Can you intuitively understand subtracting a negative number?
Can you intuitively understand multiplying by negative one?
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moorhouselondonCommented:
4/(-2) is also equal to

4 times (-1/2)

which, if you take my example, you start off from the X=0 origin and move left 0.5 a total of four times you will get to -2.
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moorhouselondonCommented:
There is nothing magical about negative numbers - they appear on all my bank statements...
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BigRatCommented:
>> Is signed division intuitive?

No, it isn't.

moorhouselondon : 4* (-1/2)

Ingenious, making the divisor into a fraction! But it doesn't work intuitively if the divisor is say pi.

One needs quite a lot of axiomatic set theory just to get to signed numbers and I doubt whether axiomatic set theory and the steps from it are "intuitive".



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ozoCommented:
Why wouldn't it work with 4*(-1/pi)?
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BigRatCommented:
I didn't say that it wouldn't work, I said it does not *work intuitively". After all what is meant by 1/pi when one understands 1/something to be a fraction like a half, a third, a quarter etc. Division by transcendentals has only real meaning in the continuum, it has no meaning in the world of portions, which is where the questioner started off from.

The question which you might ask, ozo, is, given the concept of Wheeler-Space-Time, how "intuitive" is the concept of the continuum? I sometimes feel that the e-neighbourhood concept is something mathematically constructed which is not to be executed algorithmically!
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mohananuCommented:
The concept of negative is an explanatory concept and not intuitive. The man invented zero otherwise there is no zero in the physical world. Here things change from one form to the other, you do not get a zero. We assume certain things, for example the water freezes at zero degree c. So any temperature below it is negative. And then we came up with absolute zero, a mystical number of -273 deg c called zero degree calvin or K.

Similarly the depth of oceans may be represented as negative from the ground level but from the center of the earth it is a positive measurement only.

The negative numbering is a representative system, created by man for his convenience. The physical world does not represent negatives, it can not give you more than what it holds. You sure can go red in your bank accounts but not when you are drawing water from a well, not when you are depleting the natural resources like oil and coal, nature can not go in red and give you more than what it holds. The bank also gives you the money from its own positive kitty, the moment that kitty finishes, it may borrow from other banks. But one day when everything else fails, the bank fails. There sure are numerous examples.

The division 4/-2 = -2 is more of logical consequence than intuitive one. If you just keep the negative out or
 -(4/2) = -(2) naturally 4 divided by 2 is 2 and + divided by - is -. The end result is -2.

This represtative nature of minus is very much evident in the signed and unsigned integers. The total number of these integers whether signed or unsigned is the same, however the values may differ.
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BigRatCommented:
>>nature can not go in red and give you more than what it holds.

Quantum mechanically of course it can, otherwise you would not be able to post here :)
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mohananuCommented:
Quantum mechanics is not intuitively understood. You probably studied hard and lot of explanation from your harried professors before calling it intuitive or not even today.

You sure can understand negative numbers, try explaining to your kid of five years. If he understands or you can explain it to him by the examples of the physical world, then that is intuitive.
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SunBowCommented:
> How should I think of this problem?

not sure

> How many groups of negative two are there in four?  

two

angelIII > yes, exactly.
ozo > Is signed multiplication intuitive to you?

:-))
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SunBowCommented:
> How should I think of this problem?

How about skimping the plus minus, think in terms of opposites.

Two people, four apples, two steps.

You have four apples. You give me two. Then you give me another two.

From my perspective, I end up with four, but it took me two steps to get them, two at a time.

From your perspective (opposite) you end up with four less apples, reducing quantity two at a time twice.

Any better?
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mohananuCommented:
You are left with zero apples. If you do not have any apples and you are expecting 10 apples, you give out as a future four apples but ultimately do not recieve the 10, now you are minus four apples. You need to fulfill your obligation. i believe it may be so.
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SunBowCommented:
> How many groups of negative two are there in four?

Two (steps)

> 4/2 = 2 makes sense
> 4/-2 does not make sense to me.

Change the apple example by doing one step.

Four total apples. Think of the division as taking one-half instead of numeric division.

You have the four apples, give me half of them. I have gained from the 4 (total) a times one-half (1/2) so now I have two.

From your pile, where you had four (+4) you have reduced it by one half (-1/2) so you end up with two less (-2 result).

> > How many groups of negative two are there in four?

Your reduction of those two apples can be done two times (two groups)
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SunBowCommented:
mohananu > ultimately do not recieve the 10
BigRat > of course it can, otherwise you would not be able to post

hmm.I can in one more step repeating an apple example?

You have ten apples. I have none. You feel generous, count out two, hand them to me, then I have two, you have eight. Imbalance. You count out two more and hand them to me, I now have five, and you have six.

Looking down at the piles, from one to the other, you are not sure of intuition or division. But you are a believer. Besides, you only wanted one apple for yourself anyway.

You feel generous, count out two, hand them to me, then I have seven, you have six. Imbalance. You count out two more and hand them to me, I now have ten, and you have two.

You raise your eyebrow, eat one of your apples and hand me you last one which I eat. Together we take the ten remaining apples to town to give to the poor as promised.

-------------------

Originally your ten were picked from the tree near me. While you were looking down and counting in the second half of each step of transfer, I reached up and picked one more when you were not looking.
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moorhouselondonCommented:
Ahhhhhh, now if only the apples referred to were hardware manufacturers and music producers and they could have thought of things in the same way that sunbow does, there would be less bruised apples and a few poorer lawyers in the world today.
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SunBowCommented:
And instead of whole numbers and phony divisions, we could split a nice bowl of applesauce. Anyone got some cinnamon handy? (sprinkle it in, and then tell me how we're dividing up the spice)

and               thanx,              good fortune
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