Why do negative numbers behave the way they do in arithmetic?

[goodabouthood]

I am having an issue with why negative numbers work the way they work.

For instance why does 5 - (-2) = 7

Why do two negatives make a positive when you subtract them?

For me that equation reads 5 apples and you take away -2 apples which to me doesn't really make sense because in real life you can't have -2 apples.

It's hard for me to imagine in real life what it means to take away something negative.

 

[symbolipoint]

The reason in plain common English may be expressed, subtracting a negative number is the OPPOSITE of subtracting a positive number.

 

[goodabouthood]

The reason in plain common English may be expressed, subtracting a negative number is the OPPOSITE of subtracting a positive number.

Why?

Any "real life" examples of subtracting negative numbers?

 

[Deveno]

to really understand this, one needs a better definition of number.

most of us learn numbers as standing for the "amount" of something: 5 dollars, 6 oranges, 1/3 of a pie, etc.

in this context, what could possibly be meant by "-6 oranges"? 6 oranges that somehow "aren't there" and would somehow "suck up" 6 oranges added to them? such an idea is at odds with our common experience.

so, if we are to use these so-called negative "numbers", we need a way of making them "make sense" without invalidating our idea of number as "amount". the basic idea is this:

we expand the notion of number to mean 2 things: a number, and an orientation (directionality).

we call the "old numbers" (0 and the positive numbers) numbers with the usual orientation. the newfangled "negative numbers" are still amounts, but with an "opposite" orientation (like moving to the right, instead of to the left, or going down, instead of going up).

so -2 dollars is still "2 dollars" it's just the dollars going from me to you, instead of the dollars going from you to me. we're still talking about "amounts", they can just "change direction".

now that we have some notion of what negative numbers mean, we can look at subtraction in a new way:

a - b is a + (-b).

this makes things like 3 - 5 make sense:

3 is 3 in one direction (say up, for example).
-5 is 5 in the opposite direction (down).

3 steps up, and then 5 steps down is 2 steps down, in all:

3 - 5 = 3 + (-5) = -2, 2 steps down.

so how does this help understand "subtracting a negative"?

by our "new rule" a - b = a + (-b):

a - (-b) = a + (-(-b)).

so, what we really need to know is: what is -(-b)?

a little thought should convince you that: opposite of opposite = originial (this is the principle of double negation: not not = is). for example: not not guilty = not innocent = guilty. or: if you're going up, and you "turn around", you're going down, and if you turn around again, you're going up.

so -(-b) = b. if I'm giving back what you gave back to me when i gave it to you, we might as well have skipped the "middle two" exchanges, it's the same result as if i just gave it to you (once). if you turn off a light switch, after turning it on, after turning it off, it's the same as just turning it off. this is how "one thing" and "opposite thing" works.

so...a - (-b) = a + (-(-b)) = a + b. it's a way of simplifying a complicated way of saying things, by something more direct.

you are correct: -2 apples "makes no sense". but taking back the 2 apples i gave you (-2, from me to you), is the same as as me gaining 2 apples. there's just apples involved, no mysterious "anti-apples". the sign (the little minus thing we put in front) is just a way of keeping track of the direction of the giving.

 

[goodabouthood]

I feel like asking this question is the same as asking why does 1=1 or 2=2.

It seems to me that once you start putting these equations into real life I'm not sure if they do the best job reflecting it.

3-5=-2 does not seem like it could ever be a quantity for something real.

 

[phinds]

3-5=-2 does not seem like it could ever be a quantity for something real.

Depends on how you define things. If you owe me $3 and give me $5, then you now owe me -$2

Financial balance sheets use negative numbers in the real world.

I think your over thinking the whole thing and getting worked up about nothing.

 

[Deveno]

I feel like asking this question is the same as asking why does 1=1 or 2=2.

It seems to me that once you start putting these equations into real life I'm not sure if they do the best job reflecting it.

3-5=-2 does not seem like it could ever be a quantity for something real.

is the money in one's bank account, or the bills one has to pay, not "real"?

imagine you lived in a world that had only one street. are the distances to other buildings from your home, not "real"? certainly it makes sense to speak of going east, or west. the choice of whether east or west is "positive" is, to be honest, an arbitrary one, but once decided (say east is postive), 5 miles east is +5 miles from home, 5 miles west is -5 miles from home. the "miles" aren't negative, the direction is.

there are LOTS of places where negative numbers are used to represent "real things". a negative velocity upwards, is the speed at which one is falling. a negative balance on your bank account, means you owe them (the bank) money. many things can have "opposite" directions: growth/shrinkage, up/down, left/right, profit/loss, acceleration/deceleration, giving/receiving, outward/inward.

in other words, numbers can be more than just an amount. we can give numbers a meaning of amount AND direction. -2 apples means: 2 lost apples. 2 apples means 2 gained apples.we rarely think of "quantities" as just things living in isolation, like 4 sheep in some perfect world of "sheepness" never doing anything. our world has ACTION, things move, they change. the direction of the change matters.

the proper thing to ask, when confronted with negative numbers is not: what is -x things? the proper question to ask is: what kind of opposition are we describing by a pair +/-?

 

[Mensanator]

I am having an issue with why negative numbers work the way they work.

For instance why does 5 - (-2) = 7

Why do two negatives make a positive when you subtract them?

For me that equation reads 5 apples and you take away -2 apples which to me doesn't really make sense because in real life you can't have -2 apples.

It's hard for me to imagine in real life what it means to take away something negative.

Stand on a N/S sidewalk. Call North the positive direction. Spin about
and face South (the negative direction). Now take 2 steps backwards.
That's negative motion in the negative direction, yet you end up
farther North of your starting point. Negative motion in the
negative direction yields positive displacement.

 

[goodabouthood]

I know I was over thinking this and it seems like really basic grade school math but I do appreciate these answers.

For me I have resolved my issue with addition and subtraction in the following way:

I feel as addition is moving in the same direction as the number that follows.

For instance if you have 5 + 2 you are moving 2 in the same direction from 5. Since both numbers point in the same direction you get 7.

Now if you have 5 - 2 you are moving in the direction opposite of 2. If you were to move in the opposite direction of 2 you would get 3.

Now if you have 5 - (-2) you are moving in the opposite direction of -2 and you get 7.

So addition means to move in the same direction as the next number and subtraction means to move in the opposite direction as the next number.

I know this makes things difficult and there are most likely much clearer ways to talk about addition and subtraction but this thought I laid out makes logical sense to me even if it is still a bit confusing.

 

[Dielectric]

Dear Deveno,

Thank you for your intelligible response to the OP. Your explanation has been one of the best ones that I have read. Even several math books do not properly convey the idea of numbers [negative numbers for that matter] as being conceptual. Instead most books just dictate the order of operations and such.

Anyway, your explanation has helped me to better convey this abstract notion of negative numbers with my daughter's question as to when and why one would actually use subtraction of a negative in the "real world."

 

[OldBeginning]

we expand the notion of number to mean 2 things: a number, and an orientation (directionality).

If negative numbers really are about directions, why do we call them "scalars" [e.g.: multiplying a vector by -1 is considered "multiplication of a vector by a scalar"]? Also, if they are actually vectors, why call them "numbers"?

 

[Millennial]

The negative of a number a is the solution of the equation a+x=0. Now, since subtraction is defined as a-b=a+(-b), we also need to find the inverse of x. This inverse should satisfy x+y=0. But, we know that the solution for y must be a! Hence, a's inverse's inverse is a itself. So, we obtain that the inverse of the inverse of 2 is once again 2. Applying this to your question, we get (since -(-a) is the inverse of the inverse of a:)

5-(-2)=5+(-(-2))=5+2=7

Note: "inverse" here was used as a shorthand for "additive inverse".

 

[pwsnafu]

If negative numbers really are about directions, why do we call them "scalars" [e.g.: multiplying a vector by -1 is considered "multiplication of a vector by a scalar"]?

In the vector space $\mathbb{R}^1$, the real numbers form a vector space over itself. That -1 is both a vector and a scalar.

Also, if they are actually vectors, why call them "numbers"?

There is no mathematical definition of the word "number".

 

[chiro]

There is no mathematical definition of the word "number".

That's a bit of a copout. Numbers are just a collection of objects that we give context to when we define things like ordering and algebra. They just capture variation in that the cardinality is greater than 1 element.

 

[SteveL27]

If negative numbers really are about directions, why do we call them "scalars" [e.g.: multiplying a vector by -1 is considered "multiplication of a vector by a scalar"]? Also, if they are actually vectors, why call them "numbers"?

Simply a matter of pedagogy. If we called real numbers a 1-dimensional vector space over themselves, it would be incomprehensible to students. It would be like telling third graders that the counting numbers 0, 1, 2, 3, ... are an inductive set whose existence is given by the axiom of infinity. It's true, but educationally premature.