# 'difference' and subtraction

I have a dumb question. What is the difference between "difference" and "subtraction" or is the same thing. For example, 2 subtract 2 (2-2) is 0 .. But is the difference between 2 and -2 equal to 4?

## Answers and Replies

tiny-tim
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Welcome to PF!

Hi cowah22! Welcome to PF! Yup … difference = subtraction!

Except that difference is always positive (or zero).

So you'd say "the difference between 7 and 5 is 2", and also "the difference between 5 and 7 is 2".

So it can sometimes cause confusion.

(And yes, the difference between 2 and -2, or between -2 and 2, is 4.)

How come? sorry, still noobish, but shouldn't be -2 and 2 = -4?

HallsofIvy
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I have a dumb question. What is the difference between "difference" and "subtraction" or is the same thing. For example, 2 subtract 2 (2-2) is 0 .. But is the difference between 2 and -2 equal to 4?
Yes, but you are comparing two different things. "2 subtract 2 (2-2) is 0" and "the difference between 2 and 2 is 0" (not -2). "2 subtract -2 (2-(-2)= 2+ 2) is 4" and "the difference between 2 and -2 is 4".

How come? sorry, still noobish, but shouldn't be -2 and 2 = -4?
What do you mean by "and" here? Normally "and" is interpreted as a sum: -2+ 2= 0. You seem to be thinking about -2- 2= -4.

Here is on possible distinction between "subtraction" and "difference". The "difference between a and b" is a- b. Is the "difference between b and a" b-a or is it the same as the "difference between a and b"?

It's really a matter of common English rather than mathematics (and so much vaguer) but typically by the "difference between two numbers" we mean the absolute value: the "difference between a and b" is |a- b|. That's what Tiny Tim said.

mathwonk
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a "difference" is the result of a "subtraction". i.e. subtraction is an operation, and a difference is an element of a group.

Seems odd, that: -2 + -2 = -4 (difference is 0?)
and the opposite: +2 - +2 = 0 (difference is 0?)

seems strange..

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arildno
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Seems odd, that: -2 + -2 = -4 (difference is +4?)
No. Here, you ADD the negative number (-2) with itself. The difference between a number and itself is, of course 0.
and the opposite: +2 - +2 = 0 (difference is 0?)

seems strange..
Here, you SUBTRACT the positive number 2 from itself, giving 0 as the result.

NOTE this is an idea that kinda makes sense and is a postulation
that i beleive is easily proven whith the ideas Ive mentioned.
Dont know if its actualy enough proof but its cool to think about
Nice question by the way

I just realized something
very interesting

ok draw your x-axis look
the numbers
2 and 4 the and number inbetween which is 3
the difference should signifie the length of the line betwen 2 and 4 right?
ok what about a line between 3 and 4
4and 4

It is the magnitude of line inbetween what ever integers you select
subtraction is numbers and is not geomtric because you cannot have a negative length

Therefore
Subtraction - Non geometric
difference - geometric

tiny-tim
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… classical v modern geometry …

Hi Marcwhydothe! I get your point, but I think it depends what you mean by "geometric".

I entirely agree that classical geometry, of the ancient Greek sort, wouldn't have been interested in negative distances.

But modern geometry (space-time, for example) is quite used to the coordinate system in general, and vectors in particular.

(A vector, of course, is a length and a direction.)

So I'd be more inclined to write:

difference - classical geometry
subtraction - modern geometry. No. Here, you ADD the negative number (-2) with itself. The difference between a number and itself is, of course 0.

Here, you SUBTRACT the positive number 2 from itself, giving 0 as the result.

Strange.. My calculator told me different.

Daniel Y.
Strange.. My calculator told me different.

Then either you mistold the calculator what you wanted it to do, or you need a new calculator.

Think of the values on a number line. Given 0 is the origin, and you are at a value -2, which is to say two left of the origin 0. Suppose you move two further toward the left, or -2 units, you would obviously end up four units to the left, or (as any value to the left of the origin is called), -4 units.

As for your calculator problem, on every calculator I've used there is a button that allows you to assign a negative value to a number, this is NOT the subtract button.

Well, I entered: 2,+/-, +, 2, +/-, = into the calculator and it returned -4. I tried this on several calculators and the answer was -4 each time. But then I tried it on another calculator, which converted 2,+/-, +,2,+/- into:

-(-(2)+2) which is 0 ... But if you flip this around (swap + and - signs) you get

((-2)-2) which is -4 Strange?

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Daniel Y.
Well, I entered: 2,+/-, +, 2, +/-, = into the calculator and it returned -4. I tried this on several calculators and the answer was -4 each time. But then I tried it on another calculator, which converted 2,+/-, +,2,+/- into:

-(-(2)+2) which is 0 ... But if you flip this around (swap + and - signs) you get

((-2)-2) which is -4 Strange?

How is this strange? PEMDAS. Parenthesis first, which, on your first example is -(0), which is 0.

On your second example two to the left of 0 and two more to the left -2 is -4.

How can a mathematical opposite not be the inverse, when looking at
-2 + -2

+2 - +2
There shouldn't be any multiplication here, right?

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arildno
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How can a mathematical opposite not be the inverse, when looking at
-2 + -2

+2 - +2
There shouldn't be any multiplication here, right?

You need to lay aside crude, fallacious and vague notions like "opposite".

You are doing two entirely different things here:

In the first, you ADD a number to itself.

In the second, you SUBTRACT a number from itself.

Same with multiplication:
-2 * -2 = 4 why shouldn't this be -4 ?

2 * 2 = 4

division: -2 / -2 = 1
how can a positive number come from 2 negative numbers?
2 / 2 = 1

arildno
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Same with multiplication:
-2 * -2 = 4 why shouldn't this be -4 ?

2 * 2 = 4

division: -2 / -2 = 1
how can a positive number come from 2 negative numbers?
2 / 2 = 1
Sigh. What do you mean by "come from"?

Here, I'll show you why (-1)*(-1)=1, by reference to the axioms valid for ordinary arithmetic.

1. (-1)+1=0 This is the basic definition of the "negative" of a number, i.e (-a)+a=0 for every number "a"

2. Since a=b implies c*a=c*b for numbers (expressions) a,b,c, 1. implies:
(-1)*((-1)+1)=(-1)*0

3. Since, for all numbers a,b, c we have a*(b+c)=a*b+a*c, 2. may be rewritten as:
(-1)*(-1)+(-1)*1=(-1)*0

4. Now, given any number "a", we have a*1=a and a*0, thus 3. may be rewritten as:
(-1)*(-1)+(-1)=0

5. Now, since for any numbers/expressions a=b implies a+c=b+c, 4. implies:
(-1)*(-1)+(-1)+1=0+1

6. Now, invoking 1. on the left hand side, and that 0+a=a on the right hand side, we get:
(-1)*(-1)+0=1

7. Noting that adding 0 doesn't change the value of "a", i.e, a+0=a, we finally get:
(-1)*(-1)=1

which was to be proven.

I just meant, if there isn't a positive number in an equation how can the result ever be negative.

If:
4 * 4 = 4 + 4 + 4 + 4 = 16

Why isn't:
-4 * -4 = -4 + -4 + -4 + -4 = -16

I know these seem like stupid questions.. But really, where would something like -4 * -4 = 16 ever occur in nature or physics... Which math is used to explain.

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symbolipoint
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I just meant, if there isn't a positive number in an equation how can the result ever be negative.

If:
4 * 4 = 4 + 4 + 4 + 4 = 16

Why isn't:
-4 * -4 = -4 + -4 + -4 + -4 = -16

I know these seem like stupid questions.. But really, where would something like -4 * -4 = 16 ever occur in nature or physics... Which math is used to explain.

Read and study the information in the link in post #17, and read the proof in post #18; and then you should clearly understand why the product of two negative numbers is a positive number.

arildno
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I just meant, if there isn't a positive number in an equation how can the result ever be negative.

If:
4 * 4 = 4 + 4 + 4 + 4 = 16
Correct.
Why isn't:
-4 * -4 = -4 + -4 + -4 + -4 = -16
Wrong. Here, you add -4 FOUR times with itself, rather than adding it MINUS FOUR times with itself.
YOu should see from this that multiplication naively thought of as repeated addition is simply false. (It exists a non-naive way of having that perspective, but I won't go into that)
I know these seem like stupid questions.. But really, where would something like -4 * -4 = 16 ever occur in nature or physics... Which math is used to explain.

A rather irrelevant issue.
However, here is one example:

Let the direction "to the right" be denoted as positive, "to the left" as negative.
Also, let some time instant be regarded as 0, instants after that has positive values, instants prior to zero has negative values.

Suppose a man walks (runs, actually) TO THE LEFT with the speed 4m/s.
His velocity is then -4m/s (speed is the magnitude of velocity and always positive)

Now, at the instant t=0, the man's position is at the origin.

Now, let us try to answer the question:
"Where was the man 4 seconds ago?"

Well, this is simply found by multiplying together the man's velocity (-4m/s) with the instant asked about (-4).

Thus, we get that the man's position 4 seconds ago was: -4m/s*-4s=16m

That is, 4 seconds ago, he was position 16meters on the right hand side of the origin.