# Help with explaining a maths concept

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My 10yr old has been applying the rule for years now, but she really questions rules and wants (needs?) to truly understand it and not just accept it.

One rule in question is that subtracting a negative number is the same as adding.

Here are the two ways I've tried to explain it:

- imagine you have a debt (so that's negative) and you want to remove the debt (take away that negative amount). You have to 'give' ie add

- if you have anything and you take it away, you'll have zero. You have 4 and take 4 away, you'll have zero. You have -5 and take that -5 away, you'll have zero.

So -5 - (-5) = 0

Now, this is a rule we all learn pretty early on and just roll with it and apply it.

But I need to explain it better.

Any ideas to help me??

We'll save why multiplying two negatives makes a positive for another day, eh?  ;)

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That's how I teach it.  We have negative and positive counters that came with our Hands On Equations set that show the taking away easily.   Actually, physically, taking the piece (i.e. subtracting) helped my oldest understand.

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Can you make a number line that she can stand on and physically move to add? Then talk about how subtraction means "turn around" and a negative sign before a number means "move backwards." So if she's at 4, she's facing positive infinity, and she needs to subtract three, she's going to turn around (now facing negative infinity) and then move three, taking her to 1. But if she's at 4, facing positive infinity, and needs to subtract negative three, she's going to need to turn around (now facing negative infinity) and then move backwards three, taking her to 7. (It doesn't actually matter which one is "turn around" and which one is "move backwards," which is an idea she can explore. Understanding subtraction as adding the opposite is a very useful thing.)

I hope that was clear. It'd be easier with pictures, lol! If you don't have the space to let her physically walk along a number line, you could do the same thing using a number line on paper and a token, as long as the token has a clear front and back.

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Can you make a number line that she can stand on and physically move to add? Then talk about how subtraction means "turn around" and a negative sign before a number means "move backwards." So if she's at 4, she's facing positive infinity, and she needs to subtract three, she's going to turn around (now facing negative infinity) and then move three, taking her to 1. But if she's at 4, facing positive infinity, and needs to subtract negative three, she's going to need to turn around (now facing negative infinity) and then move backwards three, taking her to 7. (It doesn't actually matter which one is "turn around" and which one is "move backwards," which is an idea she can explore. Understanding subtraction as adding the opposite is a very useful thing.)

I hope that was clear. It'd be easier with pictures, lol! If you don't have the space to let her physically walk along a number line, you could do the same thing using a number line on paper and a token, as long as the token has a clear front and back.

This is the right approach.  Trying to "take away" is an inaccurate concept - there's nothing to "take" in a continuum.

Subtraction is simply a shorthand for addition.  The left operand is the starting point, and the right operand is the delta to be applied.  Even more technically, the distance from the origin (0) to the second term is transposed along the number line until the "origin" end of the segment matches the left operand.  The other end of the transposed segment is the result of the operation.  For simplicity, you can just "walk" to the left when adding negative numbers, or to the right when adding positive numbers.

The concept is actually more significant than it appears.  When moving to other number systems, the rules may need to be redefined - such as is the case with complex numbers.  Those are two-dimensional, and though addition is relatively similar, multiplication is very different.

ETA: The REAL answer, btw: you are applying the additive inverse.  That's all that is really going on.  The transposed segment idea is a visual explanation of how a number added to its additive inverse leads you back to the additive identity (0).

Basis for this: the real numbers are closed and commutative under addition.  First, there is a number 0 such that for any real number a, a + 0 = 0 + a = a (thus, "0" is an additive identity).  For any number a, there also exists another number b such that a + b = b + a = 0.  b is the additive inverse of a.  It just so happens that for real numbers, b = -a.  Thus, a + (-a) = 0.

Edited by Mike in SA
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Ds asked me this just yesterday.  I told him it would be the opposite of the opposite (of the number).  He thought that was funny.

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https://proofwiki.org/wiki/Definition:Abelian_Group_Axioms

Subtraction is defined by
x-y:=x+(-y) for every x,y

And the axioms imply
-(-z)=z for every z

So it must be that
a-(-b)=a+(-(-b))=a+b for every a,b

"Subtracting the negative of a number is the same as adding that number."

Note that "negative numbers" (numbers x satisfying x<0) are not really relevant here because an Additive Abelian Group need not even be equipped with an inequality "<".

Instead, what is being used is the negative -b of a number b.

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https://proofwiki.org/wiki/Definition:Abelian_Group_Axioms

I was trying to avoid the group theory, but yeah, there it is.  ;)

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I used this hot air balloon game, which has a number line on it with my kids when we were exploring this....https://nrich.maths.org/9941

There is a video to teach you how to do it, and then a download file to print out the game.

I felt that they intuitively understood why after playing it for awhile.

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How about  a real world example, where negative numbers are used to measure real things?

For my kids, the tides (we live on the ocean). If the tide is dropping from +20 feet to -4 feet, then it's quite intuitive that the difference in water levels will be 24 feet, not 16. And difference is subtraction, so 20 - -4 = 24. Temperature could do the same. Say it was +10 degrees during the day then dropped to -5 overnight. How much did it cool down? I think these are less abstract than debt, and make the answer more obvious.

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I used the two-sides counters (red/yellow in our case). The yellow ones stood for positive integers and the red stood for negative. Pour some out and count the positives and negatives, and show how they cancel each other out, and figure out the net total.

Then subtract a negative number - and see that by removing the negative tokens, your net total is increasing.

Then out only yellow tokens down - say you have seven. To take away -3, you would have to first *have* -3. If you put down three reds along with three new yellows, your net total is still 7 (because those cancel each other out) but now you have three reds to take away. Your new net total is ten.

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Thanks folks for all the ideas.

I had to read a couple of those posts several times to absorb them. I'm talking to you Mike and epi!   Crikey, I'm guessing you guys are mathematicians?

I asked my 19yr old who is doing a double degree in maths and IT - she just said 'well, two negatives make a positive. Like in English, it's a double negative.'  Me: But, why?

Following all your ideas, I immediately did purple owl's numberline one, with an emphasis on direction. I think this helped.

Mike, I think this that you said is particularly helpful: The left operand is the starting point, and the right operand is the delta to be applied.

So, we start at the first number. The next sign tells us which direction to face (either toward positive infinity or negative infinity) and then the sign in front of the second number tells us whether we go forward or backward in that direction. Correct? Oh, I hope so. Otherwise, I'll need to re-read it all again.

The thing is that my 10yr old wants to think deeper than application. She gets all in knots over numbers being so abstract. My husband will say 'five fingers, that's not abstract'. Yes, but that's five fingers, like 5x or 5 apples. It's 5 of something. But my 10yr old asks, but what is 5, just 5? How to answer that? It's a change, a delta? Which means different things in different contexts?

I'm also going to explore all the other ideas - thank you all heaps for your help!

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Thanks folks for all the ideas.

I had to read a couple of those posts several times to absorb them. I'm talking to you Mike and epi!   Crikey, I'm guessing you guys are mathematicians?

We are not alone, either!

I'm going to try a slightly different approach - see what you think of this one...  It's wordy, but maybe more fun.  :)

Mathematics is an amazing human invention.  It is a purely logical model of the world around us.  Counting with natural numbers seems to be instinctive (1, 2, 3, ...), though not all people count in this way.  There are some Aboriginal tribes who have numbers only for 1, 2, and "more than two."  The PirahÃ£ people do not count at all.  As it turns out, counting is a learned behavior.

The natural numbers are just one of our countable sets.  Integers were still not universally accepted by western mathematicians even as late as the 18th century; indeed, negative numbers were considered a "false" invention in some circles, much as complex numbers were labeled "imaginary" as a form of ridicule.  The irony, of course, being that every number exists only in the imagination of the person using it to describe their world.  The "real" numbers which we are referring to in this thread were only coined in the 18th century.

Algebra, a wonderful 16th century invention (derived from Arabic studies), created a set of rules by which we could study the nature of the number systems we imagined.  This field has introduced classifications for these systems: sets (e.g., numbers), operators (e.g., addition), groups (a set + an operator), rings (a group + an additional operator + distributive operation), and so on.  It's not important to understand all of these for the moment, but the group and the operator bear discussion.

An operator - despite all the arithmetic we have had drilled into our heads - is nothing more than a mapping function.  It takes two operands from within the set, and returns a value.  Hopefully, that value is also defined in the set, but not all groups have this characteristic.  Real numbers and addition DO, but real numbers and the square root DO NOT (if you take the square root of a negative number, the result is not a real number).  If the returned value is always in the set, then the group is considered "closed" under the operation.  Further, if the operation can be conducted in any order (the operands can be interchanged without affecting the result), then the operation is "commutative."  That's the end of the vocabulary lesson for the time being...

At this point, a pause is in order.  It's interesting to consider what it means to have an operation.  Just to get the point across of how it is an invention, let's make one up.  My new operation will be called "flooby."  It takes two real numbers, and returns the second operand less the product of the operands.  So, 2 flooby 3 = 3 - (2*3) = 3 - 6 = -3.  7 flooby 1 = 1-7 = -6.

In my example, "flooby" is not commutative.  2 flooby 5 is not the same as 5 flooby 2.  However, the real numbers are still closed under "flooby," and flooby always returns a single value, so flooby is a well-defined operation.  It's now just as real as addition or multiplication.  Feel free to use it!

"Addition" has been defined as "starting from the first operand, take a walk in the direction of the sign, with the length of the walk equaling the magnitude of the number."  So, perhaps surprisingly, "magnitude" is a more fundamental operation than addition.  You can't add without it.  Still, with very little effort, you can see that addition is commutative and well-defined, and the set of real numbers is closed under addition.

Back to vocabulary...  Before we can define "subtraction," we need to define the "identity" and the "inverse."  The identity is the number which, when applied to any other number, returns the other number.  For real numbers, the additive identity is "0."  For any number "a," a+0 = a.  The "inverse" is defined for each specific number.  It is another number which, when applied to the first number, returns the identity.  For real numbers, the inverse of "a" is always "-a": a + (-a) = 0.  Under multiplication, the identity is 1, and the inverse is, well, usually, 1/a (note that 1/0 is undefined, so the real numbers are NOT closed under division!).

For "flooby," the problem is tricky, and doesn't always work (uh oh).  Let's see.

For real numbers to have a floobian identity, then there must be a number b such that a flooby b = a.  If we find it, then b is the floobian identity.

a flooby b = b - ab

if b is the identity, then a flooby b = a

therefore, b - ab = a

b(1-a) = a

b = a/(1-a)

That's ok as long as a <> 1.  Not good enough.  Kind of like our issue with division.

Now, it gets easy.  Back to real numbers and subtraction...

"Subtraction" may be defined as "adding the first operand to the inverse of the second operand."  For example, "3 - 2" is the same as "3 + (-2)".  Subtraction is NOT commutative.  When you substitute the definition for the shorthand, though, you do get back to addition, which is commutative.

So, subtraction isn't really addition, but it may be rewritten in a form which is addition.  The nice thing about that is it allows you to convert to a commutative form when problems get tough.

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Mike, you're floobin' fabulous.

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There are some Aboriginal tribes who have numbers only for 1, 2, and "more than two."  The PirahÃ£ people do not count at all.  As it turns out, counting is a learned behavior.

Wow! I never knew this! I shared it with my son and he is now off to research more about who does not a number system at all.

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Wow! I never knew this! I shared it with my son and he is now off to research more about who does not a number system at all.

We saw an amazing documentary last year called The Grammar of Happiness. Excellent show.

My memory is that the South American people in it had a language with no numbers or minimal numbers. It may even be about the Piraha people that Mike mentioned.

I want to watch it again now.

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Ds asked me this just yesterday.  I told him it would be the opposite of the opposite (of the number).  He thought that was funny.

MATH IS LOUD! :)

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The opposite of the opposite of the opposite of easy is loud! :rolleyes:

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The piles and holes and opposites formulation here is basically the same as how James Tanton explains it.

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The key idea as stated before is that X-Y is the same as X + (-Y).  In fact this is a definition of subtraction.
Then A-B = C means that C+B = A.  so A - (-B) = C means that C + (-B) = C-B =  A, which means that C = A+B.  I.e. A - (-B) = A +B.

But I myself like informal explanations such as your example that taking away a debt means adding funds.  But the above sequence is the logical derivation.

Edited by mathwonk
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or to reprise epi's explanation, -X is defined as the thing that "cancels" X, i.e. such that X + (-X) = 0.  But if -X is the thing that cancels X, then -(-X) is the thing that cancels -X, namely X.   I.e. since by definition (-X) + (-(-X)) =0, adding X to both sides, gives (-(-X)) = X.

Edited by mathwonk
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You mathemagicians are making my brain feel squidgy  :blink:

PS All help is much, much appreciated and we've definitely made progress. I tried a few of these ideas on my daughter and walking the number line seemed most helpful. Aim in this direction (the operation) and then change your position by this magnitude (the second operand, including its sign). That kind of sums it up nicely.

Thanks heaps everyone. Please know you're fabulous.

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