I have very little math education, and I will attempt to explain why. Since the issue is hard for me to understand, I'll write this "stream of consciousness" style, or something similar.

5x=10 I have a ton of issues with this when I'm trying to learn basic math; I cannot continue past this. When taught, it's explained that you need to divide 5x and the other side of the EQUATION by 5. Why? Khan Academy does not explain it, I'm expected to follow this "rule" mindlessly, and I don't like doing that.

Next, I'm supposed to visualize ten objects for the right side of the equation and five xs. The ten objects are divided into groups of two for a total of five groups of two. This is somehow equal to a distinct, separate group of objects "X" which contrast like this


How is that equal? That is not equal. Two things are not similar to me; I don't believe that two things can be equal to each other. I don't think this visualization works? Can someone prove it for me? I feel that if there is a distinction to be made if you can somehow refer to multiple things, then they are not equal. To me, it seems that ten has to be the ONLY 10, and there cannot be another 10 and that only ten is 10.

IF so how is it that 3+7 = 10? How can two separable concepts become 10, which is already a "thing"?

Maybe you see my issue. Everything seems to be everything or to be wholly unequal, and the methods to solving things are unexplained, and I don't know why equations are answered the way they are and seem wrong to me. If two things are separable, to me, they are not equal. It all seems so arbitrary and stupid.

I'm probably extremely pretentious, and I'm aware that my tag is incredibly pretentious, but I feel that my question is complex because it deals with things that I believe are complex, to me everything is complicated. Maybe I'm really stupid, but that doesn't change anything since I still need help, and probably the best person to ask are people who deal with this stuff all day, so that's why I tagged it with what I did. I feel like the LSD I once did ruin how I comprehend ideas. Everything literally seems like everything to me, and that distinctions in things are arbitrary and just for convenience to continue living, or that maybe everything is unique and that it's wrong to say anything is equal. I don't know; I feel weird about concepts.

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ Jul 17 '17 at 4:47

Short answer that may help.

You can think of the equal sign as representing two ways to describe the same thing. Then $$ 3 + 7 = 10 $$ should make sense.

If $5x$ and $10$ are equal then they are two ways to describe the same number. It follows that $5x/5$ and $10/5$ must then also be two ways to describe the same number. The first of those is $x$, the second is $2$, so $x=2$. That means $x$ is another way to describe the number $2$, so you've "solved the equation".

You can think of


as saying that five groups of two is a way to describe ten objects.

The "equals" in the sentence "one foot equals twelve inches" does not mean that two rulers are the same physical object, it means that those two quantities are two different ways to describe the same length.


Let's start with something not too complicated, that is:


What does this statement mean exactly? In order to understand it, you need to understand what each symbol means - or rather, what each symbol has been (arbitrarily) defined to mean.

I say arbitrarily because in reality the following:


is the same as


if we agree that $1$ is the same as 웃, $ $ $+$ is the same as ‽, $ $ $=$ is the same as ☎, $ $ and $2$ is the same as ⚔.

So, in reality, the symbols we use in mathematics only make sense because we've defined them to mean something.

Note that although the symbols can be described as arbitrary, the meanings associated to the countless combinations of symbols we can make are not arbitrary at all!

So what do we mean by the combination of symbols $1+1=2$? "Formally", it can be quite challenging indeed to explain this. In fact, it took some very smart people (Russell and Whitehead) some 300 pages to formally prove this proposition as a part of a three-volume work called Principia Mathematica, only about a century ago.

Intuitively though, it could be thought of as $1$ meaning an entity, a unit of something, then $+$ meaning the addition, the grouping of similar things (e.g. apples are similar things, but apples and oranges are different things), and finally $=$ would mean "the acknowledgement of the equality of two things" (at least in this context).

With these symbols defined as such, $1+1$ then means the addition (or grouping) of a unit of something with another unit of that something. $ $ Now, why not create a whole new symbol to mean exactly that? $ $ We can indeed do so and choose that symbol to be $ $ $2$.

So from now on, whenever we have the addition of a unit of something with another unit of that something, we can simply say we have "$2$ of that something". Therefore, $2$ of a something is really the same as the addition of a unit of that something with another unit of it. And so, since $=$ means the acknowledgement of the equality of two things, we can write, and make sense of


Fundamentally, the rest of mathematics isn't much different than what I've tried to show above (although, the underlying concepts and ideas have indeed been quite simplified here).

It's really about defining things, that is, giving them meaning, and then deducing other more interesting things from what we started with.

I recommend these sites for much more on the topic: https://welovephilosophy.com/2012/12/17/do-numbers-exist/


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    $\begingroup$ Hm, I don't see any good reason to downvote this. $\endgroup$ Jul 17 '17 at 0:56
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    $\begingroup$ @JairTaylor I agree. I've edited it out. I was really just referring to what the OP said himself towards the end of his question - nothing more. $\endgroup$
    – Stephen
    Jul 17 '17 at 0:59

It seems that you're saying every number is unique and therefore it couldn't be represented by anything else.

Let me say, Numbers are abstract things which can be represented by different expressions.

For example:

The number 10 can be represented by any number multiplied by another number, infinitely.

Let C be some number, they're usually called constants. Let x essentially be another constant.

Then Cx=10, is as I said, always true for some numbers; we must first choose one of them, then the other. Take C=1 then Cx=10 becomes x=10. We have replaced the C with 1.

Take C=2, then we have 2x=10. This just states that some number multiplied by 2 is 10. You're told to do whatever operation to BOTH sides so that we keep the equivalence. Here we divided by 2, and we see 2×5=10.

We can go on infinitely in this way.

Likewise with this representation we can change the number 10 to any number whatever.


Unfortunately, the doubts you are experiences cannot be solved in an easy way, and are part of a higher abstraction which normally people tends to accept, but not to "understand".

What you are facing is the difference between expressions and their values. Not too much people, even maths people, can distinguish them at some cases. Hence i dont think your concerns are pretencious in any way. And i am not surprised your tutors were unable to tackle them with you.

First of all, lets agree numbers are concepts. That is, 1 is unique, 2 is unique, 10 is unique. I cannot separate 10, this is only 10.

Second, lets agree numbers allows to represent groups of real things. That is: $10a$ represents the group: $$aaaaaaaaaa$$ And, $5c$ represent the group: $$ccccc$$

At this point we can have some sort of solid agreement. I can represent the group $dddd$ as $4d$ and the group $zzzzzzzzzzzz$ as $12z$.

Third, and if and only if, we had an agreement at our last point, we can introduce the symbol $+$. We know that:

  • $zzzz$ is represented as $4z$,
  • $zzzzzzzz$ as $8z$ and
  • $zzzzzzzzzzzz$ as $12z$,

and so we know that

  • $zzzzpppppppp$ can be represented as $4z+8p$,
  • $aaabbbbcc$ as $3a+4b+2c$.
  • this last is a classical: 3 apples plus 4 bananas plus 2 carrots.

If we have an agreement here, we could be able to represent different quantities, such as:

  • $zzzccccsssss$ as $3z+4c+5s$

Or even imply an infinity group: - $abbcccddddeeeee...$ as $1a+2b+3c+4d+5e...$

At this point our agreement is very solid and allow a huge of posibilities. You can write expressions such as $2x+4y+6z$ and you clearly know this represents the group $xxyyyyzzzzzz$, or whatever other expressions you wish.

Fourth, at this point, we finally should accept the truth that a same group could be represented in different ways.

The object: $$kkkkkkkkkkkk$$

Can be represented as $12k$. But also as $1k+11k$, or as $2k+2k+8k$. All these three expressions represent the same group.

Moreover, groups like:


Can be represented as $4x+4y$ or like $1x+3x+2y+2y$, $2x+2x+1y+3y$, among many many others.

If we have an agreement on all this, we can continue...


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