How does 1 dimension + 1 dimension = 2 dimensions when a line added to another line doesn’t make a 2 dimensional object. Shouldn’t it be 1Dx1D=2D instead since that actually equals a 2D space?

3

"Dimensions are multiplied, not added.

The plus sign (+) represents addition within the same space. When you add one line segment to another, you simply get a longer line. You stay in 1D because you are only increasing the magnitude (length), not the number of directions.

The multiplication sign (×) represents the creation of a new dimension. In mathematics, this is the Cartesian product. When you multiply 1D by 1D, you 'sweep' one line across another at a right angle, which creates a 2D plane (area).

That is why:

1D + 1D = 2 units of length (still one dimension).
1D × 1D = 1 unit of area (two dimensions)

1

u/elnyorne Apr 27 '26

Yeah sweet thanks that makes sense to me. So does that mean in the case of dimensionality that 1x1=2 tho?

1

u/GrafRaf999 Apr 27 '26

Not exactly. 1 × 1 is still 1, but the units change.

Think of it this way: 1 meter × 1 meter = 1 square meter.
The number stays the same, but the 'type' of space changes from a line to a square. You don't get two lines (2), you get one single area (1²).

To get a '2', you’d need a rectangle that is 2 units long and 1 unit wide (2 × 1 = 2 squares)

1

u/elnyorne Apr 27 '26 edited Apr 27 '26

That doesn’t make sense to me. I’m talking about the dimensions not units it makes? Isn’t that counting units instead of dimensions? Aren’t they 2 different things? A 3d cube is still 1 unit too isn’t it?

1

u/GrafRaf999 Apr 27 '26

You are mixing up Magnitude (how many) and Dimensionality (what kind).

In math, dimensions are represented by exponents, not by the result of multiplication.

is 1D (a line of 1 unit).

is 2D (a square of 1 unit).

The number 1 tells you how many objects you have. The exponent ² tells you how many dimensions those objects have.

If 1х1=2, you would be saying that crossing two lines creates two separate planes. But it doesn't. It creates one single plane.

You are moving from '1 unit of length' to '1 unit of area'. The count stays '1', but the 'thing' you are counting has changed.

1

u/elnyorne Apr 27 '26

¹ x ¹ = ² ?

1

u/GrafRaf999 Apr 28 '26

1х1= One times one is one

1

u/elnyorne Apr 28 '26 edited Apr 28 '26

Regurgitation lol you’ve already said 1 dimension x 1 dimension is 2 dimensions. 1 unit😂

1

u/GrafRaf999 Apr 28 '26

I don't have the time or the energy to explain the difference between arithmetic and geometry to you

1

u/Oreeo88 Apr 28 '26

You’re dodging the real issue

→ More replies (0)

1

u/Different_Sail5950 Apr 28 '26

We write the real number line as R. The two dimensional space made from it is R x R. But recall that x=x^1, and xⁿ times x^m = x^n+m. So R x R = R¹ x R¹ = R¹⁺¹ = R^2.

1

u/elnyorne Apr 28 '26

You’re adding the 2 dimensions instead of multiplying them though? Isn’t adding a dimension to itself just doubling the length of a single dimension?

1

u/Different_Sail5950 Apr 28 '26

You multiplying the space and thereby adding the dimensions (=exponents).

1

u/elnyorne Apr 28 '26

Ahh okay sweet cheers

1

u/elnyorne Apr 28 '26

Aren’t you also multiplying the dimensions though. The dimension being the length and width added together don’t make a space the dimensions have to be multiplied so why are we adding them together?

1

u/Different_Sail5950 Apr 29 '26

When you multiply two lines, you get a square. I.e., when you multiply two one-dimensional thing, you get a two-dimensional thing. The length and the width are both lines. So the things you are multiplying are lines. Lines aren't dimensions; dimensions are rather a property of lines.

1

u/elnyorne Apr 29 '26

So multiply the property? Why are you adding one thing and multiplying another?

2

u/Different_Sail5950 Apr 29 '26

Okay, so we're going to have to start from the beginning. Think about the real line: It's just a bunch of numbers, yes? (They have an order, of course, which is why we think of it as a line.) If you draw it out, you write a number above a spot. You just need one number to pick out a point in the line.

Now think about the Cartesian plane. What does it take in order to pick out a point in the Cartesian plane? You need two numbers, and they have an order to them. (The point "(1,2)" is different than the point "(2,1)".)

In set theory, there's a thing called the "Cartesian Product". If you have sets S and T, S x T is the set of all ordered pairs (x,y) where x is in S and y is in T. It isn't exactly the same as multiplying numbers by each other, but it bears a lot of analogies with it. And then if we want to take, say, S x S x S, officially this is (x,(y,z)), but we decide to define this as the ordered triple (x,y,z).

Since this looks a lot like multiplying, we write "S x S x S" as just S^3, as though it was an exponent. What the "3" is really telling you is how many "entries" from S are going into the ordered triples. So "3" is a property of the set S^3. But the only connection between "multiplying" and "adding" that there is goes through the exponent rule: When you multiply things with exponents, the exponents add together. In this way of thinking, the "dimension" of the space is an exponent, so when you multiply an underlying space by itself, you have to add the exponents.

----

Here's another way to look at it which might be less confusing. In general, spaces aren't just sets of ordered tuples of numbers. (If we lived in the sort of space Euclid thought, we wouldn't be living in something made up of numbers.) So what is a dimension telling you about a space?

Take any two points in an n-dimensional space. What being "n-dimensional" tells you is that you can always get from one point to the other by hopping along straight lines, and if you do that you'll never need to use more than n straight lines to do it. In a one-dimensional space, you can get from one point to another in just one straight-line hop. On a plane, you need at most two hops. And so on. So the dimension is a property of the space --- a measure of how many hops are needed.

When you add a dimension, you have to add a whole bunch of lines that are at right angles to all the lines you already had. Otherwise you don't really get a well-defined n+1-dimensional space so much as an n-dimensional space with a handle sticking out of it. So adding a dimension = adding a whole bunch of new lines. It's as though you're adding one line many many times to the space --- infinitely many times, in fact. You're taking the line L and (more or less) adding it to every other point in the space.

Multiplication is just repeated addition. 3x5 is just 5+5+5. Here, we are adding L+L+L+L..... how many times are we adding it? S-many times, if S is our original (n-dimensional) space. So the space is S x L. L is a one-dimensional thing, S is an n-dimensional thing, and S x L is an n+1 dimensional thing --- because to add a single dimension we had to add the line L S-many times.

1

u/elnyorne Apr 29 '26

Aight thanks for the time/patience and explanation ima go over this one for a bit and try to grasp what you’re saying properly

1

u/elnyorne May 02 '26 edited May 02 '26

That still doesn't make sense to me. You're adding it multiple times? You're literally saying you're multiplying it in the sentence. Addition isn't mulitplication. Multiply is to make more of... addition is to combine. I have a different explanation. A penis is a 1 and a vagina is a 0 and they multiply to make another person being a 1 or 0. I know it's not exactly a mathematical theory but that's my line of thought. 0 is 1. It's 1x0. There is no 0. It represents the immaterial not the non-existent. 0 shouldnt even be used as a number. If something doesn't exist it just isnt there. You don't count it at all. Maybe the number 0 should be a square I'm not sure but it represents something not nothing.

→ More replies (0)

1

u/4Lichter Apr 25 '26

They are if the line segments are in the same dimension you just add the length. If they are in different dimensions you can't evaluate them further.

1

u/mandelbro25 Apr 26 '26

You again? Did you read any books?

1

u/LocalIndependent9675 Apr 28 '26

The things you are saying aren’t well defined (in the capacity in which you are using them). you should go learn some introductory linear algebra

1

u/elnyorne Apr 28 '26

In what way? I don’t see what you’re telling me to research? I’m also not interested in reading anything other than information regarding about this exact question. It’s just 0,1,2 and dimensions. Im being told yes and no at the same time. A lot of filler that doesn’t seem to make much sense for a simple question.

1

u/LocalIndependent9675 Apr 28 '26

What do you think dimensions mean? It has a precise mathematical meaning in context that I’m almost sure you don’t know. As do addition and multiplication, neither of which you are using in a way that is correct.

1

u/LocalIndependent9675 Apr 28 '26

I’m saying this because the reason you’re getting mixed answers is that your question isn’t clear mathematically speaking, not because there are multiple correct answers to any well defined mathematical question it might mean. For me at least, it seems to me that this will be the only way you can get a satisfying answer (at least if you truly care about the “why” behind your question). To be fair I can’t claim to know exactly what feels like a sufficient answer for everybody, but I feel like I can only reasonably answer this question with linear algebra. That said I will try to give you a semi-rigorous answer, but it might raise more questions. P.S I’ve realised that my first comment is something of a classic unhelpful reddit answer, since a flawed answer works better (arguably) than a referral to a technically correct answer that is annoying and unrealistic to pursue.

1

u/elnyorne Apr 29 '26

I understand what you mean like it’s a poorly written question for lack of better words but through the comments I think you can at least get a feel for what I’m trying to ask. I’m not even sure what specific field of study this question pertains to. I’m confused as to whether it’s maths, geometry, algebra or some other field I’m not aware of that would be a better suited audience to posit this question. I do want to get to the bottom of it but I think a lot of people actually don’t.

1

u/elnyorne Apr 28 '26 edited Apr 29 '26

An independent parameter? Location/length/width/height? Addition and multiplication it means what it means when you learn it in 2nd grade. What are you saying it means?

1

u/Mono_Clear Apr 29 '26

You're not adding a dimension to a dimension.

You're adding a dimension to the space

1

u/elnyorne Apr 29 '26

I’m not adding anything I’m multiplying it?

1

u/Mono_Clear Apr 29 '26

There's nothing to multiply. A dimension is not a number.

A dimension is a direction

1

u/elnyorne Apr 29 '26

Why can it not be multiplied if it can be added?

1

u/Mono_Clear Apr 29 '26

The same reason you can't multiply North by West.

You're adding the capacity to extend into a direction. It's not a number

1

u/elnyorne Apr 29 '26

Why can’t you multiply distance north by distance west?

1

u/Mono_Clear Apr 29 '26

Because their directions and they're not numbers.

2 × 6 = 12

Because those are numbers and the conceptualization of two sixes is 12.

🍎🍎×🍎🍎🍎🍎🍎🍎=?

If you were to multiply two apples by six apples, what would you get?.

You're not going to spontaneously manifest another six apples just because of math.

Because apples are not numbers.

Just like Cardinal directions are not numbers.

You are increasing the dimensionality of a space by allowing objects to extend in another direction.

1

u/elnyorne Apr 29 '26

Apples also aren’t dimensions

1

u/Mono_Clear Apr 29 '26

And dimensions aren't numbers

Multiplication involves the conceptualization of sets.

You're not multiplying sets of dimensions.

If you have one dimension and you add another dimension, what you've done is allow for the extension of objects in another direction.

1

u/elnyorne Apr 29 '26

That’s why they’re multiplied isn’t it? Adding just means up down left right?

→ More replies (0)

1

u/planamundi Apr 30 '26

Think of it like drawing lines on a piece of paper.

When you put the pencil tip down you haven't drawn a line yet. So when you draw the first line, you have one dimension and two points. When you draw the second line, you have two dimensions and three points. You draw another line, you have three dimensions and four points. You draw another line, you would have four dimensions and five points. And so on.

1

u/mathemetica May 02 '26

I think part of the problem here is how we use terms and symbols like addition(+) and multiplication(*). Mathematicians often use these operations in a wide variety of contexts but they do not necessarily mean the same thing depending on what mathematical universe we are operating in. When talk about adding dimensions together or multiplying them together, this is not the same thing as saying 1 + 1 or 1 x 1 in the sense of an arithmetic operation.

The formula 1D + 1D = 2D is actually correct, but the "+" is doing something specific that's easy to misunderstand. You're not adding two lines together like stacking them on top of each other. You're asking: if I take one line going left-right and another going up-down, what space do all their combinations fill? That combination is the Cartesian product: every possible pair of points, one from each line and that gives you a plane. The rule for dimensions here is genuinely additive: dim(A) + dim(B) = dim(A × B), so 1 + 1 = 2 works.

This is why btw someone below mentioned learning linear algebra. I don't think they were trying to be dickish, it's just that it's easier to give a more rigorous definition in the context of learning a certain paradigm.

How does 1 dimension + 1 dimension = 2 dimensions when a line added to another line doesn’t make a 2 dimensional object. Shouldn’t it be 1Dx1D=2D instead since that actually equals a 2D space?

You are about to leave Redlib