What is the role of geometry and dimensional operations?

[internationallove089]

TL;DR Summary

what is multiplication, what is the nature of numbers

As I was thinking about understanding the nature of multiplication and division, understanding the nature of numbers, I thought about the idea that there is something beyond multiplication and repeated addition. Then I thought about dimensional multiplication operations, such as area and volume. I researched the use of number since primitive times. First, let's look at the concept of an apple. Here the apple is a $3D$ object. There may be billions of atoms in one apple. However, we can multiply and collect an apple as we wish. We can perform very good operations in the universe of natural numbers. When we switch to integers, we may encounter some problems. The concept of $-4$ apples is inconsistent.

Then I realized that vectors on a one-dimensional number line worked wonderfully to explain operations with integers. However, our numbers here are one-dimensional vectors, not one apple. First of all, our vector $\overrightarrow{a}$ is $1m$ long in the positive direction in the coordinate system. We express this vector with the unit meter. So it is a vector of size 4m. Likewise, if we consider the number as $4m$ in size, independent of direction, the result does not change when multiplying (with positive integers).

$$4\cdot\left| \overrightarrow{a}\right|=4m$$

At this point I'm starting to get confused. First of all, when multiplying, dividing, adding and subtracting numbers, we can perform operations between two scalar numbers (apple example). Now let's consider a square with sides $4m$ long. When calculating the perimeter, we are actually performing a multiplication operation on a single dimension. If we multiply one side by $4$, this means multiplying each unit on the coordinate system by $4$ in one dimension. This Scalar can be the product of two numbers or the product of a vector and a scalar number.

$$4\cdot\left|\overrightarrow{a}\right|\cdot4=4\cdot(0,000000000\overline{1}+0,000000000\overline{1}+.....)=16m$$

Now let's move on to calculating the area. Here we are now working in $2$ dimensions (x and z). Let's have a size that we define as $4m$. Let's consider it directionally independent. Again, we have an edge defined as 4m in the z direction. Now, while trying to understand the nature of field calculation, I think like this; Multiplying my size of 4m in the x direction by my size in the z direction actually means multiplying my size in the x direction in the z direction for each number value in the z direction. (repeated addition) The interesting thing here is that we are doing the multiplication in two dimensions.

$$4m\cdot4m=4m\cdot(0,000000000\overline{1}+0,000000000\overline{1}+.....)m=16m^{2}$$

When we multiply in $2$ dimensions, the result is like the product of $2$ scalar numbers (apple examples). However, logically, we created the area by adding the length $x$, $4m$, infinitely one after the other. Now I have stated my questions below:

- Do numbers have dimensions? Thinking of numbers as one-dimensional vectors works amazingly well when operating with integers. However, is it true that it is in the nature of numbers that we can add them and assign them to different objects, independent of size, as in the apple example (for example, we all intuitively know that the sum of $1$ apple and $1$ apple is $2$) and that it can have many different natures?

- My other question is, we can multiply a number with the vector whose unit we define on a single dimension. For example $4m\cdot4$ this is quite logical. However, we cannot make $4m\cdot4m$ in one size. But if we think in two dimensions, $4m\cdot4m=16m^{2}$. Don't we have more 4m sizes than we can count in a $16m^{2}$ area? What's strange is that we can express $16m$ and $16m^{2}$ with the same numbers. This is interesting, don't you think?

 

[anuttarasammyak]

TL;DR Summary: what is multiplication, what is the nature of numbers

- Do numbers have dimensions?

I do not think so. Let all the numbers has dimension, say m. Making a fraction, say
$$\frac{4m}{2m}=2$$
RHS number has no dimension.

 

[Mark44]

TL;DR Summary: what is multiplication, what is the nature of numbers

As I was thinking about understanding the nature of multiplication and division, understanding the nature of numbers, I thought about the idea that there is something beyond multiplication and repeated addition. Then I thought about dimensional multiplication operations, such as area and volume. I researched the use of number since primitive times. First, let's look at the concept of an apple. Here the apple is a $3D$ object. There may be billions of atoms in one apple.

It's irrelevant that it contains billions of atoms.

However, we can multiply and collect an apple as we wish. We can perform very good operations in the universe of natural numbers. When we switch to integers, we may encounter some problems. The concept of $-4$ apples is inconsistent.

Might be inconsistent with apples, but not so with, say, money. If you have $100 in your checking account but write a check for $104, your account will, in essence, have a balance of -$4. Actually, it will be lower than that, as most banks will charge you a fee for nonsufficient funds (NSF).

View attachment 342345

Then I realized that vectors on a one-dimensional number line worked wonderfully to explain operations with integers. However, our numbers here are one-dimensional vectors, not one apple.

The numbers mark possible endpoints for vectors

First of all, our vector $\overrightarrow{a}$ is $1m$ long in the positive direction in the coordinate system. We express this vector with the unit meter. So it is a vector of size 4m. Likewise, if we consider the number as $4m$ in size, independent of direction, the result does not change when multiplying (with positive integers).

$$4\cdot\left| \overrightarrow{a}\right|=4m$$

View attachment 342346

At this point I'm starting to get confused. First of all, when multiplying, dividing, adding and subtracting numbers, we can perform operations between two scalar numbers (apple example). Now let's consider a square with sides $4m$ long. When calculating the perimeter, we are actually performing a multiplication operation on a single dimension.

Or you can simply add all four sides.

If we multiply one side by $4$, this means multiplying each unit on the coordinate system by $4$ in one dimension. This Scalar can be the product of two numbers or the product of a vector and a scalar number.

$$4\cdot\left|\overrightarrow{a}\right|\cdot4=4\cdot(0,000000000\overline{1}+0,000000000\overline{1}+.....)=16m$$

What is the above supposed to mean? $4 \cdot 4 \text m = 16 \text m$

View attachment 342347

Now let's move on to calculating the area. Here we are now working in $2$ dimensions (x and z). Let's have a size that we define as $4m$. Let's consider it directionally independent. Again, we have an edge defined as 4m in the z direction. Now, while trying to understand the nature of field calculation, I think like this; Multiplying my size of 4m in the x direction by my size in the z direction actually means multiplying my size in the x direction in the z direction for each number value in the z direction. (repeated addition) The interesting thing here is that we are doing the multiplication in two dimensions.

$$4m\cdot4m=4m\cdot(0,000000000\overline{1}+0,000000000\overline{1}+.....)m=16m^{2}$$

Again, what is the meaning of the above? Particularly this part: $(0,000000000\overline{1}+0,000000000\overline{1}+.....$.

When we multiply in $2$ dimensions, the result is like the product of $2$ scalar numbers (apple examples). However, logically, we created the area by adding the length $x$, $4m$, infinitely one after the other.

No, this makes no sense at all. You can't get the area by adding an infinite number of linear segments. However, when the calculation of areas is presented in calculus one is in effect adding a large number of very thing rectangles to arrive at the area. This might be a new concept to you if you haven't studied calculus.

Now I have stated my questions below:

- Do numbers have dimensions?

No, numbers are dimensionless.

Thinking of numbers as one-dimensional vectors works amazingly well when operating with integers. However, is it true that it is in the nature of numbers that we can add them and assign them to different objects, independent of size, as in the apple example (for example, we all intuitively know that the sum of $1$ apple and $1$ apple is $2$) and that it can have many different natures?

- My other question is, we can multiply a number with the vector whose unit we define on a single dimension. For example $4m\cdot4$ this is quite logical. However, we cannot make $4m\cdot4m$ in one size. But if we think in two dimensions, $4m\cdot4m=16m^{2}$.

This is laid out in dimensional analysis.

Don't we have more 4m sizes than we can count in a $16m^{2}$ area? What's strange is that we can express $16m$ and $16m^{2}$ with the same numbers.

But the units are different. For example, $100^\circ F.$ and 100 apples both share a total count attribute, but are otherwise very different.

This is interesting, don't you think?

Yes and no.

 

[anuttarasammyak]

TL;DR Summary: what is multiplication, what is the nature of numbers

- My other question is, we can multiply a number with the vector whose unit we define on a single dimension. For example 4m⋅4 this is quite logical. However, we cannot make 4m⋅4m in one size. But if we think in two dimensions, 4m⋅4m=16m2. Don't we have more 4m sizes than we can count in a 16m2 area? What's strange is that we can express 16m and 16m2 with the same numbers. This is interesting, don't you think?

Numbers are to be multiplied or divided with numbers. Dimensions are to be multiplied or divided with dimensions. I hope in this way your question will be answered.