Looking at the most simple of operations in a visual way to help with intuition and potentially build insight.

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If you divide anything by one you’re essentially taking a line of blocks and fitting it through a point. The quotient is equal to the height of the blocks. Imagine a line going through a point...the number of times the line can go through the point is equal to the number of points in the line. Now, instead of point we can say ‘unit block’ which allows us to avoid infinitesimals. The line of blocks is constructed of unit blocks and the size of the unit block is equal to the divisor.

So, the quotient is the number of times you can move the line through the unit block (divisor). Now, you can bring division into higher dimensional movements if you wish. Let’s say you were dividing 64 by 16. Instead of a line of unit blocks with a length of 64 going through a station of length 16, you can have a station of 4 by 4 (a square) and find an object with 64 units that happens to have 4 by 4 cross sections (this turns out to be a 4 by 4 by 4 cube). So, you get the same result of a 16 unit station as a 4 by 4 station.

How does one represent division with subtraction?

Division can be seen as repetitive subtraction.

It can also be seen as leveling unit blocks onto a plane. This means you have some number of unit blocks (dividend) and you pour them onto a surface (a line) (which has a length of the divisor) and you shift them around so that they all have the same height. The resultant height is the quotient.

Square roots? You take some number of unit blocks and you pour them onto a square surface and adjust the sides of the square so that the entire surface is flat.

Cube roots is the same thing only you do it (you don’t have to) with liquid in a cube.

You have some function that defines a quantity, the square root could be a planar surface, the cube could be the cubic volume, then this function is divided by another, which means you’re taking a quantity and equally distributing values to another quantity.

The quotient will increase by 1 unit if the dividend increases by the divisor.