Today we are mixing the concepts of addition, subtraction, multiplication and division with the concept of dimensions to understand all of them better bit-by-bit. The first minute or two of the video is left fairly empty intentionally, to support the idea of reduced dimensions, but in the last part of the video we will see how insightful it can be to throw all these concepts together. We will show practically and visually, for a few examples, why mathematical rules about multiplication, factorization and simplification make sense, and how those rules look if you look past the formulas. So, please watch until the end! Background music: www.bensound.com
Our everyday lives happen in 3 dimensions, 4 if you want to include time as well, but let’s kick off with 0 dimensions. You can think about this as a point somewhere in the 3-dimensional space that you’re used to, but it’s a point with no clue about the dimensions around it. It does not have size and is not aware of any directions. From its point of view, it is the only thing in existence.
This is somewhat of a boring situation – everything is 0 and we cannot add, subtract, multiply or divide to change anything. So, let’s add one dimension. That point now becomes two points – a starting point and an endpoint, with a line in-between, which has a length. In 2 or 3 dimensions it has length and direction, but from its point of view it only has length, and is not aware of any direction. It cannot move in any direction other than on the straight line that results from extending it to both sides. But we can now add, subtract, multiply and divide. Let’s define the current length of the line as one unit, represented by the number 1, and we define the position of the starting point as 0, or the origin, with the right-hand side being the positive side and the left-hand side the negative side. When we add, the position of the endpoint moves to one side, while it moves to the other side when we subtract. Think about the long line of possible positions as the number line. Something like “plus 3” moves the endpoint 3 units to the right to lengthen it, and something like “minus 6” moves the endpoint 6 units to the left to shorten it. But what happens when the endpoint crosses the origin? The line starts lengthening again, but on the negative side. The negative numbers may feel a bit strange, but maybe think about it as “in the opposite direction”, rather than “less than nothing” or “shorter and longer”. Subtraction therefore moves the endpoint in the opposite direction than addition would. This also makes it clear why the well-known rule “a minus times a minus is a plus” holds. If you want to move the endpoint -1 units, you have to move it in the opposite direction than +1 would. If you want to move it -(-1) units, it means you have to move it in the opposite direction of the opposite direction, which is merely the original direction again.
And what about multiplication and division? They also influence the length of the line, but the basis unit with which it now is adjusted is the current length of the line, rather than the 1 unit of addition and subtraction. Let’s call the current length x instead of 1. Something like “4 times x” will make the line 4 times as long as its length at that time. This corresponds to the formula x + 3x = 4x. Something like “divide by 2” will cut the length of the line at that time in half. This corresponds to the formula 4x/2 = 2x and is equivalent to 4x * ½ = 2x. Note how multiplication and division with negative numbers flips the entire line over to the opposite side, as we see when we express 4x/(-2) or 4x*(-½) as -2x.
But someone stuck in one dimension is only aware of length, and have no clue about the concept of area. It’s as though such a person looks at a page of paper precisely from the side, making it look like only a line on one dimension, while there is no way of seeing what goes on on the surface of the paper. So, let’s add another dimension. In two dimensions we can now clearly observe the contents of the paper, or other area. Our 1-dimensional line with length now becomes a 2-dimensional square with sides 1 unit, say 1m, and area also 1 unit, or 1 meter squared. As in the 1-dimensional case, we can add, subtract, multiply and divide on the x-axis, but we can now also add, subtract, multiply and divide on the y-axis. Where any point on the 1-dimensional line is characterized only by its length, or distance from the origin, any point on this 2-dimensional plane is characterized by two bits of information: its length on the x-axis and its length on the y-axis. In this case the point x=5 and y=4.
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