Stop 1: Positivity Planet

Charlie Ahrendts (Image: Stickman Physics)

The first stop on our journey across the mathematical universe is a civilization very similar to ours – they too have developed a system of mathematics that is built around numbers and features aspects like geometry. The only difference is, however, that they have never heard of negative numbers. How can something that is less than nothing exist and have any meaning in their world?

These aliens would not be the first ones to come across this problem. The ancient Egyptians, Romans and even the Greeks did not have the concept of a negative number. The first recorded account of a mathematician mentioning negative numbers was between 200 BC and 200 AD in China, but as far as European mathematics goes, they weren’t really accepted until the 17th or 18th century. There are records of a Greek mathematician who came across the equation 4x + 20 = 4. Usually, if we want to solve it we would subtract 20 from both sides and then divide by 4, which would leave us with x = -4. Most of us have learned to solve these equations in school, but this mathematician called it absurd since negative numbers just weren’t seen as a possiblity.

For most of our existence as humans, we managed just fine without having negative numbers. And I am sure that the inhabitants of this planet do too, but there are some things they would miss out on. Almost all of today’s electronics is programmed using negative numbers. They would also need to find other ways to describe concepts like the curve of a ball that is thrown up and then falls down again under gravity, the motion of a car that is moving backwards from its starting position, or the opposite charge of proton and electron.

Projectile Motion

Today we can see how negative numbers are useful in many ways, but for a long time they were not accepted to be an actual concept, never mind useful. Fortunately, mathematicians managed to do many of the things where we now use negative numbers using other methods. Let us look at one example: projectiles. We will use the path of a ball that is thrown up and falls back down again, and we will see how we can describe that path with, and without, negative numbers.

Suppose we stand on the ground and have a ball that we throw directly upwards. We can describe the height of the ball at any time with the formula h(t) = v0 t – 12 g t2 where h(t) is the height of the ball at time t, v0 is the initial velocity at which we throw the ball up, and the term 12 g t2 accounts for the height loss due to the gravitational force of the Earth. If we want to plot the height of the ball over time we just plug in our initial velocity, let’s say 10 meters per second, and draw a graph of the resulting function. 

As we can see, this function is made up of only positive values, so it would be no problem for the ancient Greeks – and of course our friendly alien inhabitants of Positive Planet.

If, however, we are standing on the edge of a cliff when throwing the ball, things will look very different. The function that describes the height of the ball will at some point go below zero, since we usually define zero to be the height where we are standing. Now you might be thinking that you can get around this in some way, for example by shifting our coordinate system down so that zero lies at the bottom of the cliff, but we have to be careful. Changing the origin of our coordinate system means that we also need to modify our governing equation so that it has an additional +h0 at the end, where h0 describes the height at which we stand. A comparison of the two graphs is shown below.

Adding the starting height h0 is of course not really a serious issue, but things get a little more interesting when we talk about velocity. To do that, we take the time derivative of our position function – but don’t worry if you don’t know what that means the important thing is what we end up with. The equation that describes the velocity of the ball is v(t) = v0 – g t. 

Initially, the ball starts with some positive velocity v0. Once it is in the air, the ball constantly loses speed due to the Earth’s gravity. At some point, the ball has to fall down again. This is where the velocity becomes negative because the ball starts moving in the opposite direction. If we want to avoid negative numbers, we can no longer plot the velocity-time graph of our function. Instead, we can try to split it up into two parts: one where the ball is moving upwards and one where it is falling back down. We will need to make it so that the second part describes the velocity away from the point where it started falling down, or in other words the speed of the ball with no regard for the direction of its motion. The two potential graphs are shown below.

There is another workaround available which comes in the form of polar coordinates. These are a way to describe the 2D plane using a distance and angle. Every point in this coordinate system is described using its distance away from the origin (0,0) and the angle between it and what would be the positive x-axis. This way we do not need negative numbers to describe the speed. However, polar coordinates are usually used when describing rotations around a point or things similar in nature. For example a spiral can be described easily with polar coordinates, but we would usually not use it to plot the motion of our ball. 

As we’ve seen above, it is possible to model projectile motion without negative numbers, but not having them complicates even simple calculations like the one in our example. The more complex the problems get, the more we would need to go out of our way to avoid them. If we use mathematics as a tool, we should make use of its versatility. The advantage that it has over many other sciences is that it is not constricted to observations about the ‘real world’. If there exists a way to make problems easier to solve while not getting in the way of others, why should we hesitate to use it? And if we keep it around for long enough this ‘new’ tool might soon feel completely natural to those that learn about it a few decades later. 

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