Riemann Sphere:

A more complicated explanation of the relationship between Zero and Infinity occurred with what is called the Riemann Sphere. In simple terms, this is a sphere in the complex 3D plane with the south pole being at the origin of 0 and the north pole being at infinity, in line with the origin. With this idea, they could use projections and analyze things such as multiplication, division, and other complex ideas by looking at the sphere deformed and rotated. For example, multiplying by the number i is like rotating the sphere 90 degrees clockwise (Seife 143).

With this idea in mind, we can use different transformations to change how it is rotated. For example, multiplying it by a number x and replacing x with (x-1)/(x+1), we rotated the whole globe by 90 degrees such that the north and south poles are now on the equator. The most interesting idea shows the relationship between 0 and infinity best. As we multiply it by the reciprocal the sphere will flip upside down, meaning that zero becomes infinity and infinity becomes zero. This implies that 1/0=∞ and 1/∞=0 (Seife 144). This is different than multiplying by infinity or zero. As we multiply by infinity, it is like blowing more and more air into a spherical balloon until it pops and all the pieces are scattered everywhere--which is because anything times infinity is infinity, similar to zero. When the Riemann Sphere is multiplied by zero, then it is like popping the balloon and destroying the remains. Both zero and infinity cause the destruction, but in opposite ways (Seife 144).

Riemann Sums:

When thinking of Riemann Sums, and finding the area under a curve, generally we think about infinity. We are trying to increase the number of rectangles and as we do this we are decreasing the width of each of the triangles. This is the yin and yang of zero and infinity. Using the applet below, we can explore the relationship between how as we increase the number of rectangles, the width decreases, making the opposites apparent.

Achilles and the Tortoise:

The paradox of Achilles and the Tortoise is one that could not be solved without the number zero. However, when Zeno introduced this paradox, it was unsolvable. "According to Zeno, nothing in the universe could move. Of course, this is a silly statement; anyone can refute it by walking across the room" (Seife 41). In this puzzle, Zeno proves that Achilles can never catch up with a slow tortoise. For example, suppose Achilles was racing a tortoise, but he was given a 10m head start. The tortoise races at a steady pace of 1m/s while Achilles runs at a quick 10m/s. We would think that Achilles would quickly catch the tortoise. However, as Achilles reaches the 10m mark that the tortoise was given, the tortoise will have made it 1 meter ahead. As Achilles gets closer to the tortoise, the tortoise slowly inches away. Regardless of how close Achilles seems to get, the tortoise will get a little closer. Therefore, with this idea, Achilles can never catch the tortoise. This puzzles mathematicians alike for centuries. The key to solving this problem is using the number zero. However, this was not something that Greek mathematicians at the time of Zeno knew about or liked.

With the idea of zero, we know that the gap between the two will approach zero, and then ultimately pass zero, where Achilles passes the tortoise. Without the idea of approaching a distance between of zero, it cannot happen. This conflict created a separation in the beliefs of others, essentially between atheism and belief in God because of the differing solutions of the paradox. This idea is interesting to explore, but with the number zero, we can find the point in which Achilles is passing the tortoise. This can be simply done with solving the solution to two linear equations. Linked below is a lesson plan to explore this concept.

Lesson Plan:
Powerpoint- Achilles and the Tortoise
Guided Notes- Worksheet
Link to Desmos- Desmos Graphing Calculator
Link to Geogebra Applet- Achilles and the Tortoise Applet

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