At this stage in the evolution of humanity there is only so much that our brains can perceive. For instance, we can easily visualize quantities of up to several hundred. We have had experiences that make the quantity of several thousand imaginable. We have an impression of how big a million is but we really have to strain to visualize it.
This aspect of human cognition falls under the broad umbrella of numeracy. Primitive humans don’t have it to such a degree as we do. Many primitive tribes have words for one, two, and many. Modern man has mastered counting things, has developed the idea of zero as a significant quantity, and has gotten the hang of addition, subtraction, multiplication and division.
We have struggled with concepts like repeatedly multiplying a number by itself a given number of times, an operation called exponentiation. It is hardly intuitive to many people and yet it is key to the future of mankind. An interesting example is often used to illustrate exponentiation.
Take a chess board. It has sixty four squares on it. That is, eight rows of eight squares. If you place a single grain of rice on the first square, and then double that number and place two grains of rice on the second square. If you continue on in that manner, by the end of the first row of squares you will have placed 1+2+4+8+16+32+64+128 or 255 grains of rice on the board. That is equal to two to the eighth power minus one.
By the end of the second row, you will have placed 65,535 grains of rice on the board. Or at least you will have attempted to. They probably won’t fit. Assuming a large enough chess board, by the time you finished doubling the number of grains of rice on each successive square you will have place two to the sixty fourth power minus one grains of rice on the board. This is approximately equal to 1.844674407 times 10 to the 19th power.
This number is well beyond the ability of any mere human to imagine. And we got there in sixty four simple steps. Similarly, there are many natural phenomena that grow in an exponential fashion. They all exhibit the same characteristic of starting to grow slowly at first and then very rapidly growing out of the realm of the human imagination.
This is the first step in our adventure to discover how exponentiation holds the key to the future of mankind. We will continue with that adventure in the next installment of this series.