Exponents

There are many kinds of growth visible in the world around us. Mathematicians were able to evolve several mathematical models to mirror the process of growth.

Multiplicative thinking could model growth rates which were faster than Additive thinking. The following table illustrates the difference between these two; using the starting size as 10 (any other number can also be used)

In both the examples the “rate of growth” is independent of the “size”.

Exponentiation

But scientists also came across examples of very rapid growth. Take the example of a human cell after fertilization. The cell splits into 2. Each of these 2 cells split into 2 cells, making it 4 cells altogether. Hence the growth proceeds in a series which can be seen as 1, 2, 4, 8, 16 and so on, doubling at each stage.

A nuclear fission is also an example of exponential growth. Assume that each Uranium nucleus produces (on fission) 3 other neutrons, each of which in turn produces 3 each. The number of neutrons will increase in a sequence which can be written down as 1, 3, 9, 27, 81 …. causing a nuclear explosion.

Since the entire entity doubles or triples, the “rate of growth” is dependent on the size of the entity at any point of time. Hence as the size increases, the rate of growth also increases rapidly.

Mathematicians used the idea of “exponentiation” to model this kind of growth. If multiplication could be thought of as “repeated addition”, exponentiation could be thought of as “repeated multiplication”. They also introduced a notation where 3X3X3X3X3X3X3 is written as !The use of his representation to denote exponentiation was started in the 17thcentury.

We can capture the power of exponentiation by comparing it with the other two models.

Base, Exponent & Power

There are 3 mathematical terms with which we need to get familiar while talking about exponentiation.

If 256 can be written as 28 which is (2 X 2 X 2 X 2 X 2 X 2 X 2 X 2), then

1. Base - 2 is the Base

2. Exponent- 8 is the Exponent to which the Base has been raised.

3. Power- 256 is a Power of 2. All numbers which can be represented with Base 2 and different Exponents are called Powers of 2.

Power of Exponentiation

There is a famous story of a sage who asks a king just to give grains of rice in each square of the chessboard. The condition was that he wanted 1 grain on square 1, 2 on square 2, 4 on square 3 doubling the number in each subsequent square. The king was disappointed that the sage was asking such a trivial gift from him! But he was in for a huge shock and surprise!! The numbers grew so fast that very soon all the rice in the kingdom was exhausted by covering just half the number of squares!!! (it would be a good exercise for students to work out the number of grains in the first 10 squares!)

There is also a mythological story of 64 golden discs of varying diameters which are arranged on a central peg in such a way that any disc is always smaller than the one below it and bigger than the one above it. The task is to shift them from one peg to another (with an additional peg for intermediate use) such that at any time in these moves, a smaller disc does not come below a larger disc! The world is supposed to end when the transfer is complete. This problem is mathematically the same as the chessboard problem and the final figure is a truly astronomical figure of 264 − 1. The tower is called Brahma’s Tower or Tower of Hanoi.

This story also indicates that the Hindus had conceived of very huge numbers much before other civilizations. This could also have been an incentive to think of the decimal place value system.