The Irrationality of Pi and the Wonders of Number Systems
The Significance of Pi Day
For math enthusiasts, Pi Day, celebrated on March 14th (3/14), holds a special place in their hearts. This date is chosen because it resembles the first three digits of pi (3.14), a mathematical constant that represents the ratio of a circle’s circumference to its diameter. However, one of the most intriguing aspects of pi is its irrationality, meaning it cannot be expressed as a simple fraction of two integers.
The Limitations of the Decimal System
When we say pi is irrational, we are referring to its representation in the base-10, or decimal, number system. This system, which uses digits 0 through 9, was likely chosen because humans have 10 fingers to count on. Interestingly, the Latin root of “digit” is “digitus,” which means “finger.” While the fraction 22/7 is a close approximation of pi, it is not exact.
Exploring Alternative Number Systems
The question arises: could there be a number system in which pi is rational? The answer is yes. To understand how this works, let’s review the basics of number systems.
Understanding Number Systems
Imagine a primitive bean counter who assigns a unique symbol to each successive bean. This method quickly becomes cumbersome, requiring 200 symbols for 200 beans. A more advanced system, like the decimal system, uses only 10 symbols (0-9) to represent any quantity. Once you reach 9, you move one place to the left and start again, with each digit representing a multiple of 10, then 100, and so on.
For example, the number 214 can be broken down as follows:
Place values smaller than 1 are represented by negative exponents of 10. For instance, 3.14 can be expressed as:
The Binary Number System
Binary, the number system used in computing devices, employs only two digits: 0 and 1. To convert a binary number like 1010 to decimal, we use powers of 2 for each place value:
To convert a decimal number to binary, we divide by 2 repeatedly, keeping track of the remainders. For example, to convert 22 to binary:
For decimal places, we multiply by 2 instead, keeping the integer part as the binary digit. Converting 0.43 to binary looks like this:
<img alt=”A list of equations 0.43 x 2 = 0.86 arrow 0 x 2^1 0.86 x 2 = 1
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