Overview
In modular arithmetic, numbers cycle rather than increase when operations are performed. Think of a clock with numbers equal to the cycle. Extensive applications include UPC numbers, the electronic numbers on the bottom of bank checks, and public key codes.
How Does Modular Addition Work?
The most common real world example of modular arithmetic is that of a clock. A common clock goes from the numbers 1 to 12. When two numbers are added, the sum is no more than 12, because another cycle is made around the clock. The number line is a circle around the “clock”, rather than a straight line. If 10 + 6 is added in a mod 12 number system, the pointer would start at 10 and go 6 spaces, past the 12 again to end at 4. Imagine a clock that only went from 1 to 8. In that case, the cycle would be much shorter, so that adding 7 and 3 would cycle past 8 to 2. Another way to say this in equation form is 10 + 6 ≡ 4 mod 12. The symbol for equivalence looks like an equals sign with 3 lines instead of 2, and the word “clock” is dropped is dropped from the description of modular arithmetic.
What about Multiplication?
Multiplication is repeated addition, with more cycles in modular arithmetic. Suppose the problem was (3 ∙ 7) + (2 ∙ 1) + 3 in mod 10. Now, 21 can be broken down to 10 + 10 + 1. In mod 10, 10 is just another cycle. In other words, 10 ≡ 0 mod 10. So 21 + 2 + 3 ≡ 1 + 2 + 3 mod 10 (or 6) mod 10. No matter how large the number, it can always be simplified by eliminating the modulus. A number such as 101000mod 10 is equivalent to 0.
What Are UPC Numbers?
Modular arithmetic has several common applications is everyday life. One of the applications is the Universal Product Code (UPC) numbers, added as barcodes to scan everyday products. Modular arithmetic can be used to check that the UPC number is correct and in the correct order. Also, the nine-digit electronic number found at the bottom of bank checks can be verified by using modular arithmetic.
How Is Modular Arithmetic Used to Decipher and Encode Data?
Public key codes use both modular arithmetic and prime numbers to decode messages from encrypted bank data to secret messages used in espionage. Only the intended receiver knows the numbers essential to the formula deciphering the numerical message.
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