The Clock Arithmetic and Modular Systems Series

This is a series of posts that explains modular systems starting from an intuitive introduction using clocks. I wrote this for high school students of average mathematical ability.

clock arithmetic

I hope you find the series easy to read and student friendly.

The Series

Part 1: Introduction to Clock Arithmetic and Modular Systems

This post introduces modular arithmetic intuitively using the 12-hour clock mathematical operations. What happens if we add the numbers on the clock? » Read more

Finding the General Formula for the kth Polygonal Number

If we represent numbers using “dots,” there are special numbers that can form “polygons.” Numbers that can form polygons are called polygonal numbers. For example, the square numbers 1, 4, 9, 16 and so on form a “square.”  As shown below, it is easy to see that the 10th square number is a square with 10 rows and 10 columns. This means that the 10th square number is equal to 100 (has 100 dots).

polygonal numbers

The Representation of Square Numbers

Like square numbers, triangular numbers form a shape. You guessed it right, the shape is a triangle. Finding the 15th triangular number seems hard, but we have already learned that triangular numbers are connected to the sum of the first m positive integers. The mth triangular number has

1 + 2 + 3 + \cdots + m = \frac{1}{2}m(m+1) dots. » Read more

Introduction to Number Bases

In Clock Arithmetic and Modular Systems, we have learned about a different number system, a number system whose largest digit is 12.  We observed that in that system, we can only use the numbers 1 through 12. We also noticed that 12 acts as 0 since 12 added to any number is equal to that number.  If we  change 12 to 0, we can only use 0 through 11 as digits.

The number system that we use everyday, the decimal number system, uses 0 through 9 as digits. In the decimal number system, if we add 1 to the largest digit which is 9, we add 1 to the number on next place value and write 0. For example, 9 + 1 = 10 and 10 means that 1 tens and 0 ones. In the decimal number system, 325 means 3 tens squared (or hundreds), 2 tens and 5 ones. Using the expansion notation, we have » Read more

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