## What If We Have 12 Fingers?

Our number system is called the decimal system (deci means 10) because we count in groups of 10’s. This is probably because we have 10 fingers. What do I mean when I said when we count in groups of 10? Our number system has the digits 0 to 9, and then we when we reach the 10th number, we place 1 in the tens place 0 in the ones digit. In the decimal number system, 23 means that we have 2 tens and 3 ones. Similarly, the number 452 means that we have 4 groups hundreds (10 tens), 5 groups of tens and 8 ones. In fact,  if we use the expanded notation, 452 is equal to $4 \times 10^2 + 5 \times 10^1 + 2 \times 10^0$.

Notice that each number is multiplied by powers of 10. » Read more

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

The Series

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

## The Definition of Congruence in the Modular Systems

This is the fourth part of the Introduction to the Modular Number Systems Series. In the previous parts, we have learned intuitively the modular systems using a 12-hour analog clock, performed operations with its numbers and introduce the symbol for congruence,  and discussed the different number bases.  In this post, we formally define congruence. image via Wikipedia

Recall that the statement $17 \equiv 5 (\mod 12)$ means that 17 gives a remainder of 5 when divided by 12, or that 17 and 5 give the same remainder when divided by 12. We have also learned that 17, 29, and 41 are congruent since all of them give the same remainder (that is 5) when divided by 12. Notice also that since all of them are congruent, » Read more 1 2