How to Change Number Bases Part 1

I have already discussed clock arithmetic,  modulo division, and number bases. We further our discussion in this post by learning how to change numbers from one base to another.

The number system that we are using everyday is called the decimal number system or the base 10 number system (deci means 10). It is believed that this system was developed because we have 10 fingers.

In the base 10 system, the digits are composed of  0 up to 9. Adding 1 to 9, the largest digit in this system, will give us 10. That is, we replace 9 in the ones place with 0, and add 1 to the tens place which is the next larger place value.

Another way to write a number in base 10 is by multiplying its digits by powers of 10 and adding them. For example, the number 2578 can be rewritten in expanded form as

2(10^3) + 5(10^2) + 7(10^1) + 8(10^0)Continue reading

Guest Post: An Interesting Property of Prime Numbers

Although I have already discussed modulo division, I believe that this proof is beyond the reach of average high school students. To explain further, I made additional notes on Patrick’s proof . I hope these explanations would be able to help students who want to delve on the proof. 


I’ve got a prime number trick for you today.

  1. Choose any prime number p > 3.
  2. Square it.
  3. Add 5.
  4. Divide by 8.

Having no idea which prime number you chose, I can tell you this:

The remainder of your result is 6.  Continue reading

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.

modular systems

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, Continue reading

Clock Arithmetic and Modular Systems Part 2

This is the second part of the Introduction to Modular Systems Series. Please read the first part before proceeding.

Last Monday, we have learned a number system that uses numbers on the 12-hour analog clock. We have performed addition using these numbers and discovered that in that system, 12 behaves like 0. We have also observed that to add large numbers, we need to divide the number by 12 and get the remainder.


Recreating the table by replacing 12 with 0 gives us the second table in the figure above. As we can see, in this new number system, we have digits 0 through 11 as opposed 0 through 9 in the number system that we use everyday (the decimal number system).

In this new system, we have observed that there is a certain  number where numbers wrap around. The wrap around number is called the modulo. The modulo of our “clock number system” is 12, so we call it modulo 12. Continue reading

Introduction to Clock Arithmetic and Modular Systems

Most of us are familiar with 12-hour analog clocks. They are numbered 1 through 12; they have hour, minute, and second hands. In this post, we are going to experiment clock arithmetic — we are going to perform addition using the numbers on the clock.

clock arithmetic

Let us think about the following questions.

  • What if we add 3 hours after 8:00?
  • What if we add 2 hours after 3:00?
  • What if we add 4 hours after 11:00? Continue reading
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