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

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.

modulo-12

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. » Read more

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? » Read more
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