Note that we use the colon: In base 10, each digit can stand on its own. Here, the modulus is 10 and each remainder is simply equal to the last decimal digit of the number involved. And each position is 10 more than the one before it. Usually, the overbar is not allowed to cover any digits to the left of the decimal point but this usage need not be respected. Try Base 16 If we want base 16, we could do something similar: Strangely enough, these two are very strongly related see below for an explanation.

We may wonder what powers of 10 are products of two integers without any zero digits. This is one reason digital signals are so resilient to noise. From the rightmost digit the units , you subtract the 4th rightmost the thousands , add the 7th, subtract the 10th, add the 13th, etc. The first number is the ninth power of 2, the second is the ninth power of 5. Fractions and irrational numbers. We always add and never subtract. This is pretty cool, right? Recall that a number is divisible by 3 or 9 iff the sum of its digits is. So 16 in hex is: This is an example of what's called modular arithmetic: The "quick tests" of divisibility by 7 or 13 are somewhat more complex. Let's investigate a slightly less trivial question: The result is divisible by 7 if and only if the original number was. It was uphill both ways, through the snow and blazing heat. OR Form the alternating sum of blocks of three from right to left. In our number system, we use position in a similar way. Do the same thing starting with the second rightmost digit the tens , subtract the 5th, add the 8th, subtract the 11th, etc. Enter zero And what happens when we reach ten? If the result is divisible by 7 then the given number is divisible by 7. If the result is divisible by 11 the number is divisible by 11 the equivalent of divisibility by eleven in decimal. Prime numbers In this section, all numbers are written with duodecimal A natural number i. We may summarize both of the above approaches for either product thusly: It's remarkable that the same test works for both 7 and 13! With different divisors, only the coefficient applied to the last digit changes. The other "divisibilty tests" presented below are based on similar considerations. And each position is 10 more than the one before it. Modular arithmetic is, again, the key:

### Video about dozenal system:

## Why 12 is the new 10 - An introduction to the Dozenal system

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