I am a person who exists in the observable universe who has access to a computer, and that's all the information about me that I'm going to put here. For the rest of this profile page, I would talk about why tau is a better circle constant than pi, explain derivatives, or explain logarithms. However, all of those things would require pictures, so I will explain why everyone should count in base twelve instead of base ten, along with properties of times tables that I have noticed when working with both of these systems of counting.
Base ten or the decimal system is how most people count numbers today. In the decimal system, there are unique symbols for numbers zero to nine and after nine it goes to the "tens place," so the number after nine is 10. In base twelve, also known as the duodecimal or dozenal system, there are unique systems for numbers zero to eleven and the number after eleven is represented as 10.
Decimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Duodecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 10 (There are lots of proposed symbols for ten and eleven in a duodecimal system, but I'm just going to use A for ten and B for eleven.)
Why Duodecimal is betterEdit
The number ten has two factors: two and five. In the decimal system the multiples of two five follow a very easy pattern to remember. Two's multiplies always end in either 0, 2, 4, 6 or 8 while five's factors always end in either 0 or 5. Fractions in the decimal system always terminate as long as the denominator factors only into 2 and 5. One fifth is 0.2 and one fourth is 0.25. One third in the decimal system is 0.33333... because three is not a factor of ten.
Twelve on the other hand is a highly composite number, which means it has more factors than any number smaller than it. Its factors are two, three, four, and six. Because twelve has so many factors, there are a lot more regular patterns in its times table than in decimal. For example:
multiples of 2: 2, 4, 6, 8, A, 10, 12, 14, 16, 18, 1A, 20
multiples of 3: 3, 6, 9, 10, 13, 16, 19, 20, 23, 26, 29, 30
multiples of 4: 4, 8, 10, 14, 18, 20, 24, 28, 30, 34, 38, 40
multiples of 6: 6, 10, 16, 20, 26, 30, 36, 40, 46, 50, 56, 60
multiples of 8: 8, 14, 20, 28, 34, 40, 48, 54, 60, 68, 74, 80
multiples of 9: 9, 16, 23, 30, 39, 46, 53, 60, 69, 76, 83, 90
As you can clearly see, there are more regular patterns in duodecimal than decimal. Fractions in duodecimal also terminate more often. one third is 0.4, one sixth is 0.2, one ninth is 0.14, and one twelfth is 0.1. Fractions don't terminate in duodecimal if the denominator is a multiple of five, and neither the decimal or duodecimal faction terminate if the denominator factors into seven.
If you use a metric ruler, you should be able to see why the duodecimal system is better. A metric ruler can divide its units into halves, and then into fifths. Being able to divide things into fifths is not as useful as dividing things into thirds or into fourths. This is actually why I hate using meters instead of feet because I can divide a foot into thirds, halves, and halves again, while in metric I can divide things into halves and I can't even remember a time where I needed to divide into fifths using a ruler.
Keep in mind that this only changes how numbers are written, not the value of the numbers themselves. Pi is still the circumference of a circle divided by the diameter; it's just written differently. The rules for addition and subtraction remain the same, along with the rules for multiplication and division. Here's a video that also argues for the duodecimal system: https://www.youtube.com/watch?v=U6xJfP7-HCc
Other Patterns in Times TablesEdit
While I was writing a duodecimal times table by hand, along with a decimal times table, I noticed some patterns that relate to what base a numerical system is using.
- Almost any pattern that a number n in which n is less than the base follows will be inherited by numbers with n in the one's place.
- All numbers will always follow a regular pattern of numbers in the ones place that end with zero.
- Factors of the base will have patterns that are shorter than non factors
- Numbers whose value in the one's place doesn't factor into the base evenly will cycle through every possible one's place value until reaching zero.
- The number that equals the base-1 will count down in the one's place until reaching 0.
- If the base is even, then the number that equals the base-2 will count down the even numbers until reaching 0.
- If you add all of the digits of a number x and the sum of those digits is a multiple of the number n in which n = base - 1, then x is a multiple of n. This rule is not inherited by numbers with n in the one's place.
- Any factor y of the number n in which n = base - 1 will inherit the previous rule and any multiples of y will partially inherit this rule. If the number x satisfies the previous rule for a number z in which z shares a factor of n, then x must be divisible by z's other factors in order to be a multiple of z.
Stephen Hawking's A Briefer History of Time.
Brilliant Blunders by Mario Livio