Patterns and Algebra
How Base Twelve Works & Why It Matters
Different Counting Systems
Base Ten (the one you are probably familiar with)
The Base Ten system of counting, is the one most people use in modern society. It probably evolved this way because most people have ten fingers. We group things into groups of ten because we find that easiest to count. We use ten distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. After 9, we move 1 to the tens place and count from there; 10, 11, 12, 13, 14, until we reach 19 and then we change the digit in the tens place to 2 and continue. Eventually we reach 99 and have to start using the hundreds place and follow the same pattern to infinity. This all seems rather obvious, but it has to be stated in order to explain other counting systems.
The Abacus, which was used in ancient times in places like China, is an excellent tool for understanding place value. Once all beads in a row are used, they are pushed back and a bead on the next row is used to represent that set of objects.
Binary
Binary is an alternative counting system used by computer programmers. It uses base two, and so only requires two digits: 0 and 1. After 1, we start using the twos place. and continue thusly: 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111. Instead of a ten's place, hundred's place, and so on, we have a two's place, a four's place, and eight's place and so on.
Challenge: How many numbers can you write in binary? How many Binary numbers can you translate back to Base Ten?
Hexadecimal
The Hexadecimal system of counting is used by computer programmers as a more user-friendly way to denote binary. In this system, numbers up to 15 have unique symbols (letter A-F denoting numbers ten-fifteen). Only when the numbers reach sixteen does it start to use double digits. 10 represents sixteen. The same way you look at 56 and think "five tens plus six", if you were trained to think in hexadecimal, you would think "five sixteens and 6". B3 would mean eleven sixteens and 3. Programmers us this, rather than Base Ten, because it more naturally aligns with binary numbers. Each place value in binary is double the one after it, this doubling pattern from two does not include a ten but does include sixteen. If we were trained to think in Hexadecimal or Binary, translating to Base Ten would no longer be necessary. However, translating between different number bases can be a good exercise in practicing multiplication and addition.
Challenge: practice multiplication by translating numbers from Hexadecimal to Base Ten. For example to find F2 in Base Ten, you must multiply fifteen sixteens and add 2.
Base Twelve
Drawing upon the concepts explained above, you should now be able to understand how Base Twelve would work. Ten and Eleven would each have their own unique symbols and 10 would denote twelve. 20 would denote two twelves (24 in Base Ten) and 100 would denote twelve twelves (144 in Base Ten). You would look at a number like 36 and think "three twelves plus six". It is a little hard to imagine because we are so accustomed to Base Ten, but if you grew up learning this way it would come naturally.
Challenge: What number do you think would be a good base to use for counting? Draw a chart translating numbers to your chosen base. Write down reasons you think it would be a good system. What could it be ideally used for in real life?
These files contain a table of the numbers 1-100 (1-144) in Base Ten compared to Base Twelve. Every second column is Base Twelve while the other columns show the number equivalents in Base Ten. I have used X to denote ten and # to denote eleven.
If Base Twelve Were The Norm
It can be fun to imagine a world where Base Twelve is the system people are used to instead of Base Ten. Imagine society evolved to use a Base Twelve system as our primary counting system. Let us look at some ways life would be different.
How would we count using our fingers?
We could use the segments in our fingers, as pictured here, to count up to twelve. Like an abacus, we could use the other hand to keep track of each group of twelve and count all the way up to Twelve Twelves
Challenge: Try counting to 144 on your fingers. See how far you can get without mistakes.
How Would Multiplication And Division Change?
If we were trained to think in Base Twelve, we would think differently about multiplication and division. As it is, we use Ten as a base. Ten only has two numbers (excluding one) that it divides evenly into: Two and Five. This makes learning multiples of Five and Ten easy, we just take each grouping of ten and cut it in two. Our brains quickly calculate "5x5= two tens plus five". If we used Base Twelve, Five would no longer be so easy to deal with, instead Six, Three, and Four would be. We would think of 6 in the way we currently think of Five. We would also think of Three as fitting into the base grouping four times, and Four fitting in three times. We would be able to quickly multiply 4x7 by thinking "two groups of twelve plus four" (24 in Base Twelve, 28 in Base 10). Overall, I believe students would memorize more multiplication facts faster.
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The way we think about dividing large numbers into parts would change too. We would use the symbol 100 to denote what Base Ten calls 144. 60 would denote half of that. 40 would denote a third of 100. 30 would denote a quarter of 100. In my opinion, a cleaner set of divisions than our current 100, 50 (half), 33.33 (one third), 25 (one quarter). In Base Twelve, these divisions come out to round numbers ending in 0.
Currently, 120 divides neatly into 60, 40, and 30. In Base Twelve, 120 becomes X0. Now, by using ten twelves, we run into the messiness of Base Ten. Half of X0 would be 50, one third would be 34, and one quarter would be 26. Messier than our divisions of 100 in Base Twelve, but neater than when we did 100 in Base Ten. Notice these notations are very similar to what we had with dividing up Base Ten 100, yet one third has gone from a messy decimal to a whole number. In my opinion, Base Twelve makes for cleaner equations.
In my opinion, people would find math easier if Base Twelve were the norm.
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Challenge: Find some equations you have done using Base Ten and try to do them in Base Twelve. If we use Base Ten 8x8=64. In Base Twelve, 8x8 would be denoted as 54. It would be the same quantity, but represented differently.
How Would We Think Differently About Fractions, Decimals, and Percentages?
Fractions are when we take a whole and divide it into parts. Decimals are another way of representing part of an object. When using decimals, we cut 1 into ten parts, hence the root word "dec". The fraction 1/4 translates to 0.25. 1/2 translates to 0.5. 1/20 translates to 0.05. 1/5 translates to 0.2 1/3 translates to the unwieldy 0.333333.... repeated to infinity. Base Twelve would change all that. We would be accustomed to dividing 1 into twelve parts instead. Some fractions would become harder to express as decimals, while others would become easier. 1/6 would easily translate to 0.2. 1/3 would now be the easy 0.4. 1/4 would be 0.3. It all feels a little strange, but if you grew up used to Base Twelve, it would come easy. I believe that overall, more decimals would come easier. Percentages are a way of dividing a whole into a hundred parts. In this hypothetical, percentages would be out of the number that is 144 in Base Ten but 100 in Base Twelve. It would theoretically lead to slightly more precise percentages, as there would be more room to maneuver. What we currently call 25%, would be called 30% with thirty-six (30 in Base Twelve) percentage points in between to express different percentages. As you can see, more percentage points would exist within each part and more precision would be possible.
Challenge: make a chart translating percentages in Base Ten to percentages in Base Twelve. Remember, it will come out differently than translating regular numbers.
Would Measurements Change?
It is hard to speculate on what would and wouldn't change about how we measure things. The evolution of how society measures things is complex.
Let us speculate on a few types measurements and how or if they would be different.
Length and Distance
Traditional units of length measurements are based on what was useful to people at different points in history. A foot is approximately a pace, also called a footstep. A mile is eight furlongs, a furlong being the distance an oxen could travel without rest. Due to these units being based on things in nature, I believe they would still be the same regardless. However, the metric system is based around Base Ten. The belief behind the metric system is that ten is the easiest number for converting units because to multiply or divide by ten requires merely adding or taking away 0s. In Base Twelve, twelve would be the number that you would multiply with by simply adding 0s. Therefore, the metric system would probably be based on twelve. A meter, derived using Base Ten may have ended up being smaller or larger. The number of centimetres in a decimetre would probably be twelve, the same as the number of inches in a foot.
Time
How about measuring time? Well, the way we measure time is already based on numbers like twelve and sixty because they divide nicely into a lot of other numbers. Therefore, I think we would still use them for keeping time. However, sixty would be called fifty. It would still be the same quantity and divide into twelve groups of five, so I imagine our clocks wouldn't be much different than they are. We would also probably still have twelve months in a year and two sets of twelve hours in a day. It would probably feel even more natural due to twelve being a number we base counting on.
Temperature
Both the Celsius and Fahrenheit systems of measuring temperature were derived by choosing a temperature to label as 0 and another temperature to label as 100, and then cutting the scale in between into 100 equal parts. I suspect that if Base Twelve were the norm, they would have been derived in a similar manner. This would mean that, instead of what we currently think of as 100, the number of degrees in between the two anchor points would be what we currently call 144. This would make temperature readings a bit more precise, as there would be more points along the line.
Money
Most modern systems of money are based on the number 100. 100 pence to a pound in Britain. 100 cents to a dollar in places like Canada and the United States. One cent is 1% of a dollar. I suspect, if Base Twelve were the norm, a dollar would be divided into what we currently call 144 pieces. I suspect we would have 30 cent coins (the same value as our current 25 cent coins AKA quarters) and we might also have 40 cent coins, representing one third of a dollar. As mentioned above with percentages, we would have more precise prices. Possibly too precise. As it is, one cent coins are barely used. In Canada they are no longer used at all. In our hypothetical Base Twelve dollar, one cent is worth even less. I suspect we would not use one cent coins. Our smallest might be 6 cents (the same value as our current 5 cent coins), or maybe twelve cents (the same value as our current ten cent coins), or maybe even bigger.
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It is interesting to note that some monetary systems in history use different numbers. Britain once had a system where 12 pence equalled one shilling and 20 shillings equalled one pound. A penny could also be divided into 4 farthings. Since 12 divides cleanly into 2, 3 ,4, and 6 and 20 divides cleanly into 2, 4, 5, and 10, combining them made for many clean divisions. Furthermore, a pound plus a shilling equalled one guinea, meaning a guinea was 21 shillings, which divides nicely into 3 and 7.
Challenge: What are some ways you can confuse your parents by talking in Base Twelve in everyday life?
Challenge: think of something else we measure (weight, mass, volume, music, angles, acceleration, intelligence, etc...). Research the history of why it is measured the way it is and speculate how or if units of measurement would be different in a world where Base Twelve is the norm.
Challenge: What do you think of the older system of British currency? Write a persuasive essay on why it is better or worse than the currency system you currently use.
Extension: design your own ideal currency system, write a persuasive essay on why your system would be better.
What Is The Value Of Understanding Base Twelve?
Although Base Twelve arguably has advantages over Base Ten, society is not going to switch over. We are set in our ways. It would be near impossible to retrain the entire population to think in Base Twelve. So why discuss it? Well, here are some reasons I think the knowledge can help us understand math better.
Firstly, I think it is good to know that there are alternatives to the way we do things. Remembering that Base Ten is a social construct enables us to separate a number's inherent quantity for the digits used to represent it and think of numbers on a more metacognitive level.
The One-Third Paradox
When we convert 1/3 to a decimal, we get 0.333.... going on to infinity. When we add together three thirds, we get 0.9999....going on to infinity. What happened to the other piece? In fact, it has been proven multiple ways that 0.999999...... is equal to 1. This intuitively makes no sense. That is until we remember that the limitations of Base Ten are artificial. Had we happened to evolve with two extra fingers, we would be using 0.4 to represent 1/3. Suddenly, there is no paradox, just a limitation in how we denote the quantity of 1/3.
Understanding Fractions
As a tween, I managed to teach myself an important lesson about fractions, using a twelve piece chocolate bar. The pieces were arranged in three rows of four. This arrangement is perfect for helping learn about fractions. It can be cut into thirds, or quarters, or perhaps sixths. Each row is a third and each column a quarter. The visual/tactile aid helped me realize how one third of three quarters is 3/12 and that three quarters of a third is also 3/12. At the time it was a big revelation. Now I not only knew, but deeply understood how, multiplying fractions in a different order makes for the same result. I find twelve to be an optimal number of pieces to learn that lesson. Cutting one whole into twelve pieces is perfect for understanding fractions involving quarters and thirds at the same time. It is a perfect number for understanding things like the difference between one third and one quarter is 1/12. in short, one twelfth is a beautiful number for exploring relationships between halves, thirds, quarters, and sixths.
Challenge: What other concepts in math do you think you can improve your understanding of, using the number twelve?
Challenge: Do you think people would understand math better if Base Twelve were the norm? Write a persuasive essay on why or why not.