I like base 12 a lot, but Reverse Polish Notation is a mess when you get up to working with polynomials.
With polynomials, you’re moving around terms on either side of an equation, and you combine positive terms and negative terms. In essence, there’s no such thing as subtraction. (Similarly, division is a lie; you’re actually just working with numerators and denominators.)
Reverse Polish Notation makes that a mess since it separates the sign from its term.
Also, RPN draws a distinction between negative values and subtraction, but conceptually there is no subtraction with polynomials, it’s all just negative terms. (Or negating a polynomial to get its additive inverse.)
But, yeah. It’s a shame we don’t use base 12 more.
You would have numbers 0-9 plus two more digits (could use A and B, but any two symbols will do). The advantage is being able to cut things in thirds and quarters as well as in half. Cutting by a sixth is a bonus, as well.
RPN (reverse polish notation) is a different way of doing arithmetic where the order you write it naturally determines the order of operations. Do you know those Facebook memes where people get different answers for seemingly simple math equations? RPN does away with that. There is one and only one right way to interpret an RPN equation, and you don’t have to remember any order of operation rules to do it.
Take the simple expression 1 + 1. The plus sign is between the two operands; this is called infix notation and it’s what you’re probably familiar with. If we wanted to make this more complex, e.g. 1+(2*3), we’d need parentheses to say which part was supposed to be done first.
Reverse Polish Notation (RPN) means you write the two operands and then the operator, i.e. 1 1 +. Writing more complex equations is as simple as putting another expression in place of one of the numbers: 1 2 3 * +. To see how you’d evaluate this without any parentheses, imagine that as you go through an RPN expression from left to right, you keep a stack of sheets of paper, each with a number written on it, that starts out empty. As you go through, you see:
1, put that on top of the stack
2, put that on top of the stack
3, put that on top of the stack (from top to bottom, the stack is now 3, 2, 1)
multiply, take the top two numbers on the stack (2 and 3), multiply them, and put the result back on the stack (the stack now has two numbers, 6 on the top and 1 on the bottom)
add, take the top two numbers on the stack, add them, and put the result back (the stack now only has one number on it: 7)
If we wanted to rewrite that expression to be (1+2)*3 instead, we could write: 1 2 + 3 *
Simply by reordering the symbols, we change the meaning of the expression, so there’s never any need for parentheses.
As a bonus, this method of writing equations is a lot easier for computers to parse than infix notation, since they think in terms of stacks anyway. They can be (and often are) programmed to parse infix notation anyway, because infix notation is a lot easier for humans to wrap their brains around, but it’s much easier to program them to interpret RPN which is why a lot of older calculators and software (like the programming language FORTH) use RPN exclusively.
What base 12 gives you is a lot of common divisors: 2, 3, 4, and 6. Base 10 only has 2 and 5. Base 16 only has 2, 4, 8.
The practical upshot of this is that you can divide things evenly in more ways. Particularly when wanting to divide a board into thirds. Having 12 inches to a foot is actually helpful there, though it falls apart as soon as you get larger.
Base 16 is great when you’re interacting with a computer, but aside from that, not much. Only being divisible by 2 is kind of a pain in the real world.
Both are easily countable on fingers using your thumb and counting up the segments for base 12 and adding the pads for base sixteen. You can reasonably count to 144(gross) or 256 using both hands to create a two digit dozenal or hexadecimal number.
Yes. Also everyone should be required to learn how to use a slide rule before they ever get given a calculator - I think that seeing how the numbers relate to each other on a physical device can help students conceptualize them better.
The metric system should be redone in base 12, and RPN should be the norm for teaching arithmetic.
Why base 12?
See elsewhere in the thread, but basically because of the ease of dividing whole numbers.
twelve phalanges makes four digits, use the thumb to count. Also prettier.
I like base 12 a lot, but Reverse Polish Notation is a mess when you get up to working with polynomials.
With polynomials, you’re moving around terms on either side of an equation, and you combine positive terms and negative terms. In essence, there’s no such thing as subtraction. (Similarly, division is a lie; you’re actually just working with numerators and denominators.)
Reverse Polish Notation makes that a mess since it separates the sign from its term.
Also, RPN draws a distinction between negative values and subtraction, but conceptually there is no subtraction with polynomials, it’s all just negative terms. (Or negating a polynomial to get its additive inverse.)
But, yeah. It’s a shame we don’t use base 12 more.
That’s super interesting. I adore RPN on caclulators and had never heard any drawbacks well-articulated.
RPN is a great way to type things into computers – it’s easier for the computer to parse, too – but it kind of sucks for writing abstract math.
I’ve been fully on board with base 12 for years. Didn’t know about RPN until this post. I’m convinced.
Base 60 was good enough the Babylonians and it’s good enough for me!
RPN is a gateway to LISP
What’s rpn and how does base 12 work.
Don’t you still have “numbers” between 11 and 12?
You would have numbers 0-9 plus two more digits (could use A and B, but any two symbols will do). The advantage is being able to cut things in thirds and quarters as well as in half. Cutting by a sixth is a bonus, as well.
RPN (reverse polish notation) is a different way of doing arithmetic where the order you write it naturally determines the order of operations. Do you know those Facebook memes where people get different answers for seemingly simple math equations? RPN does away with that. There is one and only one right way to interpret an RPN equation, and you don’t have to remember any order of operation rules to do it.
Here’s a practical explanation of RPN.
Take the simple expression
1 + 1. The plus sign is between the two operands; this is called infix notation and it’s what you’re probably familiar with. If we wanted to make this more complex, e.g.1+(2*3), we’d need parentheses to say which part was supposed to be done first.Reverse Polish Notation (RPN) means you write the two operands and then the operator, i.e.
1 1 +. Writing more complex equations is as simple as putting another expression in place of one of the numbers:1 2 3 * +. To see how you’d evaluate this without any parentheses, imagine that as you go through an RPN expression from left to right, you keep a stack of sheets of paper, each with a number written on it, that starts out empty. As you go through, you see:If we wanted to rewrite that expression to be (1+2)*3 instead, we could write:
1 2 + 3 *Simply by reordering the symbols, we change the meaning of the expression, so there’s never any need for parentheses.
As a bonus, this method of writing equations is a lot easier for computers to parse than infix notation, since they think in terms of stacks anyway. They can be (and often are) programmed to parse infix notation anyway, because infix notation is a lot easier for humans to wrap their brains around, but it’s much easier to program them to interpret RPN which is why a lot of older calculators and software (like the programming language FORTH) use RPN exclusively.
Let me know if that explanation made sense.
Base 16 is superior and once you learn binary math, easier to divide and multiply.
What base 12 gives you is a lot of common divisors: 2, 3, 4, and 6. Base 10 only has 2 and 5. Base 16 only has 2, 4, 8.
The practical upshot of this is that you can divide things evenly in more ways. Particularly when wanting to divide a board into thirds. Having 12 inches to a foot is actually helpful there, though it falls apart as soon as you get larger.
This is incorrect, and you don’t understand why base 12 is useful. However for binary operations, hex is great. But not for general counting.
Base 16 is great when you’re interacting with a computer, but aside from that, not much. Only being divisible by 2 is kind of a pain in the real world.
I’m not experienced with RPN but at a glance think there’s a solid argument for it.
Both are easily countable on fingers using your thumb and counting up the segments for base 12 and adding the pads for base sixteen. You can reasonably count to 144(gross) or 256 using both hands to create a two digit dozenal or hexadecimal number.
Good points, 12 seems to be superior and I’ve changed my opinion.
Yes. Also everyone should be required to learn how to use a slide rule before they ever get given a calculator - I think that seeing how the numbers relate to each other on a physical device can help students conceptualize them better.
If we were supposed to use base 10, we’d have 10 figures and 10 toes!
/s
You have 12 finger joints which you can count with the thumb.
I have 2 balls so binary is more in line with my interests.
Yes, BIPM, this user here.
woah RPN is awesome