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pi + pi + pi + (.141592... of pi) ~= 9.8696 You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of

pi + pi + pi + (.141592... of pi) ~= 9.8696 You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.

bidirectional 6y ago
Pi is definitely not a fractional number... I don't think it's cheating at all, multiplication on the naturals is repeated addition, that's not the case for the reals.
- spacedcowboy 6y ago
  Interesting how computers (which only understand ‘1’ and ‘0’ can do multiplication of reals, then. Unless you’re intel, of course... (no, I will never let it go :)
- Cerium 6y ago
  Can we not consider that the algorithm taught for multiplication of real numbers is repeated multiplication of natural numbers which can be seen as repeated addition of natural numbers so we could define an addition only algorithm for multiplication of real numbers.
- horsawlarway 6y ago
  Sure it is, as long as you're willing to repeat in increments of real numbers. And that's my point, basically - By the time we're discussing real numbers, we need multiplication as an operator, because we're discussing ratios already.
- andrewprock 6y ago
  This is an interesting statement. While it's true, it obscures the fact that pi is defined as a fraction: Circumference/Diameter. That this fraction cannot be represented as a numeric fraction is one of the great insights of early mathematics.
  - DanielMcLaury 6y ago
    I wouldn't call the fact that pi is irrational an insight of "early mathematics." We knew that sqrt(2) was irrational around 500 BC if not earlier. We didn't know pi was irrational until around the time of the American revolution.
masswerk 6y ago
However, pi is already a ratio…
- horsawlarway 6y ago
  Right, that's why it's cheating. He's asking about a number set that requires the concept of ratios, and ratios are multiplication.
alentist 6y ago
It's not cheating. It's precisely the point. There are contexts where multiplication is not simply repeated addition.

bidirectional 6y ago

Pi is definitely not a fractional number... I don't think it's cheating at all, multiplication on the naturals is repeated addition, that's not the case for the reals.

spacedcowboy 6y ago
Interesting how computers (which only understand ‘1’ and ‘0’ can do multiplication of reals, then. Unless you’re intel, of course... (no, I will never let it go :)
Cerium 6y ago
Can we not consider that the algorithm taught for multiplication of real numbers is repeated multiplication of natural numbers which can be seen as repeated addition of natural numbers so we could define an addition only algorithm for multiplication of real numbers.
horsawlarway 6y ago
Sure it is, as long as you're willing to repeat in increments of real numbers. And that's my point, basically - By the time we're discussing real numbers, we need multiplication as an operator, because we're discussing ratios already.
andrewprock 6y ago
This is an interesting statement. While it's true, it obscures the fact that pi is defined as a fraction: Circumference/Diameter. That this fraction cannot be represented as a numeric fraction is one of the great insights of early mathematics.
- DanielMcLaury 6y ago
  I wouldn't call the fact that pi is irrational an insight of "early mathematics." We knew that sqrt(2) was irrational around 500 BC if not earlier. We didn't know pi was irrational until around the time of the American revolution.

masswerk 6y ago

However, pi is already a ratio…

horsawlarway 6y ago
Right, that's why it's cheating. He's asking about a number set that requires the concept of ratios, and ratios are multiplication.

alentist 6y ago

It's not cheating. It's precisely the point. There are contexts where multiplication is not simply repeated addition.