> How would you explain to a 10-year old why 3^0 = 1 beyond "it's necessary to make the algebra of powers work out".
Actually, that's exactly the reason 3^0=1: it was the definition that preserved the most identities. Agreed that this explanation doesn't really help intuition.
You add a bunch of stuff and you get a total. You add nothing and 0 is the total because it's the identity and starting point for addition. You multiply a bunch of stuff and you get a product. You multiply nothing and you get 1 because that's the identity for multiplication.
(That's the same as saying "the rules of algebra work out" but there's maybe something intuitive about multiplying nothing and getting back the thing that doesn't change the result of multiplication?)
Yeah -- that may have been the original motivation, but repeating it as an "explanation" reinforces the notion that math is a bunch of rules (vs. models you can construct and manipulate in your head).
0 probably started as a placeholder symbol for "naught", i.e. nothing to write, and the first scribes were taught "Just write a circle when you have nothing to report".
But, with greater understanding of numbers 0 evolved into its own entity and we saw numbers on a "line", a powerful mental model (why not 2d numbers? N-dimensional numbers? etc.)
Re-teaching that 3^0 = 1 "because the math is convenient" doesn't help us build a mental model of what exponents could be (I know you don't agree with this, just stating it again because the lack of intuitive explanations for math is a major pet peeve of mine).
the lack of intuitive explanations for math is a major pet peeve of mine
I'm a very visual thinker, and that is one reason I enjoy the new Art of Problem Solving textbook Prealgebra by Richard Rusczyk, David Patrick, and Ravi Boppana--
it is full of interesting visual "explanations" and substitute for proofs in a book intended for a young audience.
That said, I finally realized that I was limiting my mathematical development by insisting that every mathematical idea must appeal to my visual intuition. Some mathematical ideas are proven even if they don't appeal to visual intuition. In the words attributed to John von Neumann, "in mathematics you don't understand things. You just get used to them."
Thanks for the pointer! I found Needham's book awesome, I've barely made a dent in it but love the visualizations.
I don't think visualization is the only intuitive method -- you can have a general "sense", not sure how to put it more specifically -- I have a "sense" about growth of e without a specific diagram.
Agreed that not every concept can be understood... yet. There's a quote I love to rail on, in reference to Euler's formula:
"It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth." (Benjamin Peirce, 19th-century Mathematician)
Really? Yes, it may be baffling at first, but we can _never_ understand it? Only if that's our attitude :).
It's an amusing way to put this, but yes, it's true. An example of such situation is the case of continuum hypothesis. Both it and its negation has been proven not to lead to contradiction, so while most mathematicians ignore it, effectively treating it as being true, many set and model theorists play with things like Martin's axiom, which makes sense only if continuum hypothesis is false.
Actually, that's exactly the reason 3^0=1: it was the definition that preserved the most identities. Agreed that this explanation doesn't really help intuition.