I’m guessing such a thing probably doesn’t exist, but I’ll try asking anyway.
Let’s call 23 shorthand, and 2x2x2 longhand. Of course the answer is 8.
20 is 1, but what’s the longhand for it? Also, 21 is 2, but what’s the longhand for that?
Just curious, thanks.
The longhand for 21 is 2. Not so sure on 20 though I suspect the answer is 2/2.
I’m pretty sure that the longhand for any non-zero number raised to the power of zero is simply 1 and that 00 is undefined.
Except that as the OP has defined it, the “longhand” is the long equation represented by a number raised by a power, written out in full, not the result. In other words, as the OP has defined it, the longhand for 22 is 2x2 not 4.
I’m out of my depth but not sure that 1 is the correct long equation that leads to the result of 1.
2^4 = 1*2*2*2*2 (four twos)
2^3 = 1*2*2*2 (three twos)
2^2 = 1*2*2 (two twos)
2^1 = 1*2 (one two)
2^0 = 1 (zero twos)
And man, Markdown made it really annoying to type this post. When I want an asterisk, I type an asterisk. When I want italics, I type a code for italics.
your “longhand” for 1 is the empty product (no factors)
While I see the logic, why is one entitled to assume the “1x”?
I mean, if I said “two times four times six” you would not write that as “1x2x4x6”. It would be just “2x4x6”.
Edited to add: Well I guess DPRK’s cite explains it - it’s a convention.
The point is, you can always multiply by 1, if you like, because it never changes anything. With, say, 1*2*2, you don’t need it, because you have other things you’re multiplying… but it doesn’t hurt. But with no 2s at all, as you noted, you don’t have anything to write… unless, of course, you include that 1 that you’re always allowed to include.
At some point, though, you have to fall back on the patterns. Like, what’s 2^-1? Well, we know the pattern that 2^(a-b) = 2^a / 2^b, and we can write -1 as (1-2), so 2^-1 = 2/4 = 1/2, and so on for all negative numbers. And what’s 2^(1/2)? Well, we know the pattern that 2^(a*b) = (2^a)^b, so (2^(1/2))^2 = 2, so 2^(1/2) must be the square root of 2. And so on for all rational numbers.
Sometimes, we write exponents because it’s shorter than writing them “the long way”, but sometimes, we write exponents because we can’t write the “the long way”.
It’s a convention, but in this case a very useful convention, since it means that x^(a+b) = x^a*x^b for all integer a and b, including if either or both is zero.
We could define x^0 to be something other than 1, but then a lot of theorems would have to have exceptions for zero exponents, which they don’t need to have with this convention.
I kinda like my way of doing 20 as the OP’s longhand because if:
n(a+b) = na x nb
and per the OP
22 longhand is 2x2
then 21 longhand is 2
and
2-1 longhand is 1/2
so 21-1 is 2/2 longhand which means that
20 longhand is 2/2
Thanks everyone, espcially Chronos and Princhester.
I don’t know if there’s anybody hear old enough to remeber adding machines. Those were mechanical devices that could add or substract a number from another.
The way they worked was that after clearing the machine you had three displays all zero. In one you selected the number you wanted to add and when you cranked the wheel the answer display had that same number and the count display showed 1.
If you wanted to add another number to the first you selected the new number and cranked the wheel and presto the answer display now had the sum and the count has number 2.
To substract a number you selected the number and cranked the wheel to the opposite direction and the result was in the display and the count was decremented by 1.
My mother was working in a bank and after the bank was closed she tallied out all the slips using this kind of machine.
There was shifting possibiltiy where you shifted your selected number and instead of adding it once you could add it ten times or hundred times etc. depending how far you shifted.
So to multiply with that machine you added up until the count had the other mutiplicant. For example if you wanted to multiply 123 by 789 you either keyed in 789 and cranket three times, shifted once, cranked two times, shifted once more and cranked once and you’d have your answer using two shifts and six cranks.
The other way round you’d key in 123 shifted four times and cranked then you’d shift back once and anti cranked twice, shifted back once and anti cranked once, shifted back once and anti cranked once. Eventually calkculating 123*(1000 -211) in four shifts and five cranks. One fastly learned which way was faster to calculate the multiplications.
Now imagine the same kind of machine but one that multiplies and divides. The count display would show the exponent and your empty display should be 1. That’s because when you key your number in and crank the display should show the multiplication of the old display with the new number. And that’s possible only if the empty multiplication is 1.
Really, everything in math is all conventions. We could create a new operation that works just like addition in every way, except that 1+1 = 3. One could even imagine a society that came up with that operation before addition, and considered it a fundamental operation.
Most of our conventions are chosen the way that they are because they’re obviously much simpler and more useful than any other convention. But in some cases, there are multiple competing notions for which convention is best. And in those cases, you can in fact get different kinds of math, using different conventions.
For instance, the usual convention is to say that infinity is not a number, and division by zero is undefined. But it seems awkward to have a number that you can’t divide by, so sometimes you instead see the convention that division by zero is defined, and is equal to infinity, which in this case is considered to be a number (and likewise, a finite number divided by infinity is equal to zero). But this then breaks other conventions, like the convention that every number must have an additive inverse, because if both 1+infinity and 2+infinity are both equal to infinity, then infinity - infinity could be either of them (or in fact, any other number). And so which convention you use depends on what’s more important to you, additive invertability or multiplicative invertability.
The OP is thinking of exponents as “shorthand” for repeated multiplication. Which is one perfectly legitimate way of thinking about it, but it only makes sense if the exponent is a whole number. So there are other ways of defining exponential expressions, that give the same value when the exponent is a whole number, but which also “work” if the exponent is some other kind of number, like a negative, irrational, or imaginary number.
Kind of like how, in some contexts, multiplication can be thought of as repeated addition. So you could say that 4 * 7 is “shorthand” for 7 + 7 + 7 + 7. But then what is 4.6 * 7.9 shorthand for?
What @Chronos and @DPRK’s link said, but in slightly different words:
If you think of 2^0 as meaning “multiply zero twos together”—that is, don’t multiply any of them—it makes sense as long as you keep in mind that, with multiplication, the “starting point” or “do-nothing number” is 1 (whereas for addition it’s 0).
Good points. Thank you.
I have an old college-level algebra book (published in 1948 I think) that very carefully defines xn (for positive integers n) to be the product of “x taken as a factor n times”.
Taken literally, this still doesn’t get you x0 (that comes later), but it does get you x1
Of course. How else would you do it? [With the caveat that taking an empty product to be 1 may also be regarded as notation.]
When you define 23 = 2 × 2 × 2, that is simply notation. If you want to define xy where x and y are arbitrary, possibly irrational, real numbers, you have some work to do, not to mention prove that this generalizes the original notation.
It would be very easy but careless and sloppy to think “xn means x multiplied by itself n times”. You (where “you” might mean a beginning algebra student) could end up getting confused over whether, e.g., 23 means:
2 x 2 x 2 x 2
(See? It’s 2 multiplied by itself once:
2 x 2
twice:
2 x 2 x 2
three times:
2 x 2 x 2 x 2 )
I think the language in the book was specifically phrased to prevent that thinking.
I took beginning algebra in 9th grade in 1965, but I had learned some basics long before that (having had a Big Brother who was a math major and somewhat of a genius at it, even to this day). I vaguely remember being a bit confused like this when I first began to learn about exponents.