Can someone please explain exponents intuitively? I get what

>>9037754
Intuitively, [math]x^y[/math] is the number of functions from a y-object set to an x-object set.

For example, [math]2^3 = 8[/math] because there are 8 functions from {A, B, C} to {0, 1}, i.e. there are 8 ways to assign a number to each of the three letters.

More generally, if X and Y are sets, then [math]X^Y[/math] can be thought of as the set of all functions from Y to X. To convert between sets and numbers, simply take the cardinality of a given set to get the resulting number, and conversely given a number just take a set with that many elements.

Disclaimer: Not guaranteed to work with complex exponentiation, ordinal exponentiation, or exponents involving 0 or anything other than cardinal numbers for that matter.

>>9037754
It's literally just repeated multiplication. If you really need to visualize it, just look at a graph of x squared. The distance between each value is larger than previous.

What about them can you not imagine?

>>9037769
Well then how can you repeat multiplication in 2^i?

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3MB, 2500x1933px

>>9037896
>>9037754
https://www.youtube.com/watch?v=F_0yfvm0UoU

>>9037896
>Well then how can you repeat multiplication in 2^i?

It is pretty easy, actually. When it comes to complex numbers you should consider them in their polar form. In this form, exponents actually work as repeated multiplication for the magnitude of the number and then repeated addition for the angles.

It is really all the same with some extra meat mixed in.

>>9037754
ln(x) is the area of the function "f(a) = 1/a" from 1 to x, by definition.
exp(x) is the inverse function of ln(x), by definition.
a^x = exp(x ln(a)), by definition.
If you can't understand that, you're a brainlet. Sorry, OP.

>>9037896

This is something that happens a lot in math.

It's the same thing for binomial coefficients, which have been defined for arbitrary complex numbers including reals. Factorials, stirling numbers, multiplication, etc. have all gotten this treatment.

In math what you are doing is not based on consistency with the real world. It doesn't matter that there is no way to multiply 2 by itself i times. Math only cares about consistency within its own set of rules.

Those rules have been used to do unintuitive (but not illogical) things with the definitions of such mathematical constructs.

>>9037896
It's still repeated multiplication. You're multiplying 2 by itself an i number of times.

This is less intuitive, though, because our minds can't properly process imaginary/complex numbers.

>>9037957
If you're okay with multiplying 2 by i, I can't see how you could have a problem with 2^i.

>>9037754
Go on libgen.io (or buy) "Elements of Mathematics: From Euclid to Gödel" by John Stillwell.
http://press.princeton.edu/titles/10697.html

It's a mathematicians overview of elementary mathematics and will explain what you are looking for.