Linked Questions

0 votes
2 answers
679 views

Dimensions (squared, cubed, and more!!) [duplicate]

I know that when you square something you can visualize it as a 2d square. When you cube it you can visualize it as a 3d cube. For example: 2^2 -- a 2 by 2 square 2^3 -- a 2 by 2 by 2 cube I've ...
Zachooz's user avatar
  • 169
1 vote
1 answer
117 views

The concept of an real irrational power [duplicate]

One can understand the concept of natural power, as $x^n$, being a product of a number by itself $n$ times: $$x^n=\underbrace{x\cdot x\cdot\dotsb\cdot x}_n$$ We can also get the idea of a rational ...
user avatar
-3 votes
1 answer
152 views

What is the general definition of exponentiation? [duplicate]

The definition of exponentiation is commonly defined as: $$x^n:=\underbrace{x\cdot x\cdot x\cdot ...\cdot x}_\text{n times}$$ when $n \in \mathbb N $. Given that, what is the definition when $n \notin ...
Noya Santiago's user avatar
225 votes
10 answers
13k views

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
David G's user avatar
  • 4,277
5 votes
6 answers
408 views

Why is exponentiation defined as $x^y=e^{\ln(x)\cdot y}$?

There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?
Christian Chapman's user avatar
7 votes
3 answers
1k views

Which general physical transformation to the number space does exponentiation represent?

Addition and multiplication may be defined in two ways, one specific and one general: Addition specific: addition is repeated incrementation. This is specific and sub-optimal as while $2 + 4$ is ...
user3105485's user avatar
3 votes
2 answers
958 views

Is there a deeper meaning when a number is squared? [closed]

In my opinion, math is about more than just memorizing equations, it's about numbers that are built in a way that represents our understanding of something. So I ask this, what does it mean ...
Klik's user avatar
  • 941
4 votes
5 answers
741 views

Trying to _really_ understand exponents...

I am a programmer, but grew up with a pretty weak math education. While I can cobble together enough understanding to build something like these charts solo from raw data, my level of true math ...
Kyle Baker's user avatar
3 votes
2 answers
118 views

Is $(-3)^\sqrt{2}$ a real number ? If it a complex number, then what is its $a+bi$ form?

Is this a real number? If it a complex number, then what is its $a+bi$ form? $$(-3)^\sqrt{2}$$
Angelo Mark's user avatar
  • 5,954
1 vote
3 answers
99 views

How do Mathematicians determine when to use the constant $e$?

I'm only an undergrad so sorry if this is a dumb question, but I was studying Poisson distribution and it struck me that so many models involve "e". So it got me wondering; how/when/where/why do they ...
Jeff's user avatar
  • 23
0 votes
0 answers
58 views

Derive the p-test if p is an irrational number

I would like some help with this problem. I've been given that we have a $k^p$ s.t. $p$ is irrational and $k$ is a positive integer satisfies the following: $k^r\lt k^p$ if $r$ is positive and ...
cambelot's user avatar
  • 2,411