What is $x^y$? How to understand it?

$x+y=z$

I have a pen. He has a pen. Total is two pen. This is plus.

$x-y=z$

I had two pens. A pen was lost. So, I have a pen. Total remaining is one. This is minus.

$x\cdot y=z$

I have two pens. Three other friends have two pens. $4 \text{(persons)}\cdot 2 \text{(pens/person)}=8 \text{pens}$. This is multiply.

$x^y=z$

How to understand this?

-
How can I understand $x\cdot y=-1$? –  please stop stalking Sep 28 '12 at 8:47
By seeing enough examples. –  Berci Sep 28 '12 at 9:41
I tried to edit this question but my edit was rejected. In my opinion, you should not use $i$ in this expression, but should use $z$ instead. $i$ is reserved for the square root of $-1$. This might seem extremely picky, but imagine if instead of $x + y = i$ you wrote $e + \pi=i$. –  Matt Calhoun Sep 29 '12 at 16:11
@MattCalhoun I used the character ' i ' because javascript programmer declare mostly variable as i. For example, var i=1; That's why I used i instead of z. However, z is better. –  Spacez Ly Wang Oct 1 '12 at 4:19

I have one pen. I become really angry and I break it into $x$ pieces. Each time I get angry I break each of my pieces into $x$ even tinier pieces. I become angry $y$ times. I end up with $x^y$ pieces.

-
I like this! $\,$ –  Brian M. Scott Sep 28 '12 at 6:28
It's not exactly the best analogy but he was using pens so I had to think of something =P –  Christopher A. Wong Sep 28 '12 at 6:28
Thank you, sir. Brian M. Scott explain clearer than you. Don't be angry. –  Spacez Ly Wang Sep 28 '12 at 6:29
you are one very angry mathematician! :P –  Belgi Sep 28 '12 at 6:42
Too bad we started with pens. If we were talking about chalk instead, my scenario above would basically be describing a daily problem for me. –  Christopher A. Wong Sep 28 '12 at 7:10

I have two friend. my friend has two more friend each which doesn't know each other, and they have 4 pen each.

-

Yesterday I could not sleep. Then I imagined a pen, which could write, moving itself on the paper. That pen invented a story and wrote it down x times. So in my half-asleep imagination I had to read x stories. For one I was already too tired, but the other x-1 stories went as follows:

Yesterday I could not sleep. Then I imagined a pen, which could write, moving itself on the paper. That pen invented a story and wrote it down x times. So in my half-asleep imagination I had to read x stories. For one I was already too tired, but the other x-1 stories went as follows:

Yesterday I could not sleep. Then I imagined a pen, which could write, moving itself on the paper. That pen invented a story and wrote it down x times. So in my half-asleep imagination I had to read x stories. For one I was already too tired, but the other x-1 stories went as follows:

(...)

...zzz...

Today's morning I awoke with a strange feeling... having x^y stories read, didn't I...

-
Thank you for explanation. –  Spacez Ly Wang Oct 1 '12 at 12:36

I have a bacteria living on my pen. After a minute the bacteria splits on X microbacterias. After a minute they grow up and each of them splits on X microbacterias. After Y minutes I get $x^y$ bacterias on my pen.

-

I have a pen and a replicator device. The device is set to make $x$ copies (including the original, which gets grouped together with $x-1$ copies). I put my pen into the replicator device and collect all the pens that it produces. Then I run all of my pens through at once and collect the results. I do $y$ rounds of replication. I end up with $x^y$ pens.

-

Exponential Growth and the Legend of Paal Paysam

Exponential Growth is an immensely powerful concept. To help us grasp it better let us use an ancient Indian chess legend as an example.

The legend goes that the tradition of serving Paal Paysam to visiting pilgrims started after a game of chess between the local king and the lord Krishna himself. (picture of 18th century Miniature of Lord Krishna playing Chess against Radha from National Museum, New Delhi)

The king was a big chess enthusiast and had the habit of challenging wise visitors to a game of chess. One day a traveling sage was challenged by the king. To motivate his opponent the king offered any reward that the sage could name. The sage modestly asked just for a few grains of rice in the following manner: the king was to put a single grain of rice on the first chess square and double it on every consequent one.

Having lost the game and being a man of his word the king ordered a bag of rice to be brought to the chess board. Then he started placing rice grains according to the arrangement: 1 grain on the first square, 2 on the second, 4 on the third, 8 on the fourth and so on:

Following the exponential growth of the rice payment the king quickly realized that he was unable to fulfill his promise because on the twentieth square the king would have had to put 1,000,000 grains of rice. On the fortieth square the king would have had to put 1,000,000,000 grains of rice. And, finally on the sixty fourth square the king would have had to put more than 18,000,000,000,000,000,000 grains of rice which is equal to about 210 billion tons and is allegedly sufficient to cover the whole territory of India with a meter thick layer of rice. At ten grains of rice per square inch, the above amount requires rice fields covering twice the surface area of the Earth, oceans included.

It was at that point that the lord Krishna revealed his true identity to the king and told him that he doesn't have to pay the debt immediately but can do so over time. That is why to this day visiting pilgrims are still feasting on Paal Paysam and the king's debt to lord Krishna is still being repaid.

-
Thanks for sharing. –  Spacez Ly Wang Oct 1 '12 at 4:23

I want to give many pens away. I start my campaign asking $x$ of my friends to recruit $x$ friends each, so I have $x\cdot x=x^2$ recruits now. Then I ask each of them to recruit in turn $x$ friends, and so on. After $y$ steps I have: $x\cdot x \cdot x \ldots$, or $x^y$ recruits, so I need $x^y$ pens.

Just a detail: in this case I am discarding the original set of friends after they recruit other friends. If I kept them the result would be $x^y+x^\left(y-1\right)+\ldots$, which is a bit bigger.

Observe that I am starting what is essentially a social campaign, and the result is exponential. That is why viral campaigns grow exponentially. You might also wonder why exponential growth ever stops: if each people that receives a chain mail resends it to $n$ other people, how come the chain ever stops? There are at least a couple of answers: sometimes the population is saturated, since each friend cannot find $x$ other people who aren't already "infected" by the viral campaign. Sometimes (e.g. when there is no way to find out if new recruits are "infected") the chain just never stops: there are chain mails doing the rounds on the internet for many years.

Incidentally, true viral infections (such as the common cold) follow the same pattern: initially there are $x$ people infected, they each infect $x$ other people before they are cured, and the result is $x^2$ people infected. As time passes this grows to $x^y$. That is why social interactions that grow exponentially are called "viral". In this case saturation is reached because infected people are immunized (or just die). There are also occasions when the virus mutates (such as the common cold) and reinfects people; these have been doing the rounds also for... all of our history!

-
Thanks for sharing. It's also combination. –  Spacez Ly Wang Oct 1 '12 at 4:24
1. I have an alphabet of $x$ letters. How many $y$-letter combinations can I make? Answer: $x^y$ For example, with the 26 letter English alphabet we can make $26^5$ five-letter combinations, or 11881376.

2. Multiplication of integers is repeated addition. The notation $3\times 4$ means $3 + 3 + 3 + 3$. In this expansion, the number four appears in an encoded way: the number of times that $3$ occurs. When we introduce variables, wanting to express the multiplication of $x$ and $y$, we cannot write down $x + x \ldots + x$ such at $x$ occurs $y$ times because $y$ is abstract, and not a concrete number. We can only use the $x\times y$ notation, in which $y$ appears a symbol, rather than encoded into a number of repeated additions. So now, what is exponentiation of integers? It is repeated multiplication. $3^4 = 3\times 3\times 3\times 3$. What if variables are involved? What if we want to express that $x$ is multiplied by $x$, such that $x$ appears $y$ times? Again, we cannot write the expansion down because $y$ is not a concrete number. But the $x^y$ notation handles this situation.

3. With pens: I have a stationery store which sells $x$ different kinds of pens. $y$ customers come in, one by one, and each makes a selection of one of the $x$ kinds of pens and buys one. In how many possible ways can these sales happen, with regard to the type and sequence of pens sold?

-

Suppose you have a chessboard. On the first square you place a single pen. On each square after that you double the pens compared to the previous square. So, if you take the first square as zero, how many pens will you have on square $n$? The answer is $2^n$.

This formula is quite useful for any form of relative growth. Suppose you have $\$1,000$in the bank at$0.06%$interest rate. The money you have after one year will be equal to$\$1,000 * 1.06$. If we rewrite this to $\$1,000 * 1.06^1$a pattern emerges. The money you started with is equal to$\$1,000 * 1.06^0$. So after $n$ years you will have $\$1,000 * 1.06^n$. - I planted a tree in my yard. When I saw it the morning, it grow y times. The next day when I saw it, it grew another y times. The next day it grew yet another y times. So if its original height was 2m and it grew by 2 meters the first morning. Its height on 2nd and 3rd day was original height = 2m first day height = x^y = 2^2 = 4 m, growth = 2m second day height = 4^2 = 8m, growth = 4m 3rd day height= 8^2 = 16m, growth = 8m  In other words this is an exponential function in where the growth increase with the passage of time, again a linear function in which the grow rate is contact. It would grow 2m each day. - I have k friends. Each friend has k friends and each of my friends'friends has k friends and so on (good representation is of course a tree) and my friends'friends'friends..(n times)..friend has k pens. - Not a very good analogy - each of your friends probably has friends in common.$x^y$only works as a strict upper bound in that case. – fluffy Sep 28 '12 at 15:26 Ahah, true if you consider smallworld theories and such things, but using your imagination this example is ok! moreover let's take the pen example, which I consider very good. Probably if you don't have a really long pen the breakings won't be noticeable at all. BUT using your imagination it is possible to assume that you can (easily) break the pen in lots of pieces – Gabber Sep 28 '12 at 15:35 Each pen gives you x eggs of which next pens are born... Oh, okay... You have X pens by brand A. There brand B set a shop which have an advertising action. They exchange old "A" pen for X new brand "B" pens. But well, brand A also set a shop, where they replace old "B" pens for X new brand "A" pens. You took your X pens and start shuffling between the shops. After Y visits, how many pens (regardless of brand) would you have ? But typically powers used when breeding and generations are taking place. Like neutrons in atomic bombs or power stations. Like pigs and flies. Somethign like that: a hen (each one) while living is giving you eggs, of which X would be hens again and the rest are cocks. After that hen is old and no more can produce eggs. Probably you would eat it. You start with X hens and after Y generations how many hens would there be ? - I didn't know that pens were oviparous. – JavaMan Jan 10 '13 at 9:16 I have$y$pens of different colours, and$x$friends. In how many ways can I give each pen to some friend (without any qualms about equity, I could give them all to one of the friends if I feel like)? Answer: in$x^y$ways, since for each pen I have$x$possibilities, and the choices are independent (so the numbers for the individual pens must be multiplied together). -$2\cdot 4$: $$\underbrace{2+2+2+2}_{4\text{ twos}}$$$2^4\$: $$\underbrace{2\cdot 2\cdot 2\cdot2}_{4\text{ twos}}$$

-
Thank you, sir. Now I understand. –  Spacez Ly Wang Sep 28 '12 at 6:28
@SpacezLyWang: You’re welcome. –  Brian M. Scott Sep 28 '12 at 6:31
Interesting, considering that it is only thanks to Mathematics that I know there is an argument about whether multiplication 'is' repeated addition (for suitable meanings of 'is'...) –  AakashM Sep 28 '12 at 8:39
That's just one interpretation. One can think of it as the motivation for the concept of multiplication of whole numbers, and then multiplication of all other types of numbers as generalizations of that concept, but not necessarily the motivation. –  acjay Sep 28 '12 at 9:12
The pattern goes on: en.wikipedia.org/wiki/Tetration –  Chris Sep 28 '12 at 16:13