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# Swapping two variables

## Question

When you are programming, you will write something like

x = ___ 
y = ___

z = x # store value of x in z
x = y # put value of y in x
y = z # restore value of x in y

to swap the values of two variables `x` and `y` . What is you are out of memory and cannot declare a new variable `z` , are you able to do it?

## Experiments

You need: [handout](https://www.math.nsysu.edu.tw/~chlin/math-runway/swapping-variables.pdf)

1. Pick two positive integers from $0 \sim 511$ as `x` and `y` .
2. Follow the steps on the handout to keep track of `x` and `y` .
3. Think about why this work.
4. Think about how to use row operations to obtain $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ from $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ without swapping.

## Intuition

Although we say there are three rules for row operations, but **Rule 1** can actually be replaced by a sequence of **Rule 2** and **Rule 3** . Using this idea, one may obtain the algorithm for swapping two variables.

## More questions to think about

1. In C language, an `int` variable can only handle integers from $-32768 \sim 32767$. Adding big numbers, as we did in the algorithm, can potentially cause integer overflow. How can we deal with this issue?

## Resources

1. [YouTube: Field Definition (expanded) - Abstract Algebra by Socratica](https://youtu.be/KCSZ4QhOw0I?si=vncCBX7rw_jmKlyk)