Lesson 2. Functions and Real Numbers

Functions: if you give me something, I give you something.

Again, we want to study calculus to maximize / minimize / measure something. For example, you suppose that you know how the price $p$ (in USD) of bread you made fluctuates with respect to the quantity $q$ (in number of loafs) you produce. For example, say when $q = 1$, we have $p = 100$. When $q = 2$, we have $p = 10$. More generally, suppose that 

$p = 100^{1/q}.$

The total revenue you gain is then the product of the price of each loaf and the number of loafs you sold, namely 

$p \cdot q = 100^{1/q}q.$

Notice that when you decide the number $q$ of loafs of bread you sell, the total revenue you gain is determined. That is, when you produce $2$ loafs, you will obtain $10$ dollars by putting $q = 2$ in the right-hand side of above. This is what we call a function. There are many ways to write a function. My favorite way is to write 

$q \mapsto 100^{1/q}q.$

Very often, you would want to name your function, say $R$ for ``revenue'' and write 

$R(q) = 100^{1/q}q.$

You say the revenue function "$R$ is a function in quantity $q$". Notice that the price function $p$ was a function in quantity $q$ as well. More explicitly, we could write 

$p(q) = 100^{1/q}.$ 

Now, here is something nice to tell your parents when they inquire what you learn at school: 

"Mathematics is the study of functions.''

Real numbers: they are real, but you may not write them how you want

For calculus, we use real numbers. That is, we use numbers in our real lives. For example, we use integers 

$\dots, -2, -1, 0, 1, 2, \cdots.$

We use rational numbers (or synonymously, "ratios"), which are written as fractions of integers such as $5/2, 100/3,$ or $-20/999$. We also use irrational numbers, but unlike rational numbers, it is a bit difficult to wrap our heads around irrational numbers, which is probably why we call them "irrational" in the first place. The easiest irrational number can be given as a solution to the equation 

$x^{2} = 2.$

That is, no rational number squares to $2$. This is one of the moments where mathematicians "legally cheat". That is, we imagine such a number that can be a solution to the above equation in $x$, and call it 

$x = \sqrt{2}.$

By our imagination, such a number must satisfy 

$(\sqrt{2})^{2} = 2.$

You might say this seems fishy. You should ask: "How do you construct such a number?" Here is one trick: draw a triangle two of whose edges have length $1$ and require the triangle to have the right angle between the two edges of the same length. Then the length of the hypotenuse of the triangle is the number you want!

We now know that our imagination of such a number $\sqrt{2}$ is actually a reality, but we still feel like we are missing something. How do we write this other than this weird symbol above? Let's do some experiments. We have 

$1.4^{2} = 1.96,$

which is quite close to $2$, so it is tempting to write

"$1.4 = \sqrt{2}$",

but this is NOT true! We wanted something that squares to $2$, NOT $1.96$. No matter how you learn, one of the most important reasons you use mathematics is to be correct. The difference or error you get is $2 - 1.96 = 0.04$, so you might say that's small. However, if $2$ means "$2$ billion dollars", how much is $0.04$? I will let you answer this question.

Still, you can say $1.4$ is an estimation of $\sqrt{2}$, albeit not being equal to it. Unfortunately, it is impossible to write a decimal expression for $\sqrt{2},$ and again, that's why it is called "irrational".


Another example of irrational number is: $\pi.$

What is it? Before the "correct" answer, we should all remember it is NOT equal to $3.14$. What is it then? Well, remember how we constructed $\sqrt{2}?$

We are going to say $\pi$ is the number that is half of the circumference of the circle whose radius is $1$. It turns out that you cannot write $\pi$ as an any decimal expression. This is quite bizarre, isn't it? (Google this if you want!) Our familiar $3.14$ is just an estimation of $\pi$.

To wrap up, we have learned that a function (you give me something, I give you something back) is and what real numbers are. Some of the real numbers such as integers and rational numbers were familiar to us, while irrational numbers were esoteric. This is because we are so used to decimal expressions of numbers! Since you cannot write irrational numbers in a terminating decimal, it just looks more difficult. You do not need to know so much about irrationality unless you need it for a very particular use, so don't worry!