Chapter 1 Real Numbers

Somehow we take the real numbers for granted. “Real numbers are just points of the real lines”. We can add and multiply them exactly the same way as the integers, don’t we? Well, try to square the $\pi$. Even, if you want to add $\pi$ and $\frac{1}{9}$, you can get into trouble. You imagine (and that is actually a good thing) $\pi$ as an infinite decimal. You can consider $\frac{1}{9}$ as an infinite decimal $0.\overline{1}$. When you add two decimals you first find the “left end” of them and start there. These guys have no “left ends”. You face the same difficulty if you try to multiply $\pi$ by $2$, you want to start the multiplication on the “left hand”. Squaring the $\pi$ means infinitely many multiplications (none of them seem to be doable) and then somehow adding up infinitely many infinite decimals. Dividing to infinite decimals is even worse. The fact of the matter is that our addition, multiplication, division algorithms work only for integers and for some neat rationals. The goals of this chapter is:

• •

  Define the addition and multiplication of the real numbers.

• •

  Show that these operations behave “nicely” e.g. $(a+b)c=ac+bc$.

Before getting into details let us see the basic idea. OK, we cannot square $\pi$. But, what is $\pi$? It starts like that:

$3.141592653589793238462643383279502884197169399375105820974944592307816406286\mathrm{\dots }$

$3^2=9$, ${(3.1)}^2=9.61$, ${(3.14)}^2=9.8596$, ${(3.141)}^2=9.865881$, ${(3.1415)}^2=9.86902225$
${(3.14159)}^2=9.8695877281$, ${(3.141592)}^2=9.86960029446$, ${(3.1415926)}^2=9.86960406437$.

Since ${(3.1415927)}^2=9.86960469269$, you can be sure that if we continue to square the “starting parts” of the $\pi$ you will be always in between $9.8696040$ and $9.8696047$. Also, you can hope that “after a while” all the squares above will start with the same $10000$ numbers.

Yes, it means that if you square the “starting parts” of the $\pi$ they are getting closer and closer to each other and hopefully, they will be closer to closer to a decimal, that we can call the square of the $\pi$ even if we cannot apply the standard procedure of multiplication.

“After a while”. “Closer and closer”. We need to “mathematize” these expressions and also, we need to learn how to “see”, how to visualize mathematical formulas about the real numbers.

There is another important issue: we know that there is no rational number $q$ such that $q^2=2$, why are we so sure that there is a decimal number $r$ for which $r^2=2$? Why does $\sqrt{2}$ actually exist?

You learned about exponentiation, you can work with the function $x\to 2^x$ you learned about the properties such as $2^x\times 2^y=2^{(x+y)}$, but it is based on belief. Why does the function $2^x$ really exist?