1.1 The rationals and the decimals

You learned about the rational numbers in high school. Rational numbers are nothing but fractions $\frac{m}{n}$, where $m$ and $n$ are integers. We should be a bit cautious, though… A rational number can be represented by a lot of fractions: $\frac{6}{10}=\frac{9}{15}=\frac{3}{5}=\frac{-3}{-5}$. We say that the rational number $q=\frac{m}{n}$ is in its lowest terms if $m$ and $n$ have no common factor.

So far, so good. We can add up two rational numbers:

We can even multiply two rationals numbers

and even divide a rational number by a non-zero rational number:

What else we can do with rational numbers? We can compare (or, in other word: order) them. Say, we have two positive rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ ($a,b,c,d>0$), then if , $\frac{a}{b}>\frac{c}{d}$ if $ad>bc$ (do not forget that $\frac{a}{b}=\frac{c}{d}$ if $ad=bc$).

Proof:  Let us suppose that $q^2=2$, $q=\frac{a}{b}$, where $q$ is in its lowest terms. Then

If $a$ is odd, then we get a contradiction immediately, since $a^2$ is odd and $2b^2$ is even. If $a$ is even, then $a^2$ can be divided by $4$. On the other hand, $b$ must be odd, since $q$ is in its lowest terms. So, $a^2\ne 2b^2$. $\mathrm{\square }$

You also learned about the decimals. Decimals are in the form of $x=x_0.x_1{x}_2{x}_3\mathrm{\dots }$, where $x_0$ is an integer number and all the $x_k$’s are digits in between $0$ and $9$. We will call $x_0$ the integer part of the decimal $x$ and we will call $x_k$ the $k$-th digit (note that in number theory, they call $x_0-1$ the integer part of a negative $x$).

Are the real numbers just the decimals?

Basically, yes. Clearly, there is nothing special in the number $10$, there are many ways to define the real numbers, even without referring to any kind of digit sequences. On the other hand, choosing the decimals to represent the real numbers seems to be a good idea. Finite decimals were first used in the XVI-th century by Vieta and Stevin, the infinite decimals were introduced by Newton. On the other hand, it was known by the ancient Greeks that the length of the diagonal of the unit square (that is $\sqrt{2}$) cannot be expressed by the ratio of two integers.

You learned long division in school. If I give you a rational number, say, $\frac{2}{7}$, you give me a decimal. This is called the decimal expression of the rational number.

Proof:  When we make a step of the long division to calculate $\frac{a}{b}$ we always get a remainder that is an integer in between $0$ and $a$. The next digit we calculate depends only on this remainder. Eventually, we will get a remainder that occured before. From this moment the digits must repeat forever. On the other hand, if we have a decimal expression like $x_0.x_1{x}_2\mathrm{\dots }{x}_r\overline{y_1{y}_2\mathrm{\dots }{y}_s}$, where $x_0$ is an integer number, then clearly $A=x_0.x_1{x}_2\mathrm{\dots }{x}_r$ and $B=0.y_1{y}_2\mathrm{\dots }{y}_s$ are rational numbers. It is easy to check that the decimal expression of the rational number $A+\frac{B}{{10}^r}\frac{{10}^s}{{10}^s-1}$ is exactly $x_0.x_1{x}_2\mathrm{\dots }{x}_r\overline{y_1{y}_2\mathrm{\dots }{y}_s}$. $\mathrm{\square }$

If we use the proposition to write $1.\overline{9}$ as a rational number, we get something interesting.

We need to make a convention: The decimal expressions $x_0.x_1{x}_2\mathrm{\dots }{x}_r\overline{9}$ and $x_0.x_1{x}_2\mathrm{\dots }y$, where $x_r\ne 9$ and $y=x_r+1$ represent the SAME rational number. So

$43.6\overline{9}=43.7$

$-0.\overline{9}=-1$

We end this section with a crucial definition.

We need to write the previous definition in a well, more mathematical form. What “closer and closer” means in English? It means that “eventually” the numbers will be as close as we wish. Suppose that we wish for being not further than ${10}^{-100}$. The word “eventually” means that if $n$ is large enough, then $|q_n-q|\le {10}^{-100}$. It does not mean that $|q_{100}-q|\le {10}^{-100}$, it only means that “after a while” $|q_n-q|\le {10}^{-100}$. So, there exists some number $N>0$ such that $|q_n-q|\le {10}^{-100}$ provided that $n\ge N$. The slight problem is that for different wishes, we might need different $N$. Imagine that the same $N$ would be good for all wishes. Then, not only $|q_n-q|\le {10}^{-1000}$ if $n\ge N$, but $|q_n-q|\le {10}^{-1000000}$ and even

That is, all the $q_n$’s must be equal to $q$, and that is too much to ask for. So, we need a trickier definition.