Real number
A real number is a rational or irrational number. Usually when people say "number" they usually mean "real number". The official symbol for real numbers is a bold R or a blackboard bold Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbb{R}} .
Some real numbers are called positive. A positive number is "bigger than zero". You can think of the real numbers as an infinitely long ruler. There is a mark for zero and every other number, in order of size. Unlike a ruler, there are numbers below zero. These are called negative real numbers. Negative numbers are "smaller than zero". They are like a mirror image of the positive numbers, except they are given minus signs (–) so that they are labeled differently from the positive numbers.
There are infinitely many real numbers. There is no smallest or biggest real number. No matter how many real numbers are counted, there are always more which need to be counted. There are no empty spaces between real numbers. This means that if two different real numbers are taken, there will always be a third real number between them, no matter how close together the first two numbers are.
If a positive number is added to another positive number, that number gets bigger. Zero is also a real number. If zero is added to a number, that number does not change. If a negative number is added to another number, that number gets smaller.
The real numbers are uncountable. That means that there is no way to put all the real numbers into a sequence. Any sequence of real numbers will miss out a real number, even if the sequence is infinite. This makes the real numbers special. Even though there are infinitely many real numbers and infinitely many integers, we can say that there are "more" real numbers than integers because the integers are countable and the real numbers are uncountable.
Some simpler number systems are inside the real numbers. For example, the rational numbers and integers are all in the real numbers. There are also more complicated number systems than the real numbers, such as the complex numbers. Every real number is a complex number, but not every complex number is a real number.
Different types of real numbers
There are different types of real numbers. Sometimes all the real numbers are not talked about at once. Sometimes only special, smaller sets of them are talked about. These sets have special names. They are:
- Natural numbers: These are real numbers that have no decimal and are bigger than zero.
- Whole numbers: These are positive real numbers that have no decimals, and also zero. Natural numbers are also whole numbers.
- Integers: These are real numbers that have no decimals. These include both positive and negative numbers. Whole numbers are also integers.
- Rational numbers: These are real numbers that can be written down as fractions of integers. Integers are also rational numbers.
- Transcendental numbers cannot be obtained by solving an equation with integer components.
- Irrational numbers: These are real numbers that can not be written as a fraction of integers. Transcendental numbers are also irrational.
The number 0 (zero) is special. Sometimes it is taken as part of the subset to be considered, and at other times it is not. It is the Identity element for addition and subtraction. That means that adding or subtracting zero does not change the original number. For multiplication and division, the identity element is 1.
One real number that is not rational is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \sqrt{2}} . This number is irrational. If a square is drawn with sides that are one unit long, the length of the line between its opposite corners will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \sqrt{2}} .
Proof that the square root of 2 is not rational
The number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \sqrt{2}} is not rational. Here is the proof.
- Assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \sqrt{2}} is rational. So there are some numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a, b} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a/b=\sqrt{2}} .
- We can choose a and b so that either a or b is odd. If a and b were both even, then the fraction could be simplified (for example, instead of writing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \frac{2}{4}} , we could write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \frac{1}{2}} instead).
- If both sides of the equation are squared, then we get a^{2} / b^{2} = 2 and a^{2} = 2 b^{2}.
- The right side is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle 2b^2} . This number is even. So the left side must be even too. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a^2} is even. If an odd number is squared, then an odd number will be the result. And if an even number is squared, an even number would be the result too. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a} is even.
- Because a is even, it can be written as: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a=2k} .
- The equation from the step 3 is used. We get 2b^{2} = (2k)^{2}
- An exponentiation rule can be used (see the article) – the result is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle 2b^2=4k^2} .
- Both sides are divided by 2. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle b^2=2k^2} . This means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle b} is even.
- In step 2, we said that a is odd or b is odd. But in step 4, it was said that a is even, and in step 7, it was said that b is even. If the assumption we made in step 1 is true, then all these other things have to be true, but since they disagree with each other they can not all be true; that means that our assumption is not true.
It is not true that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \sqrt{2}} is a rational number. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \sqrt{2}} is irrational.