2101c45386
Added more content, showing why the rationals are nice, and why they are interesting (and of use!) to study
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Rational Numbers
A rational number is a real number that can be written as a fraction of two integers.
10/5 is a fraction, and is equal to 2. 8/3 is a fraction, and is equal to 2.66666.... Since the denominator of a fraction can be 1, every integer is a rational number. Note that as 2/1 = 4/2 = 6/3 = ..., each rational number can be represented by different fractions, but every rational number has a unique representation as an irreducible fraction.
The decimal representation of a rational number either terminates or is infinite with a pattern that repeats, e.g., 1/2 = 0.5 or 1/3 = 0.333.... Note that the decimal representation -- similar to a fractional representation -- is not necessarily unique, e.g., 0.999... = 1.
Not all real numbers are rational. A real number that is not rational is called irrational, such as sqrt(2), pi and e. While it is easy to see a real number is rational (either express it as a fraction of integers or compute the decimal representation until it repeats) it is usually a lot more difficult to show a number is not rational.
The rational numbers have many nice properties. For example,
- Addition of any two rational numbers
a/b,c/dgives another rational number(ad + bc)/(bd); - Multiplication of any two rational numbers
a/b,c/dgives another rational number(ac)/(bd); - Any rational number
qhas an additive inverse,-q; - Any non-zero rational number
qhas a multiplicative inverse,1/q; - Between any two distinct rational numbers there exists another rational number (e.g., the mean of the two rationals).
- Between any two distinct rational numbers there exists an irrational number (e.g., for almost any pair of rationals, the square root of their product is irrational).
- Between any two distinct irrational numbers there exists a rational number (e.g., look at the decimal representations of both irrational numbers and pick a finite decimal between them).
- There are ''as many'' rational numbers as there are positive integers!
Properties 1-4 give way to the rational numbers being a particularly nice mathematical structure called a field. Properties 5-7 show that the rationals are a dense subset of the real numbers, making them quite useful for studying the set of all real numbers. Property 8 states the rational numbers are countable.
The study of algebraic structures over the rational numbers instead of the real numbers is a very rich area of investigation including Arithmetic Geometry, Arithmetic Dynamics, Cryptography, Galois Theory, Number Theory, and more. There are also multiple $1 million prizes for solving some of the deeper and most challenging problems involving the rational, so while they are well understood, there is still quite a lot unknown.