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Put in the repeating decimal construction. This is accessible to junior high students.
*The "[[complex number]]s", which give solutions to polynomial equations
*The "[[complex number]]s", which give solutions to polynomial equations
Real numbers are typically represented by a decimal (or any other base) representation, as in 3.1416. It can be shown that any decimal representation that either terminates or gets into an endless repeating pattern is rational. The other numbers are real numbers that are ''irrational''. Examples are <math>\sqrt{10} = 3.162277660168...\,</math> and <math>\pi = 3.1415926535...\,</math>. These decimal representations neither repeat nor terminate.
Real numbers are typically represented by a decimal (or any other base) representation, as in 3.1416. It can be shown (see below) that any decimal representation that either terminates or gets into an endless repeating pattern is rational. The other numbers are real numbers that are ''irrational''. Examples are <math>\sqrt{10} = 3.162277660168...\,</math> and <math>\pi = 3.1415926535...\,</math>. These decimal representations neither repeat nor terminate.
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<!-- [This paragraph is all wrong, and doesn't belong here in any case. The topic should be covered in the compter arithmetic page, perhaps pointing here.] It is sometimes convenient to have a set of numbers that ''does'' include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 754-1985; this system is built in to modern processor chips. It provides for values which print out as INF and -INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or -INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."-->
<!-- [This paragraph is all wrong, and doesn't belong here in any case. The topic should be covered in the compter arithmetic page, perhaps pointing here.] It is sometimes convenient to have a set of numbers that ''does'' include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 754-1985; this system is built in to modern processor chips. It provides for values which print out as INF and -INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or -INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."-->
==How do we know that irrational numbers never get into an endless repetition of digits, or, equivalently, that any number that gets into an endless repetition is rational?==
Suppose we know that a number is .0039571428571428571428571428.... with the sequence "571428" repeating forever. (Astute readers will recognize instantly where this construction is going.) Take the number .99999, that is, with a number of "9" digits equal to the length of the repeating pattern. Take its reciprocal, that is (1/.999999). When working out the long division, it will be seen that the result is 1.000001000001000001... repeating forever. Multiply that by the pattern, getting 571428 * (1/.999999) = 571428.571428571428571428571428571428.... Multiply by the appropriate power of 10 (in this case 10^(-10) ) and add the necessary digits for the initial non-repeating part, in this case .0039. All of these operations were plain arithmetical operations, so the result is rational.
==Topological properties==
==Topological properties==