Difference between revisions of "Real numbers"
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(oww, the pain, the pain. this is so bad. will try to clean it up. PAIN!) |
(It's for teenagers, Joshua, teenagers. I hope this effort only causes you a splitting headache, not a migraine level.) |
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| − | Real numbers can be thought of as numbers which can be represented by some infinite decimal representation. | + | Real numbers can be thought of as numbers which can be represented by some finite or infinite decimal representation. |
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| + | In classical physics, measurements of things that can vary smoothly and continuously, like speed or temperature, are treated as real numbers. | ||
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| + | In a number line representation, the real numbers correspond to all the points on a geometric line. The distance between any two points on a line is a real number. | ||
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| + | The real numbers include within them all of these other kinds of numbers: | ||
| + | |||
| + | *The "natural numbers" or positive integers, 1, 2, 3, ... | ||
| + | *Zero and the negative integers | ||
| + | *Fractions, like 355/113 | ||
| + | *Any decimal representation which terminates (comes to an end), like 6.023, because this is just a way of writing a fraction (in this case, 6023/1000) | ||
| + | *Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractions | ||
| + | *Irrational numbers, like &sqrt;10 = 3.162277660168... π = 3.1415926525..., whose decimal representations never repeat or terminate. | ||
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| + | Formally, real numbers are defined as the unique [[Field (mathematics)|field]] which is [[Ordered]], [[Complete (mathematics)|Complete]], and [[Archimedean]]. The reals can be constructed from the rationals by means of [[Dedekind cuts]] or [[Cauchy Sequences]]. | ||
| + | |||
| + | ==Notes and references== | ||
| + | <references/> | ||
Revision as of 16:00, February 24, 2007
Real numbers can be thought of as numbers which can be represented by some finite or infinite decimal representation.
In classical physics, measurements of things that can vary smoothly and continuously, like speed or temperature, are treated as real numbers.
In a number line representation, the real numbers correspond to all the points on a geometric line. The distance between any two points on a line is a real number.
The real numbers include within them all of these other kinds of numbers:
- The "natural numbers" or positive integers, 1, 2, 3, ...
- Zero and the negative integers
- Fractions, like 355/113
- Any decimal representation which terminates (comes to an end), like 6.023, because this is just a way of writing a fraction (in this case, 6023/1000)
- Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractions
- Irrational numbers, like &sqrt;10 = 3.162277660168... π = 3.1415926525..., whose decimal representations never repeat or terminate.
Formally, real numbers are defined as the unique field which is Ordered, Complete, and Archimedean. The reals can be constructed from the rationals by means of Dedekind cuts or Cauchy Sequences.