Rational & Irrational Numbers

                                                                                        (Golden Ratio)

What are rational and irrational numbers?

   Rational numbers are numbers that can be written as a ratio of two intergers, as a fraction. For example, 2 is a rational number since 2 = 2/1. The fraction has to become a repeating or terminating decimal when divided. A terminating decimal number is not infinite, and ends at one time or another, such as 0.8, which is equivalent to 8/10. A repeating decimal is a number such as 0.888888888888......, which is equivalent to 8/9.

                                                The number line

We can think of a rational number as a distance from 0 along the number line.  For, we use the rational numbers for measuring.

   Irrational numbers are numbers that can't be turned into a fraction.  For example, is an irrational number. Most people think that would be rational since it is usually rounded to 3.14, yet the real value is infinite (3.14159......). Most square roots, cube roots and other roots are irrational numbers (but not all, for example the square root of 4 =2, which is a rational number).  Irrational numbers go on forever, and do not repeat.

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What are some more examples of rational and irrational numbers?

 Rational Numbers:                      

    Number            Fraction

Irrational Numbers:

Number	Decimal value	Notes
	1.4142135623730950488016887242097 (etc..)	Many, but not all, square roots are irrational numbers.
	3.1415926535897932384626433832795 (etc...)	Pi is a famous irrational number.
	2.7182818284590452353602874713527 (etc ...)	The number e (Euler's Number) is another famous irrational number.
√3	1.7320508075688772935274463415059 (etc...)	Since it is an irrational number, it cannot be expressed as a fraction.
	1.61803398874989484829... (keeps on going, without any pattern)	Many artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

Many buildings and works of art include the Golden Ratio in them,

such as the Parthenon in Greece.

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What is one way that rational and irrational numbers are similar?

   Rational and irrational numbers are real numbers, which means they can be expressed as a decimal. For example, 1/3 is a rational number, and in decimal form is 0.3333333333...(etc.) 23/47 is a rational number, and is equivalent to 0.489361702....(etc.) To find the decimal value of a rational number, you have to divide the numerator by the denominator.

   All irrational numbers are real numbers too. For example, the square root of 7 is an irrational number, and can be expressed as 2.645751311...(etc.)  Another example is the Golden Ratio, which is in decimal form 1.6180339887..(etc.)

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TEST

Here are a few numbers for you to decide if they are rational or irrational. The answers can be found at the bottom of the page, but try to answer the questions without looking first!

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  Answers to test

                                                                               1)Golden ratio= 1.6180339887...(etc.),                  IRRATIONAL

                                                                               2) 7= 7/1,                                                               RATIONAL

                                                                               3) Square root of 17= 4.123105626...(etc.),          IRRATIONAL

                                                                               4) Square root of 9= 3/1,                                       RATIONAL (not all square roots are irrational!)

                                                                               5) 2947/6948 = 0.424150834...(etc.),                   IRRATIONAL

                                                                               6) Square root of 456,798= 675.8683304...(etc.)  IRRATIONAL

                                                                               7) 1/3 = 0.33333333333333333 (repeating)         RATIONAL

                                                                               8) To what does the word "rational" refer?           The ratio of two natural numbers.

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 These links below are helpful to this subject!

      Links: http://www.regentsprep.org/Regents/math/rational/Lrat.htm

               http://www.regentsprep.org/Regents/math/rational/Prat.htm

               http://www.mathsisfun.com/irrational-numbers.htm 

               http://www.mathsisfun.com/numbers/golden-ratio.html

               http://www.themathpage.com/aReal/rational-numbers.htm