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Irrational numbers are real numbers that cannot be expressed as the ratio of two integers.

Common irrational numbers are nonrepeating and nonterminating decimals.These include the roots of any prime and most radicals.

Pythagoras's Proof that the [square root of two] is irrational

Given below is Pythagoras's proof of the existence of irrational numbers using the [square root of two] as an example.

Simplifying Radicals

The symbol , is called a radical. The number underneath the symbol is the radicand. n is the root index, indicatingwhat the root is. When no number appears, 2 meaning square root is assumed.

[square root of two] canalso be written as 2½. In general, xa/b means the bth root of xa.Such rational exponents still follow the rules given inlesson 4.

[square root of two]= a/bProof by contradiction: assume true what we are proving false
2=a2/b2   2b2=a2 Square both sides (expressions remain equal)
a and b have no common factors assumed without loss of generality:
a/b represents reduced fraction
If a is odd, a2 is odd,
but 2b2 is clearly even, a contradiction
odd times odd is odd, a cannot be both even and odd simultaneously.
If a is even, let a=2ceven can be factored into 2 and another number
even (2) times anything is even
a2 = a·a = 4c2 = 2b2Substitution of equals into product (twice)
2c2= b2Division Property of Equality
So b is even; hence a,b have the common factor 2, a contradiction.Q.E.D.   (quod erat demonstrandum:
    Latin for which was to be proved.)

When simplifying radicals, break the radicand into factors of perfect squares, cubes, etc. ( 9is the perfect square of 3, 4 is the perfect square of 2, 27 is the cube of 3). Separate the factors into separate radicals. Then express the roots of the radicals with perfect squares, cubes...For example:

[radical expression 1

[radical expression 2

[radical expression 3

Multiplying Radicals

When multiplying radicals, multiply the radicands of like root indexes and then simplify the product. Usually, the easiest way is to simplify as you go along so that you don't end up with large products to factor. Examples:
[radical expression 4

[radical expression 5

[radical expression 6

Compare the next two examples and notice how they differ. Both methods are correct. Choose the one which saves you the most time.
[radical expression 7

[radical expression 8

Note when the radicals have different root indexes:
[radical expression 9

Rationalizing Denominators

Common practice is to simplify expressions to get rid of radicals in the denominator of fractions. Historically, this was all but necessary before calculators.(Imagine dividing [square root of two] by the[square root of three] by long division!)In order to rationalize the demoninator, the common practice of multiplying by one is used. One comes in many forms: anything divided by itself is one. So multiply the fractionby the square root that is in the denominator over itself.For example:
[radical expression 10
[radical expression 11
[radical expression 12

Extracting Roots

The [square root of two] can be approximated on your calculator. Before calculators were developed, the following methodwas widely taught and used. It is based on Newton's Method which will be taught in calculus.Since the decimal representation of [square root of two]goes on forever without terminating or repeating,calculators can only give you a fairly precise decimal approximation.

Whenever you use the decimal approximation of a radical, you should note that it is an approximation and not exact by the use of the symbol[approximately equal to].

  1. Separate the number into groups of two digits going each way from the decimal point.
  2. Estimate the largest square which will go into the first group.
  3. This number goes both in the normal divisor's location for long division and above the first group as in long division.
  4. Double this digit and bring it down for the next step (see example below).
  5. Also bring down the next group of digits as in long division.
  6. Estimate how many times the two digit number formed using this doubled digit and the number of times...will go into the number.
  7. Repeat steps 4-6 above, but now the number down will be 2, 3, 4 digits,etc. Continue until the desired accuracy is achieved.

Example of extracting root 2.

Step 1:
?/2. 00 00 00 00 00 00

Find an integer that squared goes into 2:

1/2. 00 00 00 00 00 00