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Numbers and Their Application - Lesson 10

Beyond Rationality

Lesson Overview

Irrational Numbers

It was widely believed that all numbers were rational, expressible as the ratio of two integers, until the Pythagorean school (around 500 B.C.) discovered otherwise. Today, such numbers are called irrational numbers. Since then irrational has become an adjective meaning lacking normal logical clarity! The [square root of two] was the first irrational number discovered. It is the solution to the simple problem x2 = 2.

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, indicating what the root is. When no number appears, 2 meaning square root is assumed.

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

StatementsReasons
[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. ( 9 is 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 fraction by 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 method was 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_______________
1/2. 00 00 00 00 00 00

Double the quotient and bring down to be the divisor. Another digit will follow.
1.?_______________
1 / 2. 00 00 00 00 00 00
1
2? /1 00

Find the number,?, so that 2? will go into 100. (We find that it is 4: 24•4>100>25•5)
1. 4_ ______________
1/2. 00 00 00 00 00 00
1
24/1 00
96
4
We continue to repeat the steps: double the quotient and find the last digit until we get the precision we need.
1.    4   1__________
1 / 2. 00 00 00 00 00 00
1
24 / 1 00
96
281 4 00
281
1 19
How long would it take you to verify for accuracy the following level of precision?

[square root of two] = 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799 07324 78462 10703 88503 87534 32764 15727 35013 84623 ...

Golden Ratio

Another curious irrational number is [phi is half the sum of 1 and the square root of 5] [approximately equal to] 1.618... and his partner [phi prime is half the difference of the square root of 5 and 1] [approximately equal to] 0.618.... These are known as the Golden Ratio and symbolized by Ø, the Greek letter capital phi. Notice how things like 3"×5" cards often assume these proportions. Notice also how ratios of consecutive Fibonnaci numbers approach the Golden Ratio as seen in Homework 7. The Golden Ratio is also one of the roots of the quadratic equation x2 - x - 1 = 0. If you change the 2's in the continued fraction given in lesson 7, to 1's, you will have yet another representation!

Ø= 1.61803 39887 49894 84820 45868 34365 63811 77203 09180 ...

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