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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
is irrational
Given below is Pythagoras's proof of the existence of irrational numbers using the ![[square root of two]](sqrt2_2.gif)
Simplifying Radicals
The symbol ![]() |
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.
Statements | Reasons |
---|---|
![]() | Proof 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=2c | even can be factored into 2 and another number even (2) times anything is even |
a2 = a·a = 4c2 = 2b2 | Substitution of equals into product (twice) |
2c2= b2 | Division 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:
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](numb0904_2.gif)
Compare the next two examples and notice how they differ. Both methods are correct. Choose the one which saves you the most time.
Note when the radicals have different root indexes:
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]](sqrt2_2.gif)
![[square root of three]](sqrt3_2.gif)
![[radical expression 10](numb0910_2.gif)
![[radical expression 11](numb0911_2.gif)
![[radical expression 12](numb0912_2.gif)
Extracting Roots
The![[square root of two]](sqrt2_2.gif)
![[square root of two]](sqrt2_2.gif)
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![]() |
- Separate the number into groups of two digits going each way from the decimal point.
- Estimate the largest square which will go into the first group.
- This number goes both in the normal divisor's location for long division and above the first group as in long division.
- Double this digit and bring it down for the next step (see example below).
- Also bring down the next group of digits as in long division.
- Estimate how many times the two digit number formed using this doubled digit and the number of times...will go into the number.
- 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.
?.??????______________ | ||
? | / | 2. 00 00 00 00 00 00 |
Find an integer that squared goes into 2:
1_______________ | ||
1 | / | 2. 00 00 00 00 00 00 |