Monday, February 25, 2013

Solving Irrational

solving irrational numbers:

Irrational numbers in difference to rational numbers aren’t presented a fraction of the shape: m / n,  where m and n are integers. There are numbers of a new type, which are calculated with any accuracy, but can’t be changed by a rational number. They can appear as outcome of geometrical measurements,

for example:

- a ratio of a square diagonal duration to its side length is equal to √2.

- a ratio of a circumference duration to its diameter length is an irrational number  Pi



Where can you find special irrational numbers?(solving irrational)


The answer to this depends on what you consider 'special'.  Mathematicians have proved that definite special numbers are irrational, for example Pi and e.  The number e is the base of natural logarithms.   It is irrational, just like Pi, and has the value 2.718281828459045235306....

It's not easy to just "come up" with such special numbers.  But you can easily discover more irrational numbers after you've found that most square roots are irrational.  For example, what do you think of √2 + 1?  Is the result of that addition a rational or an irrational number?  How can you know?  What about other sums where you add one irrational number and one rational number, for example √5 + 1/4?

You can also add two irrational numbers, and the amount will be many times irrational.  Not always though; for example, e + (-e) = 0, and 0 is rational even though both e and -e are irrational.  Or, take 1 + √3 and 1 - √3 and add these two irrational numbers - what do you obtain?

Or, increase/separate an irrational number by a rational number, and you get an irrational number.  For example, √7/10000 is an irrational number that is moderately close to zero.  Yet another possibility to discover irrational numbers is to multiply square roots or other irrational numbers.  Sometimes that outcome in a rational number though (when?).  Mathematicians have also calculated what happens if you raise an irrational number to a rational or irrational power.

Yet more irrational numbers begin when you take logarithms, or calculate sines, cosines, and tangents.  They don't have any particular names, but are just called "sine of 70 degrees" or "base 10 logarithm of 5" etc.  Your calculator will give you decimal approximation to these.

Examples of Irrational Number


`Pi`,  √10 Are few examples of irrational numbers.

Solved Example on Irrational Number

Identify the irrational number.


Choices:
A. 0.3
B. 3 / 5
C. √6
D. √25
Correct Answer: C

Solution:
Step 1: The value of  is 2.4494897427831780981972840747059...
Step 2: The value of  is 5.
Step 3: The value of  is 0.6.
Step 4: Among the choices given, only  is an irrational number

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