Math Is Fun Forum

aleclarsen12

Member

Registered: 2008-06-01

Posts: 36

irrational fractions

A couple of weeks ago, my math teacher was explaining that ONLY square root and logs could have irrational values and all irrational  values could be expressed with one of the 2. He also stated that no fractions are irrational in their native forum.

So I started thinking.... in therory.... pi is equel to circumpherance divided by diameter... isn't that a fraction... and isn't pi irrational.

BINGO!

Now, correct me if I wrong, but is my TEXT BOOK and MATH TEACHER wrong?

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Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,427

Re: irrational fractions

1. Not only square roots but nth roots of a number where n>2 and n is Natural number can are, mostly, Irrational numbers. For example,

etc.

2. A rational number or a fraction as defined in Arithmetic is a number of the form

and where p and q are integers.

is an irrational number, and it IS the ratio of circumference of a circle to its diameter. But do circumference and diameter belong to the set of integers?

There are some other irrational numbers too which are neither roots nor log values.
a) e, the natural logarithm base is an irration number. Its value is given by

b) The Liouville constant

which is  0.110001000000000000000001000...... approximately is also an irrational number.

The Golden ratio

is an irrational number too, but since it uses the square root, it is NOT an exception to what has been said to you.

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: irrational fractions

Now, correct me if I wrong

Very well, I shall correct you because you wrong. Either your teacher did not properly explain to you what a fraction is, or you didn’t pay attention in class.

By fraction, we usually mean a ratio of two integers. As long as we stick to ratios of integers, then fractions will always be rational.

In fact, ratios of rational numbers are also rational. However, if we take a ratio of two numbers at least one of which is not rational, then the ratio may not be rational. In the case of a circle, we can never find a circle whose circumference and diameter are both rational numbers. Their ratio is π and π is irrational.

Thus fractions are not just any ratios, but ratios of integers – or at any rate ratios of rational numbers (which can always be reduced to ratios of integers). Ratios of irrational numbers may be irrational or rational. In fact, since the real numbers are an extension field of the rational numbers, any nonzero real number can be expressed as the ratio of two irrational numbers. For instance:

Last edited by JaneFairfax (2009-01-04 01:34:50)