Pi is Rational (Circumference/Diameter)
Date: Summer 2014
My mathematics teacher says pi is irrational, but in my text book it is mentioned that pi is
equal to the ratio of the circumference and the diameter of a circle. This should make pi rational (a ratio).
How is this possible as an irrational number cannot be represented by a ratio?
Even rearranging the formula for the circumference of a circle we get
22/7 is a recurring number and also is not irrational.
For a number to be rational, it must be expressible as a ratio of two integers. There are no two integers whose ratio is exactly equal to pi. 22/7 is only an approximation. For that matter, there are other pairs of integers whose ratios are closer to the true value of pi, but they are larger than 22 and 7.
Richard E. Barrans Jr., Ph.D., M.Ed.
Department of Physics and Astronomy
Not all ratios are rational numbers, only ratios of integers.
Knowing that PI is irrational, if the diameter of a circle is an integer (in some units), then the circumference is not an integer in those units.
A ratio is a comparison between two numbers or quantities.
If the ratio can be written using integers, no matter how large, it is rational. The ratio 11,234,544 /45,673,234,432 is rational. The decimal you get from dividing the two numbers either ends or repeats.
Irrational ratios, like Pi, cannot be represented by two integers, one divided by the other. The decimal neither ends nor repeats.
So ratios can take any form and can be irrational or rational.
Hope this helps.
A number is rational only if it can be expressed as a ration of two INTEGERS. If the diameter of a circle is rational (such as 37/100 meters), then the circumference of the circle will not be a rational quantity. We cannot measure an irrational length exactly. We can only find an approximation that will be extremely close to pi times the measured diameter.
Dr. Ken Mellendorf
Illinois Central College
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