Hi. I'm a seventh grade teacher. I have a question about Pi. If Pi is defined as a ratio of circumference/diameter of a circle how can it be an irrational number? The definition has Pi defined as a rational number but Pi is used as an example of an irrational number. What am I missing?

Thanks- Shelley Collier

Hi Shelley,

Numbers that can be written in the form p/q where p and q are integers, (q not 0), are known as rational numbers. What you are missing is that p and q must be integers. The fact that Pi is irrational means that you can't have a circle with both the circumference and diameter being integers. In fact you can't even have the circumference and diameter both rational since the quotient of two rationals is again a rational.

"Ratios" of lengths can yield other irrational numbers also. For instance, in any square the ratio of the length of the diagonal to the length of a side is the square root of 2 which is irrational. Historically, the first conceptual difficulty with irrational ratios of lengths was to accept that these are numbers. The ratio diameter/radius = 2/1 in a circle means that if you line up two copies of the radius, you get the same lenght as the diameter. Also, a ratio lenght1/lenght2 = 3/5 means that if you line up 5 copies of lenght1, you get the same lenght as if you line up 3 copies of lenght2. An irrational ratio such as circumference/diameter = Pi means that no matter how many copies of lengths equal to the circumference you line up, you will never get exactly the same length as if you had lined up copies of the diameter.

In the days of Pythagoras (570-490 BC) the beauty of the rational numbers was revered. Pythagoras refused to accept the existence of the irrational numbers and it is said that one of his students, Hippasus, was sentenced to death for his refusal to deny their existence.

Claude and Penny
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