# Video: Identifying Whether a Given Number Is Rational, Irrational, or Whole

According to the US Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by π. Is the circumference of a quarter a whole number, a rational number, or an irrational number?

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### Video Transcript

According to the US Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by π. Is the circumference of a quarter a whole number, a rational number, or an irrational number?

So, here we have the quarter. Weβre told that the diameter is 0.955 inches. Thatβs the distance from one side of the circle to the other through the center. And weβre told that the circumference is equal to π times the diameter. And so, we can say that the circumference of this quarter will be 0.955π. Weβre asked to work out if this is a whole number, a rational number, or an irrational number. We can recall that a decimal approximation for π begins 3.141592654 and so on. And therefore, when we multiply that by 0.955, weβre definitely not going to get a whole number. So, letβs look at a rational number.

A rational number can be expressed as a fraction π over π where π and π are integers and π is not equal to zero. If we cannot express a number as a fraction π over π, then it would be irrational. We can simply say that an irrational number is a number that is not rational. We may recall that π is an irrational number. And thatβs because we cannot express it as a fraction π over π. We know this to be the case because the decimal value for π does not terminate, and it is not a repeating decimal. So, here we have π, an irrational number, multiplied by 0.955, which is a rational number. We can say that itβs rational because itβs equivalent to the fraction 955 over 1000.

And therefore, weβre multiplying a rational number by an irrational number, which will give an irrational number, which is true for all cases except for when the rational number is a zero. So, our answer is that the circumference of the quarter 0.955π is an irrational number.