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

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. We’re told that the diameter is 0.955 inches. That’s the distance from one side of the circle to the other through the center. And we’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. We’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, we’re definitely not going to get a whole number. So, let’s look at a rational number.

A rational number can be expressed as a fraction 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to zero. If we cannot express a number as a fraction 𝑝 over 𝑞, 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 that’s because we cannot express it as a fraction 𝑝 over 𝑞. 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 it’s rational because it’s equivalent to the fraction 955 over 1000.

And therefore, we’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.