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.
