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While working on some homework tonight, I entered the following expression into my TI-84 Plus:

$$\frac{(90×0.006)}{(\pi/4×0.006^4)}$$

I did a double-take when I saw that the output was exactly 530,516,477. I was under the assumption that any (non-trivial) arithmetic operation involving $\pi$ could never evaluate to a rational number. Had I just stumbled upon a rational representation of pi?

A quick check on WolframAlpha confirmed my original assumption: the exact answer is a never-ending string of decimals, of course.

But this piqued my curiosity: Why does the TI-84 Plus simplify this expression to a whole number? This solution is only 9 digits, while there are room for 16 characters on-screen. Is there some quirk of the TI-84 software that may simplify expressions in unexpected ways, similar to optimizations in a C compiler?

Bonus points if you happen to recognize the formula, I'm using :)

Misha Lavrov
  • 142,276
leavesy
  • 23

1 Answers1

5

The calculator does not actually consider this expression to be equal to 530,516,477, which we can see by typing the following:

90*0.006/(π/4*0.
006^4)
       530516477
Ans-530516477
         -.02702

Another option to "uncover the truth" is the following:

90*0.006/(π/4*0.
006^4)
       530516477
fPart(Ans)
          .97298

So, in fact, the calculator "knows" more digits of this decimal: it is 530516476.97298. (In general, the calculator does floating-point arithmetic with 14 decimal digits of precision in the mantissa.)

The calculator has two rules limiting how many digits it displays. First, it will never display more than 10 significant digits. Second, it will never display more than 14 symbols total. (The number of symbols can end up being more than the number of significant digits due to decimal points, negative signs, and the exponent of numbers written in scientific notation.)

In the case of 530516476.97298, the calculator decides to limit the result to 10 decimal digits, but it does so by rounding - and 530516476.97 is rounded up to 530516477.0. The calculator leaves off the ending 0, and displays only 530516477.

Misha Lavrov
  • 142,276
  • 1
    A more dramatic example: if you enter 3983π^3 into the calculator, you get 123498, because the calculator rounds off 123498.00001763 to ten decimal places. – Misha Lavrov Oct 03 '23 at 18:14