How this instrument works
Every digit after a decimal point sits at a particular place, tenths first, then hundredths, then thousandths, and each place carries a different power of ten. Pulling out one digit uses a three-step trick: digit = ⌊x × 10^place⌋ mod 10. Multiplying by 10^place shifts the wanted digit all the way over into the ones position; flooring throws away whatever fractional remainder trails behind it; and taking the result mod 10 strips away every digit sitting to its left, leaving that single figure standing alone between 0 and 9.
Walk it through with 3.14159 and place=2, meaning hundredths. Multiplying by 100 gives 314.159, and flooring drops the .159 tail to leave 314. Taking 314 mod 10 keeps only the remainder after dividing by 10, which is 4 — and sure enough, reading 3.14159's digits after the point in order, 1, 4, 1, 5, 9, the second one is indeed 4. Change place to 1 instead and the same sequence lands on 1, the very first digit after the point.
The same mechanic underlies how any positional number system stores and retrieves its digits, not just base-ten decimals. This site's Babylonian, Mayan, and Binary conversion pages all rely on numbers built from place values that are powers of some base, sixty, twenty, or two instead of ten, and isolating one digit in any of those systems only ever swaps the 10 in this formula for that system's own base. Understanding the floor-and-mod trick here is understanding the mechanism behind every one of those pages at once.
- Enter x, the decimal number you want to inspect, into the Decimal number field.
- Choose which Decimal place to read: tenths (1), hundredths (2), or thousandths (3).
- Read digit: the single figure sitting at that chosen place, isolated automatically from the rest of the number.
Worked example — reading two digits of 3.14159
Set x=3.14159 and place=2 for hundredths: multiplying gives 3.14159×100=314.159, flooring gives 314, and 314 mod 10 leaves 4, so the hundredths figure is 4. Switch place to 1 for tenths on that same number: 3.14159×10=31.4159, flooring gives 31, and 31 mod 10 leaves 1, matching the very first figure after the point.
Now try x=0.5 with place=3, thousandths: 0.5×1000=500 exactly, flooring changes nothing since 500 has no fractional part, and 500 mod 10 leaves 0. That zero makes sense once you write 0.5 out fully as 0.500 — every position beyond the tenths spot is simply empty, so any place chosen past the first will always come back 0 for this particular number.
Questions
How does multiplying by a power of ten isolate one figure?
It shifts the target position over into the ones spot, since each place further right represents one smaller power of ten. Multiplying by 10 raised to the place number cancels that smaller power out, leaving the digit you want sitting right where a floor operation can grab it cleanly.
Why does the flooring step matter here?
Flooring discards every digit that trails past the one you're after, keeping only whole numbers. Without it, the result would still carry a leftover fractional tail from digits further along, and the later mod 10 step would no longer isolate a single clean figure.
What does taking the remainder after dividing by 10 actually remove?
It strips away every digit sitting to the left of the one you want, since dividing by 10 and keeping only the remainder discards whole multiples of ten. What's left over is exactly one figure, somewhere between 0 and 9, with nothing else attached.
Why is 0.5's thousandths figure 0?
Because 0.5 written out fully is 0.500 — there is genuinely nothing beyond the tenths spot, so every later position, hundredths, thousandths, and beyond, comes back as an empty zero rather than some hidden nonzero figure.
Does this same approach work for other number bases besides ten?
Yes — swap the 10 in the formula for whichever base is in play, and the identical multiply, floor, and remainder sequence isolates a single figure there too. This site's Babylonian, Mayan, and Binary pages all extract digits this exact way, just built around sixty, twenty, or two instead of ten.