How this instrument works
An integer is a whole number with no fractional part left over — 5, 0, and −12 all qualify, while 5.5, −3.25, and any figure that stops partway between two whole numbers do not. Integers continue without end in both directions, through zero, and every one of them sits exactly on a whole-number mark on the number line, never between two marks.
This calculator finds the whole number a value sits on top of, or just below, using the floor function ⌊x⌋, then subtracts that floor from the original figure: test = x − ⌊x⌋. When x already lands exactly on a whole number, its floor equals itself, so the subtraction leaves a remainder of exactly 0 — proof of a genuine integer. Any leftover above 0 means x fell somewhere between two whole-number marks, and the size of that leftover shows precisely how far past the lower mark it landed.
Floor does not simply chop off whatever follows the decimal point — it always rounds down, toward negative infinity, and for a negative number 'down' points further from zero rather than toward it. Trim the digits after the point from −3.25 and instinct suggests −3 ought to come out, but the floor of −3.25 is actually −4, one whole step further left on the number line. That single difference is why a negative non-integer's remainder still comes out positive: subtracting the smaller −4 from −3.25, rather than the larger −3, leaves 0.75 over, not −0.25 or some other negative figure.
- Type the figure to be tested into the Number field — any real value, positive, negative, or zero.
- Read the Fractional remainder (0 = whole integer) field: a result of 0 confirms Number is a genuine integer.
- Any value above 0 in that field means Number fell between two whole numbers, and the figure itself shows how far past the lower one it landed.
- Try a negative Number, such as −3.25, to see the floor round further down rather than merely toward zero.
- Compare a whole Number like 5 against a near neighbor like 5.5 and watch the remainder jump from 0 to 0.5.
Worked example — testing 5, 5.5, and −3.25
Enter Number = 5 and its floor is also 5, since 5 already sits on a whole-number mark — the Fractional remainder reads test = 5 − 5 = 0, confirming a genuine integer with nothing left over. Change Number to 5.5 and the floor stays at 5, the whole number just beneath it, so the remainder becomes test = 5.5 − 5 = 0.5, a clean sign that 5.5 is not an integer.
Enter Number = −3.25 and the floor is not −3, the figure produced by simply trimming the digits after the decimal point — floor rounds down toward negative infinity, and the whole number just below −3.25 is −4, one full step further left. The Fractional remainder therefore reads test = −3.25 − (−4) = 0.75, a positive outcome even though the input itself was negative, because the subtraction runs against −4 rather than −3.
Questions
What exactly counts as an integer?
Any whole number with no fractional part — positive, negative, or zero — counts as an integer: −12, 0, and 5 all qualify, while 5.5 and −3.25 do not, since both sit between two whole-number marks rather than on one. Integers run without end in both directions along the number line, with no smallest or largest member.
Why does the calculator rely on the floor function instead of just cutting off the decimal digits?
Because cutting off decimal digits gives a wrong answer for negative figures — trimming −3.25 down to −3 feels natural but is not how floor behaves. Floor always rounds toward negative infinity, so the floor of −3.25 is −4, not −3, and building the test on floor rather than plain truncation keeps the result consistent for positive and negative inputs alike.
Why does a negative Number like −3.25 produce a positive remainder?
Because floor rounds down past −3.25 to −4, and subtracting that more negative −4 from −3.25 leaves 0.75, a positive gap. Floor never rounds toward zero for a negative input — it always moves further from zero — so the remainder for a negative non-integer comes out positive every time, the same as it does for a positive one.
What does a Fractional remainder of exactly 0 tell me?
It confirms Number sits exactly on a whole-number mark, with nothing left over once its own floor is subtracted — a genuine integer. Any figure above 0 in that field instead measures the distance between Number and the whole number just beneath it.
Does zero itself count as an integer?
Yes — zero is a whole number with no fractional part, so it passes the test cleanly: floor(0) = 0, and 0 − 0 = 0. Zero sits right at the boundary between the positive and negative integers, belonging to neither side exclusively yet still counted among the integers.
Can this test flag a number written as a fraction, such as 7/2?
Yes, once 7/2 is entered in decimal form as 3.5 — the floor of 3.5 is 3, and the remainder comes back as 0.5, showing it is not an integer. Any ratio that fails to divide evenly, like 7/2, produces a nonzero remainder here, while one that does divide evenly, like 8/2 = 4, returns exactly 0.