How this instrument works
A harmonic number, written Hn, is the running total of unit fractions with sequential denominators starting at 1: H1 = 1, H2 = 1 + 1⁄2 = 1.5, and H3 = 1 + 1⁄2 + 1⁄3 = 11⁄6, roughly 1.8333. The pattern keeps climbing forever, one shrinking fraction at a time, and it grows without bound even though each individual term added gets smaller and smaller — a slow, famous divergence that surprises most people meeting it for the first time.
This sheet loosens the strict definition on purpose: rather than locking the three denominators to 1, 2, and 3 in that exact order, it accepts any three positive values for a, b, and c and simply reports 1⁄a + 1⁄b + 1⁄c. Entering 1, 2, 3 reproduces the true third harmonic number exactly, but the same addition also answers questions the sequential definition was never meant to cover.
The reason a generalized version is worth having shows up outside pure mathematics entirely. Three resistors wired in parallel combine so that 1 divided by the total resistance equals 1⁄R1 + 1⁄R2 + 1⁄R3 — the identical sum of reciprocals computed here, just with resistances standing in for a, b, and c. Combined work-rate problems follow the same shape: three workers who alone finish a job in a, b, and c hours complete it together at a rate of 1⁄a + 1⁄b + 1⁄c jobs per hour, adding their speeds rather than their times.
Two contrasting cases make the pattern easy to check by eye. Three identical denominators simply triple the one unit fraction — 1⁄2 three times over totals 1.5, no surprise there. Denominators that are not sequential at all can still land on a clean whole number: 1⁄2 + 1⁄3 + 1⁄6 sums to exactly 1, a tidy result of three unequal fractions happening to complete each other perfectly.
- Enter a positive value into First denominator (a) — the sheet reads it as the bottom of the fraction 1⁄a.
- Enter a positive value into Second denominator (b) for the fraction 1⁄b.
- Enter a positive value into Third denominator (c) for the fraction 1⁄c.
- Read Sum of the three unit fractions for the added total, 1⁄a + 1⁄b + 1⁄c.
- Enter 1, 2, and 3 in that order once to confirm the sheet reproduces the true third harmonic number, 11⁄6.
Worked example — 1⁄1 + 1⁄2 + 1⁄3
Set a = 1, b = 2, and c = 3, the sequential order a true harmonic number requires. Converting to a shared denominator of 6 gives 6⁄6 + 3⁄6 + 2⁄6 = 11⁄6, and the sheet reports the decimal form directly as 1.8333333333333333 — the exact third harmonic number, matching 11 divided by 6 to as many digits as the display carries.
Two other entries show how far the same addition reaches beyond that one sequential case. Setting a = b = c = 2 gives three identical halves, 0.5 + 0.5 + 0.5 = 1.5, simple tripling with nothing sequential about it. Setting a = 2, b = 3, c = 6 instead gives 0.5 + 0.333... + 0.1666... = 1.0 exactly, three unrelated-looking fractions with non-sequential denominators closing into a clean whole number.
Questions
What is the third harmonic number?
11⁄6, roughly 1.8333. It is the sum of the first three unit fractions taken in sequential order — 1⁄1 + 1⁄2 + 1⁄3 — and entering a = 1, b = 2, c = 3 into this sheet reproduces that exact value.
Does this calculator require sequential denominators like a true harmonic number?
No — it accepts any three positive denominators, not only 1, 2, 3 in order. Feeding it the sequential case reproduces the true third harmonic number exactly, but the same 1⁄a + 1⁄b + 1⁄c addition also answers unit-fraction problems, like parallel resistance, that have nothing to do with strict harmonic numbers.
What is 1⁄1 + 1⁄2 + 1⁄3?
11⁄6, or 1.8333333333333333 as a decimal. Converting each fraction to sixths gives 6⁄6 + 3⁄6 + 2⁄6, and adding the numerators across a shared denominator of 6 gives 11⁄6 exactly, the third harmonic number.
Why do three denominators of 2 give a total of 1.5?
Because 1⁄2 + 1⁄2 + 1⁄2 is the same half added three times, and 0.5 tripled is 1.5. Identical denominators always produce the same unit fraction three times over, so the total is simply that one fraction multiplied by three.
How does this relate to parallel resistors or combined work rates?
Both add reciprocals the same way this sheet does. Three resistors in parallel combine so that 1 over the total resistance equals 1⁄R1 + 1⁄R2 + 1⁄R3, and three workers who each finish a job alone in a, b, and c hours finish it together at a rate of 1⁄a + 1⁄b + 1⁄c jobs per hour — the identical sum of unit fractions, applied to physical quantities instead of pure numbers.
What is a harmonic number in general?
The sum of the reciprocals of every whole number from 1 up to some n: Hn = 1 + 1⁄2 + 1⁄3 + ... + 1⁄n. The total grows forever as n increases, even though each new term added shrinks toward zero — a well-known divergent sum that this sheet's three-term version only samples a small piece of.