Electronics · Math shelf · Optional — for the curious · build 2026.09.23-1644
The idea
Every component is a promise with a plus-or-minus, and every measurement has its own wiggle. Engineering isn't expecting exact numbers — it's knowing how far off is fine.
The fourth stripe on a resistor states its tolerance — how far the true value may sit from the labeled one. Gold means ±5% (silver ±10%, brown ±1% on fancier parts). Our kit is gold, so a "220 Ω" resistor is really a promise: somewhere between 209 and 231 Ω.
| Labeled (nominal) | ±5% is… | The promise |
|---|---|---|
| 220 Ω | ±11 Ω | 209 – 231 Ω |
| 1 kΩ | ±50 Ω | 950 – 1,050 Ω |
| 4.7 kΩ | ±235 Ω | 4,465 – 4,935 Ω |
| 10 kΩ | ±500 Ω | 9,500 – 10,500 Ω |
The labeled value is called the nominal value — the name-value. Ohm a handful of "1 kΩ" resistors from the assortment and you'll read something like 987, 1,004, 992: all different, all keeping the promise.
% diff = (measured − nominal) ÷ nominal × 100
Worked: you measure 1,032 Ω on a nominal 1 kΩ. (1,032 − 1,000) ÷ 1,000 = 0.032 → +3.2%. Inside ±5%: the part is healthy. This one formula turns "is that close?" from a feeling into a number.
The meter's spec sheet gives its own honesty range — hobby meters typically promise about ±(0.5% + 2 digits) on DC volts. The "2 digits" means the last displayed digit may be off by two counts: a true 5.00 V may read 4.97 to 5.03. So the final digit of any reading is always a little soft — watching it flutter is normal, not a bug.
| Measured vs. expected | Verdict |
|---|---|
| within a few % | normal — tolerance and meter wiggle |
| off by 2× | not tolerance: wrong resistor grabbed, or a circuit bug |
| off by ~1,000× | read the display's unit again — kΩ vs Ω (the Milli, Kilo, Mega sheet) |
Tolerance never explains a factor of ten. Small disagreements are physics; big ones are information — that's the drop check's whole philosophy on the Debugging Loop card.
Build a divider from two ±5% parts and the prediction inherits their spread. Worst case for a 1 kΩ / 1 kΩ divider: one part at 950, the other at 1,050 puts the tap anywhere from 2.38 to 2.63 V instead of exactly 2.50. In practice both parts usually sit much closer — but this is why a measured 2.46 V against a predicted 2.50 is a success, not a miss.
Which sets the reporting rule: don't write more digits than your parts can back up. Predicting "2.51694 V" from ±5% resistors is fiction past the first three digits; "≈ 2.5 V" is the honest answer.
The math, for the curious — why kit values look so odd
1.0 — 2.2 — 4.7 — 10: strange numbers, until you see the design. Standard resistor values are spaced by equal ratios, not equal gaps, so the ±-ranges tile the number line with no orphan values. The full E12 series puts twelve steps in each ×10 decade, each step ×1.21 — the twelfth root of 10 — exactly the way a piano splits each octave into twelve equal-ratio semitones. Resistor drawers and keyboards are tuned the same way.
Answers at the bottom — and problems 1–3 are checkable against real parts with the meter.
1. ±23.5 → 446.5 – 493.5 Ω. 2. (4,550 − 4,700) ÷ 4,700 ≈ −3.2% — healthy. 3. No — that's 10×. Either the display said kΩ and you wrote Ω, or the stripes are red-red-red (2.2 kΩ), not red-red-brown. Check both before blaming the part. 4. +4.4% — inside the gold promise, though barely; a part that close to the edge is still a kept promise.