Light a single candle in a pitch-dark room and the whole room changes. Light one more candle in a room that already has a hundred burning, and nobody notices. The extra light is exactly the same both times. What changes is what it is being compared with.
In the 1830s the German physiologist Ernst Heinrich Weber turned that everyday hunch into a rule. He had people compare weights, lengths and other quantities and asked: how big must a difference be before someone can reliably tell? The answer was never a fixed amount. It was a fixed fraction. Double the starting weight and you need double the extra weight before anyone can feel the change.
That constant is now called the Weber fraction, and the rule is Weber's law. In 1860 the physicist Gustav Fechner built a whole science, psychophysics, on top of it. It still holds up remarkably well, with some bending at the extremes, for brightness, loudness, weight, length, time and even for something you might not expect: how many things you see at a glance.
Below, two boxes flash a cloud of blue and yellow dots for three quarters of a second, too short to count. Tap the box you think held more. The game adapts: every time you get two right in a row, the difference shrinks; every miss makes it bigger. After 24 rounds it estimates your threshold, the ratio at which you stop being reliable.
Psychologists Justin Halberda and Lisa Feigenson ran almost exactly this game in 2008, with children aged three to six and with adults. Three-year-olds could reliably pick the bigger cloud when the numbers differed by a ratio of about 3:4, say 9 dots against 12. Adults managed about 10:11, say 20 against 22. Every age group in between was on a smooth path from one to the other.
Notice what does not matter: the size of the numbers. If you can tell 10 from 11, you can tell 100 from 110, but not 100 from 101. That is Weber's law again, applied to quantity instead of weight. Your brain seems to store "how many" on a squashed, roughly logarithmic scale, where equal ratios sit equal distances apart.
Four years later Halberda and colleagues put a version of the test online and more than 10,000 people aged 11 to 85 played it. Precision kept improving through the school years and was best, on average, at around age 30, then slowly declined.
You can feel the ratio rule directly. Both boxes below always differ by exactly five dots. Drag the slider to raise the starting number and watch those same five extra dots go from obvious to invisible.
The rule leaks out of the lab and into money. In a famous 1981 experiment, Amos Tversky and Daniel Kahneman asked people to imagine buying a $15 calculator, then hearing it cost $10 at a store 20 minutes away. 68% said they would make the trip. When the calculator cost $125 and the other store sold it for $120, only 29% would go. Same $5, same drive. What people felt was the fraction saved, not the dollars.
Once you look for it, you find it everywhere:
The world spans huge ranges. Bright sunlight is hundreds of thousands of times brighter than a moonlit night, and you care about a handful of berries as much as about a herd of hundreds. A sensor that measured absolute differences would either saturate in bright light or be useless in dim light. Measuring relative change lets the same neurons stay useful across that whole range.
The price is that small changes on top of big amounts slip by unnoticed. That is harmless for dots and candles, and occasionally expensive at the checkout. Next time a "only €5 more" upgrade sounds trivial, ask whether you would pay those five euros on their own.
Sources: Weber, De Tactu (1834); Fechner, Elemente der Psychophysik (1860); Halberda & Feigenson, Developmental Psychology (2008); Halberda et al., PNAS (2012); Tversky & Kahneman, Science (1981).