A number acquires authority the moment it is written down

Figures are read as conclusions, not as claims

A sentence makes an argument that a reader can weigh. A number appears to report a fact. That difference in how the two are received is the whole problem, because a figure produced in the same way as the surrounding prose carries none of the guarantees that a figure normally implies.

Percentages are the worst case. They arrive with two decimal places, and precision reads as evidence of measurement. A quantity expressed as roughly a third invites a question about where it came from; the same quantity as 33.4 per cent doesn’t.

The result is that numbers get lifted out of drafts and into slides, summaries and reports, at which point they are separated from any context that might have signalled how they were arrived at.

The characteristic arithmetic failures

Where a calculation is performed rather than looked up, the errors have a pattern. Percentages of percentages. Unit conversions, particularly between systems and between scales. Totals that do not equal the sum of their parts. Averages taken over the wrong denominator. Compounding treated as addition.

Multi-step calculations are the most fragile, because an early slip propagates through every subsequent line and the result remains internally consistent. Consistency is the trap: each step follows from the one before, so a check that reads the working sees a coherent chain built on a wrong second line.

A number can also be correct and answer a different question from the one asked, which is harder to catch than an arithmetic slip and considerably more common in practice. The figure is right; it is a proportion of the wrong population.

Rounding, ranges and false precision

Watch what happens to precision through a chain of operations. A rough input can emerge as a precise-looking output, because nothing in the process tracks how uncertain the starting figures were and there is no convention being applied to prevent it.

The practical correction is to state the precision you want and to round deliberately at the point of reporting. If the inputs were approximate, the output should be approximate in the same degree, and a range is usually more honest than a point.

This matters for how the number is used afterwards. A stated range invites the reader to treat it as an estimate. A precise figure invites them to build on it, and a figure built upon is very difficult to correct later.

It is also worth asking what a plausible range would even be before looking at the answer. A rough expectation formed in advance catches order-of-magnitude errors instantly, and order-of-magnitude errors are the ones that do real damage.

Every figure needs a traceable origin

The requirement that makes numbers manageable is provenance. Each figure should carry where it came from: a source document with a location, a calculation with its inputs shown, or an explicit statement that it is an estimate and on what basis.

Ask for the working rather than the result, then check the working rather than reading it. Those are different actions — recalculating one step independently tells you something, while following the explanation tells you only that the explanation is coherent, which it will be either way.

For anything that repeats, do the arithmetic outside the language step. A spreadsheet or a script computes exactly, identically, every time, and the language part should be extracting the inputs and describing the result rather than performing the sums.

Numbers that leave the document

The most consequential moment is the one where a figure is copied. It arrives in a new context with no trace of how it was made, is repeated by someone who assumes it was checked, and becomes a fact through circulation rather than through verification.

Anything a figure is quoted in should say where it came from, in a form short enough that people will actually keep it: the source, the date, and whether it is measured or estimated. That one line survives copying far better than any amount of caution in the surrounding prose.

And where a number will be published, relied upon financially, or submitted to anyone, it needs to be verified against a primary source by a person who understands what it measures. That isn’t a matter of caution about tools; it is what has always been required of a number, and nothing about how it was produced changes it.