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How calculators work

How Calculators Count Significant Figures

Significant-figure mode is not a digit counter. It preserves the digits that describe scale and measurement, then rounds the displayed result to that count.

The zeros in 0.00420 are not all doing the same job. Some locate the decimal point. One of them records measured precision. A calculator in SCI mode has to tell those roles apart before it can decide how many digits to show.

Significant figures describe the meaningful resolution of a number. They are a communication rule for measurements, not a claim that the calculator has performed a more accurate computation.

leading zeroslocate the scalesignificant digitscarry the measurement meaning

Scientific notation makes the count visible

In scientific notation, the digits in the coefficient are significant and the power of ten sets the scale. A display of 1.230 x 10^3 communicates four significant figures; the exponent does not add another meaningful digit.

Casio distinguishes FIX, SCI, and NORM settings. In SCI mode, the chosen number controls significant digits in the displayed result. That is different from fixed decimal mode, which counts positions after the decimal point.

A calculator display with meaningful digits highlighted while leading zeros fade into the background
Scientific notation separates scale-setting zeros from digits that describe precision.
0.00420three significant figures4, 2, and the trailing zero
4.20 x 10^-3same quantity, clearer countthe coefficient shows the intended precision

The input carries information the calculator cannot invent

Entering 2.0 and entering 2.000 are different signals in a measurement context, even though they represent the same mathematical quantity. The extra zeros tell a reader how finely the value was recorded. If the original measurement was only known to the nearest unit, displaying six digits after a calculation does not restore lost certainty.

A measured number passing through a precision filter that keeps meaningful digits and drops false certainty
The output precision should follow the evidence in the input, not the length of the screen.
Multiplication and divisionthe result is usually limited by the factor with the fewest significant figures.Addition and subtractionthe result is usually limited by the least precise decimal place.Exact countsdefined or counted quantities do not behave like uncertain measurements.

Resolution and accuracy are different words

A calculator may show ten significant digits while the sensor, ruler, or survey behind the input supports far fewer. Resolution describes how finely a value is represented; accuracy describes how close it is to the quantity being estimated. A display setting can control the first and prove neither the second nor the third.

Two rulers with different measurement resolution beside the same calculator result
More visible digits are not automatically more trustworthy digits.
A calculator can count digits. It cannot decide which digits your measurement earned.

Sources and further reading