In physics, every measurement has a small amount of uncertainty called an error.

When you use these measured values in a math formula, those errors combine to affect your final result. Understanding these rules ensures your scientific conclusions remain accurate.

Errors in Addition and Subtraction

When you add or subtract two measurements, their absolute errors—the actual amount of uncertainty in a measurement—always add up. If you measure two lengths as A ± ΔA and B ± ΔB, the error in the result (Z) will be the sum of the individual errors (ΔZ = ΔA + ΔB). Even if you are subtracting the values, the errors still add together because the uncertainty grows larger with each step.

Errors in Multiplication and Division

When you multiply or divide quantities, we look at relative error. Relative error is the ratio of the absolute error to the measured value. When quantities are multiplied or divided, the relative errors add up. For example, if Z = A × B or Z = A / B, then the relative error in Z is the sum of the relative errors in A and B (ΔZ/Z = ΔA/A + ΔB/B).

Errors in Powers

Sometimes, a measurement is raised to a power, like A² or A³. To find the error, multiply the exponent by the relative error of the original measurement. If Z = Aⁿ, then the relative error ΔZ/Z = n × (ΔA/A). This shows that powers significantly increase the final uncertainty in your calculated result.

Why This Matters

• Precision: It tells us how reliable our calculated data is.
• Safety: Engineers use these rules to ensure buildings and bridges can handle loads despite tiny measurement variations.
• Accuracy: It helps scientists estimate the range within which the true value lies.