Why Do We Add Percentage Uncertainties in Multiplication, Not Absolute Uncertainties?

Introduction

In scientific and engineering calculations, understanding the propagation of uncertainties is crucial. When dealing with uncertainties, two common strategies are used: adding percentage uncertainties for multiplications and additions, and adding absolute uncertainties for subtractions and divisions. However, the choice between these methods is often misunderstood. This article aims to clarify why we add percentage uncertainties in multiplication and not absolute uncertainties, providing a deeper understanding of error propagation and its practical implications.

The Nature of Uncertainties

Before delving into the specifics of multiplication, it is important to understand the nature of uncertainties. Uncertainties can be expressed in two primary ways: absolute and relative (percentage).

Absolute Uncertainty: This is the quantified error in a given measurement, often represented as a range around the measured value. For example, if a length is measured as 10 units with an absolute uncertainty of 0.5 units, the actual value lies between 9.5 and 10.5 units.

Relative Uncertainty (Percentage): This is the relative portion of the actual value that the uncertainty represents. It is calculated as the ratio of the absolute uncertainty to the actual value, multiplied by 100. For example, if a length of 10 units has an absolute uncertainty of 0.5 units, the relative uncertainty is 5%, as (0.5 / 10) * 100 5%.

Propagation of Uncertainties in Multiplication

When combining measurements in a multiplication, the relative uncertainties, not the absolute uncertainties, should be added. This is because multiplications yield results where the dominant source of error is multiplicative in nature. Let's explore why this is the case with a practical example.

Example 1: Calculating the Error in Multiplication

Suppose we need to multiply the values 17 and 43, each having an uncertainty of up to 3 (in absolute terms), meaning the actual values lie between 14 and 20 for 17, and 40 and 46 for 43.

Your estimated product, 17*43, is 731, with an expected absolute error of 12 (calculated as (12 - 725, 737 - 731)). However, the smallest possible answer is 560 (14*40) and the largest is 920 (20*46). The difference between these extremes is 360, illustrating a significant underestimation of the error with the initial method.

Explanation of Why Percentage Uncertainty is Added

The reason for adding relative uncertainties lies in the nature of the error amplification in multiplication. If we consider the relative uncertainties, the situation becomes clearer:

If the relative uncertainty of 17 is 17.65% (from 17/100 0.1765 and ((17 - 14) / 17) * 100) and the relative uncertainty of 43 is 10.23% (from 43/40.56 and ((43 - 40) / 43) * 100), the combined relative uncertainty is approximately 17.65% 10.23% 27.88%.

The relative error of the product 731 can be calculated as 731 * 0.2788 203.95, which is approximately 27.88% of 731. This prediction matches the actual range (560 to 920) better than the initial method did.

Logarithmic Interpretation of Errors

A deeper understanding of this phenomenon can be gained by considering the logarithmic nature of error propagation. When we multiply two uncertain values, we can use the properties of logarithms to add the relative uncertainties:

Let x and y be the uncertain values with relative uncertainties Δx/x and Δy/y. The error in the logarithm of the product y is approximately the sum of the logarithmic errors:

Δ(Δx/x Δy/y) ≈ Δ(Δx/x) Δ(Δy/y).

This is because the logarithm of the product is the sum of the logarithms, and the error in the sum is approximately the sum of the errors.

In other words, the relative errors in x and y are additive in the logarithmic domain, which translates to multiplying them in the linear domain.

Conclusion

The addition of relative uncertainties in multiplication is a fundamental concept in error propagation, offering a more accurate and realistic estimation of the final error compared to the addition of absolute uncertainties. This approach is both mathematically sound and practically useful in a wide range of scientific and engineering applications.