Mathematics · Chapter 7
Study notes aligned to the official NEB syllabus.
Computational methods (numerical methods) are techniques for finding approximate numerical answers to problems that are hard or impossible to solve exactly by algebra. A calculator or computer stores only finitely many digits, so almost every non-trivial calculation carries a small error. This chapter explains how numbers are represented to a limited accuracy, how the resulting errors are measured, and one standard iterative method, the bisection method, for locating the root of an equation that cannot be solved by a formula. The recurring theme is honest bookkeeping of accuracy: an answer without a sense of its error is incomplete.
The significant figures of a number are the digits that carry real information about its accuracy. The rules are:
Significant figures matter because they advertise how accurately a quantity is known: writing a length as $2.5,\text{m}$ claims less precision than $2.500,\text{m}$.
Rounding a number to a required number of significant figures (or decimal places) replaces it by the nearest such number. Look at the first digit to be dropped:
For example, to three significant figures $3.14159 \approx 3.14$, while $2.7182 \approx 2.72$.
Truncation (chopping) is different: it simply discards the unwanted digits without adjusting the last kept digit, so $3.14159$ truncated to three significant figures is $3.14$ and $2.7182$ truncated is $2.71$. Rounding is generally more accurate than truncation because it uses the discarded part to decide the last digit.
Computational methods (numerical methods) are techniques for finding approximate numerical answers to problems that are hard or impossible to solve exactly by algebra. A calculator or computer stores only finitely many digits, so almost every non-trivial calculation carries a small error. This chapter explains how numbers are represented to a limited accuracy, how the resulting errors are measured, and one standard iterative method, the bisection method, for locating the root of an equation that cannot be solved by a formula. The recurring theme is honest bookkeeping of accuracy: an answer without a sense of its error is incomplete.
The significant figures of a number are the digits that carry real information about its accuracy. The rules are:
Significant figures matter because they advertise how accurately a quantity is known: writing a length as claims less precision than .
Rounding a number to a required number of significant figures (or decimal places) replaces it by the nearest such number. Look at the first digit to be dropped:
For example, to three significant figures , while .
Truncation (chopping) is different: it simply discards the unwanted digits without adjusting the last kept digit, so truncated to three significant figures is and truncated is . Rounding is generally more accurate than truncation because it uses the discarded part to decide the last digit.