Estimation and rounding

前置知识

4 项必需的前置知识（一直追溯到底）

你将学会

• Round whole numbers and decimals to a given place, and explain the convention for halfway cases.
• Estimate sums, differences and products before calculating, and use estimates to check answers.
• Give lower and upper bounds for a quantity, such as 'between 300 and 400'.

定义

• Rounding replaces a number by a nearby simpler number, such as the nearest multiple of 10.
• An estimate is an approximate value; its error is the difference between the estimate and the exact value.
• A lower bound and an upper bound are numbers known to be at most and at least the quantity.

定理

• Rounding a whole number to the nearest ten changes it by at most 5.
• If 30 ≤ a ≤ 40 and 20 ≤ b ≤ 30, then 50 ≤ a + b ≤ 70.
• Rounding errors can accumulate: rounding many numbers before adding can be further off than rounding the total once.

常见误解

• Thinking an estimate is a wild guess rather than a reasoned approximation.
• Believing that rounding at every step of a calculation never changes the final rounded answer.

可选发现

• Introductory calculus Approximating more and more closely, with an error that can be made as small as we like, is the idea behind limits.
• Numerical methods Computers round almost every non-integer calculation; numerical analysis tracks how such errors grow.
• ← Measuring length and time