「粗略估算」在工程院校中是标准课程，对多数从业工程师来说则是谋生的必备技能。

## 基本技巧

1. 两个答案比一个答案好。也就是说要做检验。
2. 快速检验。注意量纲的统一，更不要忘记了常识性的东西。
3. 经验法则。「72 法则」是说：假设以年利率 r% 投资一笔钱 y 年，如果 $r\times y = 72$，那么你的投资差不多会翻倍。
4. 实践。多做多想。

## 习题

1. While Bell Labs is about a thousand miles from the mighty Mississippi, we are only a couple of miles from the usually peaceful Passaic River. After a particularly heavy week of rains, the June 10, 1992, edition of the Star-Ledger quoted an engineer as saying that “the river was traveling about 200 miles per hour, about five times faster than average”. Any comments?

2. At what distances can a courier on a bicycle with removable media be a more rapid carrier of information than a high-speed data line?

3. How long would it take you to fill a floppy disk by typing?

$1.44 \times 1024 \times 1024 \div 300 \div 60 = 83\ \text{(hours)}$

4. Suppose the world is slowed down by a factor of a million. How long does it take for your computer to execute an instruction? Your disk to rotate once? Your disk arm to seek across the disk? You to type your name?

5. Prove why “casting out nines” correctly tests addition. How can you further test the Rule of 72? What can you prove about it?

\begin{aligned} \left(100X+10Y+Z\right)\mod 9 &= \left[\left(99+1\right)X + \left(9+1\right)Y + Z\right]\mod 9\\ &=\left[\left(99X+9Y\right)+\left(X+Y+Z\right)\right]\mod 9\\ &=\left(X+Y+Z\right)\mod 9 \end{aligned}

$\text{FV} = \text{PV}\times \left(1 + r\right)^y$

\begin{aligned} 2 = \left(1+r\right)^y\\ y = \ln 2 \div \ln \left(1+r\right)\\ \ln 2 \approx 0.693127\\ y \approx 0.693147 \div r \end{aligned}

6. A United Nations estimate put the 1998 world population at 5.9 billion and the annual growth rate at 1.33 percent. Were this rate to continue, what would the population be in 2050?

9. Suppose that a system makes 100 disk accesses to process a transaction (although some systems need fewer, some systems require several hundred disk accesses per transaction). How many transactions per hour per disk can the system handle?

11. [P. J. Denning] Sketch a proof of Little’s Law.

12. You read in a newspaper article that a United States quarter-dollar coin has “an average life of 30 years”. How can you check that claim?