APPENDIX
SIGNIFICANT FIGURES
The idea of rounding a number to one, two or three significant figures is a recurring theme in this book. Here’s a reminder of how it’s done.
Let’s take the height of the Matterhorn in the Alps, which according to most sources is 4,478 metres. It’s likely that surveyors have readings that give them a height to the nearest centimetre, but given that mountains are always on the move, they have understandably rounded the Matterhorn’s height to a four-digit number that is correct to the nearest metre. The statistic therefore has four significant figures.
To round the number to three, two, or one significant figure, lop off the number at the end and replace it with a zero – but with one proviso: if the digit being removed is 5 or more, the digit before it should be rounded up by one.
Here’s what happens when we round the Matterhorn height:
| Rounded to … | |
| Three significant figures | 4,480 (note that the 7 has been rounded up to 8) |
| Two significant figures | 4,500 (the second 4 is rounded up to 5) |
| One significant figure | 4,000 (note that 4,478 rounds down) |
The first significant figure of a number is always its first non-zero digit. So, for example, the first significant figure of 0.0063 is 6. It’s possible for a significant figure to be zero, including a number’s final digit. For example, if an athlete runs 100 metres in 10.28 seconds, that is 10.3 seconds to three significant figures, and 10 seconds to two significant figures.
WHERE THE RULE OF 72 COMES FROM
The ‘Rule of 72’ is found by working out how many iterations (years, for example) it takes for a number to double if it is growing at a fixed rate. To follow the derivation, you do need to be familiar with natural logarithms.
Let’s call the annual interest rate R per cent. What we are looking for is the number of years, N, that it will take our starting pot of money, A, to double; i.e. after N years we will have an amount 2A:
A × (1 + R)N = 2A
Cancelling A on both sides:
(1 + R)N = 2
Take logarithms of both sides:
N.ln(1 + R)= ln 2 = 0.69 (= 69%)
There is a rule of thumb familiar to mathematicians that if R is small then ln(1 + R) ≈ R (this is accurate to within 5% if R < 10%). In other words:
N × R = 0.69
N = 69% ÷ R.
This is why it should really be the Rule of 69. The number is adjusted to 72 because 72 is a multiple of many standard interest rates: 1%, 2%, 3%, 4%, 6%, 8% and so on.
WHO WANTS TO BE A MILLIONAIRE? (PART 2)
The ocean is the Arctic. John estimated that the Atlantic is about 30 million square miles, using similar calculations to the one at the bottom of here. That’s getting on for 10 times bigger than the 4.7-metre-square-mile ocean in the question. The Indian Ocean is a similar size to the Atlantic, and the Pacific is bigger than both of them. No doubt the reason most of the audience voted for the Pacific is that 4.7 million is a really big number, and they knew that the Pacific is also really big. But, of course, ‘really big’ and ‘really really big’ are not the same thing.
RANGE OF A SHOT PUT
The range of a shot put can be worked out from this complex-looking formula:
where:
R is the range of the shot put;
υ is the speed of the shot put when released;
g is the acceleration due to gravity;
θ is the angle relative to the horizontal of the shot put when it is released;
y0 is the height above the ground at which the shot put is released;
If gravity is the only thing that changes, then the range increases roughly as I indicated here. In reality, in lower gravity it should be possible to release the shot at a higher speed (the shot will feel less heavy, so you can push it faster). This will increase the range, so my estimate of the advantage of Mexico City is on the low side.
HOW LONG TO COUNT TO A MILLION?
‘Seven’ and ‘Eight’ are the longest words from 0 to 9, and ‘seventy’ is the longest tens word, so the smallest number to exceed 280 characters is going to depend heavily on sevens. 777,777,777,777,777,777 (seven hundred seventy-seven quintillion seven hundred seventy-seven quadrillion seven hundred seventy-seven trillion seven hundred seventy-seven billion seven hundred seventy-seven million seven hundred seventy-seven thousand seven hundred seventy-seven!) uses 255 characters. We have 25 characters left. In front of that number we put the sextillions (sextillion has 11 characters including the space). The smallest number to exceed 25 characters is: one hundred one sextillion. So the Count will be frustrated when he gets to 101,777,777,777,777,777,777.
Phew!