The idea: You already do order-of-magnitude
calculation. It is quite useful. You can make it even more so.
- a "you are already doing it" introduction
How tall are you? (As for me, I'm about 6 feet.)
How far away is school? (As for me, now, its about 5 miles.)
Why didn't I say my height was 5 feet 11 and 3/4 inches?
Or that school was 5.012 miles away? Or perhaps 26400 feet?
You are doing two things:
- Choosing a good unit... Big things are measured
in big units (say miles). Small things are measured in small units
(say feet or inches).
- Ignoring details... If you don't know
or don't care about the exact number, you don't worry about
(Sometimes you do end up needing the details, and you have to go
back and figure them out. But usually you don't. And even when you
do, a rough estimate goes quickly and helps avoid making mistakes in
figuring the details.)
Making it easier
Here are two ideas to make things easier: figuring with tens; and
avoiding having lots of different units.
Use Powers of Ten to multiply & divide easily
Ten is an easy number to multiply and divide by.
Multiplying just slides the number left. Dividing slides it to the right.
We can say how `slid' a number is.
|x 10 =||10|
|x 10 =||100|
|x 10 =||1,000|
|x 100 =||100,000|
|/ 10 =||10,000|
|/ 10 =||1,000|
|/ 100 =||10|
|/ 10 =||1|
|/ 10 =||0||.1|
|/ 10 =||0||.01|
|/ 10 =||0||.001|
|x 10 =||0||.01|
|x 100 =||1|
1 is not slid. 10 is slid left by 1. 100 has a slide of 2.
Similarly, 0.1 (ie 1/10) is slid right by 1, which we will call a slide
of -1. So 0.01 (1/100) is slid -2, and 0.001 has a slide of -3.
These slides are called powers of ten.
Powers of Ten
|5|| 100,000||100 thousand|
|4|| 10,000||10 thousand|
|3|| 1,000||1 thousand|
|-1||0||.1 ||a tenth|
|-2||0||.01 ||a hundredth|
|-3||0||.001 ||one thousandth|
|-4||0||.0001 ||one 10-thousandth|
|-5||0||.00001 ||one 100-thousandth|
Names go by threes:
We can write 1,000 as 103 (exponential notation)
or as 1E3 (engineering notation). 100 is 102 or
1E2. And 200 is 2x102 or 2E2.
What is 13? Well, 10 is 101 (1E1) and 1 is
100 (1E0), so 13 is either 1.3x101 (1.3E1) or
13x100 (13E0). Usually folks prefer 1.3x101
(1.3E1). That way it looks similar to 10. It seems easier to think
of 13 as 10 with some excess than as a whole bunch of ones. Thus,
1900 is 1.9x103 (1.9E3).
Confused? Why bother?
Well, now we can do multiplication just by adding, and division just
For example, ten thousand divided by 100 is just 100 since (10,000 =
104 (1E4)), and (100 = 102 (1E2)), and 4-2 = 2,
and 102 (1E2) is just 100.
A trillion dollar national debt $1012 ($1E12) divided by
the a US population of a few 100 million people (108 (1E8))
is just, well 12 - 8, so 104 (1E4) dollars per person.
$10,000 per person.
If your head (1 foot) were the earth (4E7 feet) (thus a scale
difference of 4E7), then the moon (1E7 feet) would be 3 inches big
(1E7 / 4E7 = 1/4 foot). Its 1E9 feet away, so thats (1E9 / 4E7 =
1/4E2 or 1/4x100) 25 feet away. So if your head where the earth, the
moon would be a big mouthful sitting across the room.
Avoiding extra units
How long is 6307200 seconds? 1752 hours? 105120 minutes? 73 days?
10 weeks? 2.5 months?
They are all the same. Its 1/5 of a year.
Using many different units makes it difficult to see what is going
Instead, we can choose one unit, and use powers of ten.
[Need an example here... But I am having trouble coming up with
a pithy one...]
But we remember things in different units.
Maybe you can hold your breath for one minute. "One minute", thats easy to
Its also 2x10-6(2E-6) years.
(365 x 24 x 60 = 525600
minutes in a year, so 1 / 525600 is 0.0000019 or about 2E-6 (1E-6 being
But thats harder to remember. A week from now I'll mutter - "Hmm... was
that 10-5? or was it 10-8? or...".
["So we take what we remember, make it into powers of ten, and
then figure out what is going on..."
Need another example.]
Something vaguely like...
Say someone asks, About how many times does your heart beat during your life?
Where do we start? How often does our heart beat? How many times in
a minute? Well, counting heart beats while looking at my watch
gives... something like 80 beats per minute.
Since we are being
quick and dirty, lets just call it 100 (102) beats per
minute. What's the next question? How many minutes in a life?
Well, there are 400 x 25 x 50 = 500,000 (5 x 105) minutes in a year.
(Huh? Since we are being approximate, 400 x 25 x 50 is about
365 x 24 x 60... and much easier to multiply in my head.)
Ok, well, how many years old are you? Well, since you are reading
this you are, atleast, more like 10 than like 1. And since you are alive,
you are less than 100 years old.
So lets just say you are some tens (101) of years old. Might be about one 10, or might
be a few 10's.
How many heart beats in your life?
Well, 2 + 5 + 1 = 8, so ...
102 x 105 x 101 = 10 2 + 5 + 1 =
108. 100 million.
Oh, 5 x 108 ( left out the 5 from 500,000).
500 million. Half a billion.
So something like a billion heart beats in your life. Yipes.
[Need wrapup . . . ]
Comments encouraged. -
1997.Mar.11 This has been sitting around for too long. Lightly dusted. Released.
1997.Jan.22 Partial draft.