Energy and Human Ambitions on a Finite Planet, 2021a

A.1 Relax on the Decimals 360
What about √ 10 ∼ 3? 3 This implies that 3 × 3 ∼ 10, which is true 3: Really √ 10 ≈ 3.162.
enough (because 9 <strong>and</strong> 10 are very close; only 10% different). This
means if you pay me $30 per day for a month, I know immediately
that’s about $1,000. Is the month 30 days, or 31? Who cares? Knowing
I’ll have an extra ∼$1,000 is a good enough basis to make reasonable
plans, so it’s very useful, if not precise.
How about 10 3 ∼ 3?4 This one is actually pretty similar to saying that 4: Really it’s about 3.333.
√
10 ∼ 3, since both imply 3 × 3 ∼ 10. To use another example, let’s say
you l<strong>and</strong> a $100,000/yr job, but can only work for 4 months (a third of
a year). 5 If 10 3
∼ 3, then you’d expect to get about $30,000. Why mess
around being more precise? Taxes will be larger than the imprecision
anyway. Again, it’s good enough to have a sense, <strong>and</strong> make plans.
Much like we have multiplication tables stamped into our heads, it is
often very useful to have a few reciprocals floating around to help us do
quick mental math. Some examples are given in Table A.1 that multiply
to 10. Students are encouraged to add more examples to the table, filling
in the gaps with their favorite numbers.
The values in Table A.1 are selected to multiply to 10, which is an arbitrary
but convenient choice. This lets us “wrap around” the table <strong>and</strong> continue
past three down to 2.5, 2, etc. <strong>and</strong> learn that the entry for 1.5 would
be 6.67. To make effective use of the table, forget where the decimal point
is located! Think of the reciprocal of 8 as being “1.25–like,” meaning it
might be 0.125, 12.5, or some other cousin. The essential feature is 125.
Likewise, the reciprocal of 2.5 is going to start with a 4.
5: Money examples often seem easier to
mentally grasp because we deal with money
all the time. To the extent that money examples
are easier, it says that the math itself
isn’t hard: the unfamiliar context is often
what trips students up.
Table A.1: Reciprocals, multiplying to 10.
Number
Reciprocal
8 1.25
6.67 1.5
6 1.67
5 2
4 2.5
3 ∼3
Example A.1.2 How is Table A.1 useful to us? We can turn division
problems, which tend to be mentally challenging, into more intuitive
multiplication problems. Several examples may highlight their
usefulness.
What is one eighth of 1,000? Rather than carry out division, just
multiply 1,000 by the reciprocal—a “125–like” number. In this case,
the answer is 125. We can use common sense <strong>and</strong> intuition to reject
1,250 as 1 8
of 1,000, as we know the answer should be significantly
smaller than 1,000. But 12.5 goes too far. Also, we can recognize that 1 8
is not too far from one-tenth, 6 one-tenth of 1,000 would be 100, close
to 125.
6: This is where “blurry” numbers are useful:
8 ∼ 10 if you squint.
How many hours is one minute? Now we are looking for one-sixtieth
( 1
60
) of an hour, so we pull out the “167–like” reciprocal <strong>and</strong> weigh
1
1
the choices 0.167, 0.0167, 0.00167, etc. Well,
60
can’t be too far from
100 ,
which would be 0.01. We expect the result to be a little bigger than 1
100 ,
leaving us to have one minute as 0.0167 of an hour.
Now we do a few quick statements that may not match our table
exactly for all cases, but you should be able to “read between the lines”
using blurry numbers to reconcile the statements. One out of seven
Study Table A.1 for each of these statements
to see how it might fit in.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.