Energy and Human Ambitions on a Finite Planet, 2021a

Math <strong>and</strong> Equations
A
Depending on background, math <strong>and</strong> equations may be an intimidating
“foreign language” to some students. This brief appendix aims to
offer a refresher on techniques, <strong>and</strong> hopefully inspire a more peaceful
relationship for students.
A.1 Relax on the Decimals
First, we can form a more natural, forgiving relationship with numbers.
Like your friends, they need not be held to exacting st<strong>and</strong>ards: they
are simply trying to tell you something useful. Remembering that π is
roughly 3 is far more important than committing any further decimals to
memory. If a friend traced out a circle in the s<strong>and</strong> <strong>and</strong> asked how much
area it had, 1 the poorly-defined <strong>and</strong> irregular boundary defies precise
measurement, so why carry extra digits. Maybe just recognize that the
radius is roughly one meter, so the area is about 3 square meters. Done. 2
The message here is to give yourself a break <strong>and</strong> just not over-represent
the precision (number of decimals) in your answer.
Part of the reason students have a rigid relationship with numbers is
because homework <strong>and</strong> test problems tend to come pre-loaded with
numbers assumed to be exactly known. But the real world is seldom
so generous, leaving us to forage for approximate numbers <strong>and</strong> estimations.
By being approximate in our use of numbers, we are liberated to do
math in our heads more readily. Practice can make this into a life-long
skill that becomes second nature. It is helpful to know some shortcuts.
A.1 Relax on the Decimals ... 359
A.2 Forget the Rules ....... 361
A.3 Areas <strong>and</strong> Volumes ..... 362
A.4 Fractions ............ 363
A.5 Integer Powers ........ 365
A.6 Fractional Powers ...... 366
A.7 Scientific Notation ...... 367
A.8 Equation Hunting ...... 368
A.9 Equation Manipulation .. 369
A.10 Units Manipulation ..... 370
A.11 Just the Start ......... 373
1: Now that’s a quality friend!
2: Also notice that the circle fits within a
square 2 meters on a side, so the area should
be less than 4 m 2 : it hangs together.
Some classes formalize the concept of significant
digits, which is all well <strong>and</strong> good.
But such systems can add to the stress of
students learning the material (one more
thing that can be wrong!).
Example A.1.1 To explore the flavor of approximate math, let’s consider
the statement:
π ∼ √ 10 ∼ 10
3 ∼ 3.
Your calculator will disagree, but that’s why we use the ∼ symbol
(similar) instead of (equal). Another common option is ≈ (approximately).
Your calculator is not as clever as you, <strong>and</strong> can’t appreciate
when things are close. It’s pedantic. Literal. You can be better.
How do we use this loose association? We saw one example of using
π ∼ 3 before, so will not repeat here.