3
C H A P T E R 1
Tools for Understanding Space
e universe is big. But it is also very fine-grained, and so there is important detail at both
extremely tiny and unimaginably huge scales of size and distance. To fit this into the human
mind, we need some tools. One of the most important of such tools is the ability to relax:
abandon the idea that it can all make sense at once, and in detail. Instead, we learn how to split
it up into pieces that are manageable to the human mind, and we then learn how to relate those
piece to each other. In the sections that follow, I describe several methods that are useful for
making sense of both the bigness and the tininess, while still keeping track of how it all fits
together.
roughout e Big Picture I make use of both scientific notation and the SI system of physical
units. For readers unfamiliar with either of these tools, Appendix A provides details.
1.1 POWERS OF TEN
e diameter of a grape is roughly 10 mm greater than that of a pea; the diameter of an apple is
approximately 90 mm greater still.
1
But to say that the diameter of Earth is 6;384;000;000 mm
more than that of an apple tells us little. To meaningfully compare sizes that are so disparate, it
is not how much but rather how many times bigger that is important.
e logarithm is a mathematical tool that compares numbers by factors rather than dif-
ferences. Its foundation is the exponential function. As an example of an exponential function, a
sequence of numbers such as 1, 2, 4, 8, 16, 32, … can be written instead as follows: 2
0
, 2
1
, 2
2
,
2
3
, 2
4
, 2
5
, …. And so we can describe the sequence with the exponential function y D 2
x
.
is particular example is an exponential function with a base of 2, but other bases may
be used; the most common is the base 10 exponential function:
y D 10
x
: (1.1)
And so values for x of 1, 2, 3, 4, … simply represent y values of 10, 100, 1000, 10,000, …, or
the number’s power of 10. e inverse of this particular exponential function is called a common
logarithm, and it is defined such that:
log.10
x
/ D x: (1.2)
And so log 10 D 1, log 100 D 2, log 1000 D 3, …. For numbers smaller than one, loga-
rithms have negative values. For example, 0:01 D 10
´
2
, and so from Equation (1.2), we must
1
Parts of this section appeared, in a somewhat different form, in Beaver [2018b, Sec. 1.6].
4 1. TOOLS FOR UNDERSTANDING SPACE
-3
-2
-1
0
1
2
0 20 40 60 80 100
y = log x
Figure 1.1: A graph of the common logarithm (log x) is defined in terms of powers of 10, and
so depicts 0.01 as ´2, 0.1 as ´1, 10 as 1, and 100 as 2.
have log 0:01 D ´2. But although the logarithm of a number can be negative, there is no such
thing as the logarithm of a negative number. Furthermore, there is no value, x, for which
10
x
D 0. And so although a logarithm can be zero (log 1 D log 10
0
D 0), the logarithm of zero
is undefined. A graph of the common log function can be seen in Figure 1.1.
Logarithms have interesting and useful mathematical properties, the most important of
which are the following:
log.ab/ D log a C log b (1.3)
log a
b
D b log a: (1.4)
Because of Equation (1.4), logarithms “undo” exponents, turning them into simple factors. And
Equation (1.3) shows that logarithms turn multiplication into addition. For this reason, pub-
lished tables of logarithms were important in pre-computer times, since addition (and subtrac-
tion) is easier to perform by hand than multiplication (or division). is led to the old joke that
certain types of snakes only reproduce when placed on a wooden table.
2
Apart from that, some
species of rodent naturally think in terms of logarithms; see Figure 1.2.
e left side of Figure 1.3 shows the exponential function y D 10
x
plotted vs. x, while the
right-hand graph shows that same function plotted with a logarithmic scale for the y-axis. On a
logarithmic scale, each tic mark of the graph represents not an amount, but rather a factor. For
the example shown here, each tic is 10 times greater than the tic below it. Since log.10
x
/ D x,
this is simply a straight line. And so use of a logarithmic scale on the vertical axis of a graph has
the effect of turning an exponential function into a straight line. Notice how the logarithmic
2
Even adders can multiply on a log table.
1.1. POWERS OF TEN 5
Figure 1.2: Of all members of the order Rodentia, Castor canadensis is most attuned to thinking
in terms of powers of ten (Beaver image CC BY-SA 2.0).
y = 10
x
= 10
Figure 1.3: e exponential function y D 10
x
plotted on an ordinary linear scale (left), and a
logarithmic scale (right). e logarithmic scale straightens the exponential function and makes
a large range of values more manageable.
scale compresses an enormous range of values to a much smaller scale. Most of the direct graph
of the exponential function, as shown on the left, is almost useless; at the scale of the graph,
90% of it is either indistinguishable from zero or nearly vertical.
And so a logarithmic scale is especially useful when trying to compare numbers to each
other that encompass an enormous range of values. We represent numbers by their exponents,
using a logarithmic scale to compare their powers of ten, rather than the numbers themselves.
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