In astronomy, we often deal with very large and very small numbers. It is often convenient to use scientific notation to work with such numbers. Such numbers can be expressed as a decimal number times 10 raised to some exponent or power. Let’s look at a few simple examples. You should verify these using your calculator.

10 = 10 = 10¹

100 = 10 x 10 = 10²

1,000 = 10 x 10 x 10 = 10³

1,000,000=10 x 10 x 10 x 10 x 10 x 10 = 10⁶

And so on.

Notice however, that

1,000 x 1,000 = 1,000,000

It would be equivalent to saying that

10³ x 10³ = 10⁶.

Notice that the exponent "6" is just the sum of the other two exponents "3" and "3"! This, in fact is a basic rule whenever you multiply two numbers with the same "base" (in this case, 10). For ANY values a and b, it is always true that

10^a x 10^b = 10^a+b

Now, the exponent can also be a negative number. In this way, we can write

1/10 = 10^-1

1/100 = 10^-2

and so on.

Suppose a+b =0. What then? Let’s figure it out by taking a specific example:

10 x 1/10 = 10¹ x 10^-1

What is one-tenth of ten (or equivalently ten divided by ten)? Answer: 1

What number do you get when you add the two exponents (1 and —1)? Answer: 0

So then, what number is the same as 10⁰ = 10^1+(-1)? Answer: 1 = 10⁰

Suppose we divide two numbers:

100,000/1,000 = 100 — i.e., 10⁵/10³=10² = 10^5-3. This suggests that when dividing two numbers with the same base, the result can be found by subtracting the exponents:

10^a/10^b = 10^a-b.

Let’s look at an earlier example again:

1,000 x 1,000 = 1,000,000

It would be equivalent to saying that

10³ x 10³ = (10³)² = 10⁶ = 10^3x2

Thus, when you raise a number to a higher power, the final number can also be expressed in the same way, with the final exponent being equal to the original exponent TIMES that power.

(10^a)^b = 10^(axb)

The reverse process also works. If the square of 1,000 is 1,000,000, then the square root of 1,000,000 is 1,000. That is,

10³ = 10^6/2 = 10^6x(1/2) = (10⁶)^1/2

Most numbers are not simple multiples of 10, but they still can be expressed using powers-of-10 notation. I.e.,

26 = 2.6x10 = 2.6x10¹

45,221 = 4.5221x10,000 = 4.5221x10⁴

0.786 = 7.86x(1/10) = 7.86x10^-1

We generally write these as was done in this last case, a decimal number between 1 and 10 time the power of 10.

Likewise, we can multiply and divide numbers as long as we follow the rules we discovered before:

2x10³ x 4x 0³² = 2 x 4 x 10³ x 10³² = (2x4)x10³⁺³² = 8x10³⁵

Here, we used another rule of algebra you probably heard about in school: the "commutative property (or rule) of multiplication", a fancy was to say that it doesn’t matter what order you multiply two numbers in, the product is the same either way (2x4 = 4x2 = 8).

We can divide numbers:

(4x10³²)/(2x10³) = (4/2)x(10³²/10³) = 2x10²⁹

We can raise numbers to exponents:

(4x10³²)² = (4x10³²)x(4x10³²) = (4x4)x(10³²x10³²) = 16x10⁶⁴ = 1.6x10⁶⁵

It is often useful to use words to describe exponents, and this is a common practice when using metric units. For example, the standard unit of length is the meter, abbreviated "m".

1000 meters = 1 kilometer = 1 km

1/100 meter = 1 centimeter = 1 cm

1/1000 meter = 1 millimeter = 1mm

1/1,000,000 meter = 1 micrometer = 1 μ m

The prefixes that are usually used, by tradition, are almost always in multiples of 1000 (kilo-, milli-, micro-). The centimeter is an exception to this "rule", but is used so commonly that it is included here.

Which brings us to the use of units. In physics (including astronomy), the "official" units are the so-called S.I. metric units of the kilogram, meter, and second for mass, length, and time — "mks". Many textbooks use a variant — gram, centimeter, and second — "cgs".

In astronomy, we often find it useful to define special units. The semi-major axis of the Earth’s orbit around the Sun is about 150 million km = 1.5x10¹¹m = 1 "Astronomical Unit" = 1 AU. The semi-major axis of Jupiter’s orbit is about 5.2 AU. It is more easy to visualize distance scales in the solar system using units whose values are "familiar" numbers such as 1 and 5. Some scientific calculators (like the one I have) actually have the value of the AU programmed into it by the manufacturer!