Chapter 1: How Science Works

This is an example of e notation. It is most often seen on a calculator where the e stands for x10 and the number to the right of the e is the power which the 10 is raised to. This specific number is known as Avogadro's number.

In science, and particularly in astronomy, there are many extraordinary numbers to describe distances, ages, temperatures, and densities. It is clearly not very convenient to write the distance that light travels in a year as 6,000,000,000,000 miles or the density of interstellar space as 0.000000000000000000000002 g/cm³. Astronomers and other scientists have developed a shorthand system for writing such very large and very small numbers called "Scientific Notation."

 

In this system, a number has two parts, separated by the multiplication sign. The part in front of the multiplication sign is called the coefficient, and it is typically a number like 5.2 or 6.987 that only has one digit to the left of the decimal point. The power of ten that follows the multiplication sign is called the exponent. (Sometimes scientific notation is also called exponential notation or just "powers of ten" notation.) As an example, let's rewrite the two numbers above in scientific notation: 6,000,000,000,000 miles = 6 × 10¹² miles, 0.000000000000000000000002 g/cm³ = 2 × 10^-24 g/cm³. Scientific notation is a much more compact way to express large and small numbers.

The exponent of a number written in scientific notation tells you how many places to shift the decimal point to the left or the right to form the number. In other words, 2.51 × 10⁵ means shift the decimal point plus five places, or five places to the right, giving 251,000. On the other hand, 6.8 × 10^-7 means shift the decimal point minus seven places, or seven places to the left, giving 0.00000068. If there is a zero in the exponent, the decimal point does not move (which makes sense since 10⁰ = 1). Here are some examples of numbers written in normal notation (on the left) and scientific notation (on the right):

• 0.000490372 = 4.90372 × 10^-4

• 3001 = 3.001 × 10³

• 0.000002 = 2 × 10^-6 (or 2.0 × 10^-6)

• 100,000,000 = 10⁸ (or 1.0 × 10⁸)

• 0.887 = 8.87 × 10^-1

• 148,400 = 1.484 × 10⁵

 This is a visualisation of powers of ten from one to 1 billion. Each block after the cube on top is ten times the size of the previous block, giving a visualisation of the size of a cube one billion times the size of the original cube.

Remember that the exponent is the number of factors of ten that are multiplied together to get the quantity. For example, 10⁰ = 1, 10¹ = 10, 10² = 10 × 10 = 100, 10³ = 10 × 10 × 10 = 1000, and so on. The most often used large numbers are one thousand (10³), one million (10⁶), one billion (10⁹), and one trillion (10¹²). The exponent in any power of ten just gives the number of zeros that follows the one.

 

When units are involved, scientific notation may get consumed into the prefix of a given unit, such as a kilometer (10³m) and millimeter (10^-3m). For example:

• 0.0000673 m = 6.73 × 10^-5 m = 0.0673 mm = 67.3 µm

• 56,000 J = 5.6× 10⁴ J = 56 kJ = 0.056 MJ

• 127,000,000 pc = 1.27 × 10⁸ pc = 127 Mpc = 0.127 Gpc

• 0.0000000899 s = 8.99 × 10^-8 s = 0.00899 ms = 0.899 ns

• 145 g = 1.45 × 10² g = 145,000 mg = 0.145 kg

• 51,000,000,000 W = 5.1 × 10¹⁰ W = 51 GW = 0.051 TW

To work with large and small numbers, you need to know the rules for multiplying and dividing in scientific notation. To multiply two numbers together, you multiply the coefficients and add the exponents. (Be careful to preserve signs.) Here are some examples:

• (7.91 × 10⁴) × (2 × 10⁷) = (7.91 × 2) × 10⁴⁺⁷ = 15.82 × 10¹¹ = 1.582 × 10¹²

• (6.9 × 10⁸) × (1.1 × 10^-5) = (6.9 × 1.1) × 10^8+(-5) = 7.59 × 10^8-5 = 7.59 × 10³

• (4 × 10^-6) × (5.8 × 10^-11) = (4 × 5.8) × 10^-6+(-11) = 23.2 × 10^-17 = 2.32 × 10^-16

To divide two numbers, divide the coefficients and subtract the exponents. (Once again, be careful to preserve signs when you subtract exponents.) Here are some examples:

• (3 × 10⁶) / (6.3 × 10⁴) = (3 / 6.3) × 10^6-4 = 0.48 × 10² = 4.8 × 10¹ (or just 48)

• (8.35 × 10⁶) / (2.7 × 10^-6) = (8.35 / 2.7) × 10^6-(-6) = 3.07 × 10⁶⁺⁶ = 3.07 × 10¹²

• (7.5 × 10^-8) / (9 × 10^-7) = (7.5 / 9) × 10^-8-7 = 0.83 × 10^-15 = 8.3 × 10^-14

The standard order of operations in arithmetic must be kept in mind when dealing with numbers in scientfic notation.

Parentheses are used to group items and operations in mathematics. Always complete the operations within each set of parentheses before moving outside the parentheses. (In other words, do not just work out an equation from left to right.) If a complex problem involves more than two large or small numbers, the principles are the same. Just group all the coefficients and multiply or divide them, and group all the exponents and add or subtract them. Look at the following example:

 

• [(3.4 × 10⁴) × (6 × 10^-8)] / [(1.6 × 10^-9) × (4.7 × 10⁵)]
= [(3.4 × 6) / (1.6 × 4.7)] × 10^{4+(-8)-(-9)-5}
= (20.4 / 7.5) × 10^4-8+9-5
= 2.7 × 10⁰ (or just 2.7)

Astronomers often use the skill of estimation, where they combine large and small numbers to get a rough estimate of an important quantity. Preserving a lot of significant digits is not important in these calculations — estimation is only intended to be accurate to about a factor of two. For such "back of the envelope" calculations, often only the exponent is retained.