Scientific Notation & Units

Introduction

Adapted from http://casswww.ucsd.edu/physics/ph7/Units.htmli (H.E. Smith)

Since Astronomy is one of the oldest sciences, it has developed many traditions over the years. While an astronomer would not hesitate for an instant to apply cutting-edge technology at the telescope or computer, the language and units of measurement in the field change much more slowly. So, as you set out to investigate the Universe with us, let us introduce you to a few words, units and short-hand that you'll encounter here.

Scientific Notation

As we have seen in class, the important numbers in Astronomy span almost 40 orders of magnitude in size. Consider the mass of the Sun:

M_Sun = 1,989,000,000,000,000,000,000,000,000,000,000 grams

It's cumbersome, to say the least, having to write out all of those zeros. Even kilograms (eliminate 3 zeros) or metric tons (eliminate 6 zeros) don't help much. Furthermore, we really don't know the Sun's mass beyond the accuracy of the fourth digit. All those zeros are just place-keepers, carrying no useful information. For this reason, scientists use a short-hand called Scientific Notation to express very large or very small numbers.

In scientific notation the Sun's mass becomes:
M_Sun = 1.989 x 10³³ gm.
The number above the ten, called the power of ten or exponent, stands for the number of decimal places. If it is positive, as in the mass of the Sun, the decimal places are in front ofthe decimal point. So, 10³³ means "move the decimal point 33 places to the right and fill the empty places with zeros" (or, more mathematically, multiply by ten 33 times).

For very small numbers, such as the mass of the proton,

M_p⁺ = 0.000000000000000000000001673 grams
we use negative powers of 10. The mass of the proton becomes
M_p⁺ = 1.673 x 10^-24 grams
For negative exponents, the powers of 10 are after the decimal point; 10^-24 means "move the decimal point 24 places to the left and fill in with zeros" (or divide by ten 24 times).

In both of the examples above, the coefficient (the part before the times sign), contains 4 digits. This means that there are 4 significant figures in the number. If, for example, we knew the Sun's mass to 6 significant figures, we would say that its mass is 1.98900³³ grams.

There are several good web pages about Scientific Notation. If you would like to read a bit more, try out the University of Maryland's Astronomy Programs site, with a Scientific Notation Exercise and an Astronomical Distance Calculator.

Units

Adapted from http://casswww.ucsd.edu/physics/ph7/Units.htmli (H.E. Smith)

A centimeter is pretty small - not a very practical unit for the enormous distances in the Universe. If Astronomer A had to use centimeters to tell Astronomer B the distance to the Sun, it would look like this:

14,959,850,000,000 cm

There are three special units of distance used by astronomers. These are the astronomical unit (AU), the light-year and the parsec. The astronomical unit is the average distance of the Earth from the Sun shown above.

1 AU = 1.5 x 10¹³ cm = 150 million km = 93 million miles = 8.3 "light-minutes"

A light-year (ly) sounds like a measure of time, but it is a length - the distance light travels in one year.(We can use a light-year as a unit of measure because ALL light travels at the same speed; it is a fundamental constant of the Universe. More about this later...) So, in one year, light travels:

The name parsec comes from the technique of measuring distance called parallax, and will be introduced later.

Arithmetic Using Scientific Notation

From http://www.astro.lsa.umich.edu/Course/Bernstein102/Scinot/scinot.html

A number written in scientific notation consists of a coefficient (the part before the times sign) and an exponent (the power of 10 by which the coefficient is multiplied. For example, in 4.3 x 10⁶ (which equals 4,300,000; four million three hundred thousand), 4.3 is the coefficient and 6 is the exponent. Sometimes the "times" symbol "x" is replaced by a dot, for example 4.3∙10⁶.

When you multiply two numbers, you multiply the coefficients and add the exponents. For example,

4.3 x 10⁶ x 2 x 10² = 8.6 x 10⁸

4.3 x 10⁶ x 2 x 10^-2 = 8.6 x 10⁴

When you divide two numbers, you divide the coefficients and subtract the exponents. For example,

4.2 x 10⁶ ¸ 2 x 10² = 2.1 x 10⁴

4.2 x 10⁶ ¸ 2 x 10^-2 = 2.1 x 10⁸

When you move the decimal place in the coefficient one position to the left (i.e. you divide the coefficient by 10), you add one to the exponent. For example,

42 x 10⁶ = 4.2 x 10⁷

4200 x 10⁶ = 4.2 x 10⁹

42 x 10^-6 = 4.2 x 10^-5

When you move the decimal place in the coefficient one position to the right (i.e. you multiply the coefficient by 10), you subtract one from the exponent. For example,

0.42 x 10⁶ = 4.2 x 10⁵

0.000043 x 10⁶ = 4.3 x 10¹

0.42 x 10^-6 = 4.2 x 10^-7

You should always adjust the decimal place in the coefficient so that the coefficient is always greater than one but less than ten. Mathematically it doesn't make any difference, but that is the standard practice, and it does make a number easier to read.

When you add two numbers, you need to make their exponents equal. Take the number with the smaller exponent and move the decimal point to the left until its exponent matches the larger. Then add the coefficients and keep the (matching) exponent. For example,

4.2 x 10⁶ + 6.4 x 10⁵= 4.2 x 10⁶ + 0.64 x 10⁶= 4.84 x 10⁶

4.2 x 10^-6 + 6.4 x 10^-5= 0.42 x 10^-5 + 6.4 x 10^-5= 6.82 x 10^-5

9.2 x 10¹¹ + 9.4 x 10¹⁰= 9.2 x 10¹¹ + 0.94 x 10¹¹= 10.14 x 10¹¹ = 1.014 x 10¹²

When you subtract two numbers, you again need to make their exponents equal. Take the number with the smaller exponent and move the decimal point to the left until its exponent matches the larger. Then subtract the coefficients and keep the (matching) exponent. Note that you might have to adjust the exponent when you are done to get into "standard form." For example,

4.2 x 10⁶ - 6.4 x 10⁵= 4.2 x 10⁶ - 0.64 x 10⁶= 3.56 x 10⁶

4.2 x 10^-6 - 6.4 x 10^-5= 0.42 x 10^-5 - 6.4 x 10^-5= -6.38 x 10^-5

1.2 x 10¹¹ - 9.4 x 10¹⁰= 1.2 x 10¹¹ + 0.94 x 10¹¹= 0.26 x 10¹¹ = 2.6 x 10¹⁰

WARNING TO PEOPLE WHO USE CALCULATORS: Many calculators handle scientific notation. The exponent is usually displayed all the way on the right, with a space between it and the coefficient. To enter a number in scientific notation, you enter the coefficient, press the EXP key (one some calculators it is labeled EE) and enter the exponent. For example, 4. 05 means 4 x 10⁵. To enter the number 10³, you have to enter 1. EXP 03. DO NOT enter 10. EXP 03, since that equals 10 x 10³or 1 x 10⁴. This is a common mistake. See your calculator instruction booklet for more help.