Basic Math Concepts

Geometry

Circles

Astronomy makes extensive use of circles and fractions of circles to describe positions on earth and in space. It's useful, therefore, to discuss just what's involved in the geometry of circles and angles.

A circle just needs one number to describe it: the radius. Pick a center point (describing where that point is takes more numbers, but we'll get to that later). Pick a distance, which we'll call the radius "R". A circle is, simply, everything in two dimensions (on a flat surface, for instance, like a piece of paper) that lies exactly at the distance "R" from the center point. The distance across the circle in a straight line is called the diameter; you can see that a diameter is simply two times the radius of the circle, 2R. The distance around the edge of the circle is called the circumference, and the region enclosed by the circle is called its area. So far, so good.

Some smart-aleck ancient mathematicians found out that there was an exact relationship between the circumference and the diameter of any circle; the circumference divided by the diameter is always the same number. This number starts as 3.14159... and keeps on going to an infinite number of digits. Since that many digits was a serious pain to write, they symbolized that number with the greek letter "pi" ().

We can write that relationship down as a formula. Restating the definition of , we can write: the circumference of a circle equals the diameter times . Mathematically,

circumference = diameter x = 2 R

The area of a circle also depends on : area = R² .

Spheres

A sphere extends the concept of a circle into three dimensions; that is, a sphere is all the points in three-dimensional space that are a distance R from its center. The surface area of the sphere is, essentially, the amount of area you would have if the sphere were cut open and laid flat. The volume, of course, is how much space there is within the sphere.

surface area of a sphere = 4 R²

volume of a sphere = (4/3) R³

Angles

An angle is a fraction of a circle. One of the most common units for angles is the degree (derived from the ancient Babylonian units, which used a numerical system based on 60). Start at any point along the edge of a circle, then trace the circle around to your starting point; you will have gone through 360 degrees (360^o). One quarter of the way around the circle is 90^o, one half of the way around is 180^o, and three quarters of the way around is 270^o.

Each degree can be divided up further. Every degree is made up of 60 minutes of arc (60'). Every minute of arc is made up of 60 seconds of arc (60"). So, for instance, 3^o = 3 x 60' = 180', and 180' = 180 x 60" = 10800". Angles are thus written in degrees, minutes of arc, and seconds of arc; 3^o 4' 21" is 4 minutes of arc plus 21 seconds of arc more than 3^o.

Algebra

Proportionality

You may sometimes see equations of the shape Y X, or "Y is proportional to X". This is a way of showing the relationship between quantities without worrying about fiddling with the arithmetic. When you've got a mix of constants and variables, you can look at the important thing - the way some quantities change as other quantities do - without needing to plug in the numbers.

First look at direct proportionality. This has the form "Y X", which is the same as saying that Y equals some constant times X. Thus, as X increases, Y increases; as X decreases, Y decreases.

Example: The section above shows that the circumference of a circle is proportional to its radius. That is, C = 2 R. For quick approximations, we can simply say that C R; thus, if R is increased by a factor of 2, C is also increased by a factor of 2.

OK, now let's try using exponentials in direct proportionality (see Exponential Notation below). This has the form of "Y X^a", or "Y is proportional to X to the a power." Saying "Y X²" means that if X increases, Y increases a lot; in fact, as the square of X. Likewise, saying "Y X³" means that if X increases, Y increases even more than when Y X²; Y goes as the cube of X.

Example: Once again drawing from the section above, we see that the area of a circle is proportional to its radius squared. That is, A = R², or A R². Thusly, if R is increased by a factor of 2, A is increased by a factor of 2², that is, a factor of 4.

We can do the same thing for inverse proportionality. This takes the forms of "Y 1/X", meaning Y is equal to a constant times one over X. In this cases, if X increases, then Y decreases, and if X decreases, then Y increases.

We can just as easily use exponentials in inverse proportionality. That will take the form "Y 1/X^a". Thus, "Y 1/X²" means that if X increases, Y decreases lots - by the square of X. If Y 1/X³, Y decreases even more rapidly as X increases.

Example: The gravitational force between two objects depends on their masses and the distance (D) between them; if the masses are constant, then this force F 1/D². So, if D is increased by a factor of 10 (the masses are moved further apart), the gravitational force between the two objects decreases by 10², that is, a factor of 100. The force for the more distant objects is 1/100 of what it was when they were closer together.

Simplifying Equations

Suppose you have the equation, A + B = X² Y - Z. You're told that you need to solve for X. How do you find it? The answer is, by simplifying the equation so that X is shown in terms of all the other variables. The way to do this is to do the same arithmetic operations to both sides of the equals sign. In this case, the objective is to get X by itself on one side, and everything else on the other side. Let's take it in steps.

• A + B = X² Y - Z
• A + B + Z = X² Y - Z + Z
• A + B + Z = X² Y
• (1/Y) (A + B + Z) = X² Y (1/Y)
• (1/Y) (A + B + Z) = X²
• [(1/Y) (A + B + Z)] = [X²]
• [(1/Y) (A + B + Z)] = X

The operations you can usually use to simplify an equation are:

1. Add the same thing to both sides
2. Subtract the same thing from both sides
3. Multiply both sides by the same thing
4. Divide both sides by the same thing
5. Square (or cube, etc.) both sides
6. Take the square root (or cube root, etc.) of both sides

Numbers and Notations

Exponential Notation

Astronomy works with some very large numbers, and with some extremely small numbers. In order to deal with these numbers easily, astronomers (and other scientists) express them as powers of ten, in what is called "Scientific Notation" or "Exponential Notation."

Here's how it works. Suppose you have the speed of light, in meters per second. That number is:

299,800,000 m/sec

That's a bit of a nuisance to write out over and over. Instead, we express it as powers of ten:

299,800,000 = 29,980,000 x 10
      = 2,998,000 x 100  = 2,998,000 x 10 x 10
      = 299,800 x 1,000  = 299,800 x 10 x 10 x 10
      = 29,980 x 10,000  = 29,980 x 10 x 10 x 10 x 10
      = 2,998 x 100,000   = 2,998 x 10 x 10 x 10 x 10 x 10

Wait! We've run out of zeros! However, we can still keep going, by using a decimal point:

      = 299.8 x 1,000,000    = 299.8 x 10 x 10 x 10 x 10 x 10 x 10
      = 29.98 x 10,000,000  = 29.98 x 10 x 10 x 10 x 10 x 10 x 10 x 10
      = 2.998 x 100,000,000 = 2.998 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

Now, you may well ask, "How does that make things easier? That's much larger than the original number!" Well, we can sum that up by using exponentials:

299,800,000 = 2.998 x 100,000,000 = 2.998 x 10 ⁸

This is simpler! Something similar can be done for small numbers:

0.000000356 = 3.56 x 0.0000001 = 3.56 x 10 ^-7

A simple way to convert back and forth from normal notation to exponential notation is to just count the digits. For a number greater than zero, simply count all the digits between the first digit (which is not counted), and the decimal point. So since 299,800,000 has eight digits after the first digit (the "2"), the exponent is 8 and the final number is 2.998 x 10 ⁸.

Significant Figures

Suppose you divide 2.0 by 3.0. Your calculator will display something like 0.666666667. But are all those digits really necessary? To answer that question, we turn to the rules for significant figures. Significant figures are all the digits that are not used only as place holders. A place holder is always a zero, although not all zeros are place holders. For instance, in the number 0.35, the zero is only there in order to hold the "ones place;" it is only intended to show that the entire number will be smaller than one. Therefore, only the ".35" is really significant; the number 0.35 has two significant figures. In the case of a number like 0.0000316, all five zeros are place holders, showing that there is nothing in the ones place, the tenths place, the hundredths place, the thousandths place, or even the ten-thousandths place. You have to go all the way to the hundred-thousandths place to find any numbers. Thus, all five zeros are insignificant figures. The number 0.0000316 has only three significant figures.

On the other hand, zeros that come after the decimal and after at least one other significant digit are significant. Thus, 0.00003016 has four significant figures, and so does 0.00003160. The number 0.000031600 has five significant figures, and so on. If there are only zeros after the decimal point, then the zeros are significant: 1.000 has four significant figures.

What about numbers that come before the decimal place? These are a little more tricky. Zeros that come between two nonzero digits are always significant. If there is a decimal point, any zeros in front of that decimal point are significant. Therefore, 10.1 has three significant figures. Sometimes a decimal point alone is used to indicate that all the zeros before the decimal point are significant, even though there are no digits after the decimal. So, 19,000. has five significant figures. However, if there are no decimal points, it can be hard to tell which zeros are not significant. Does the number 1000 have one significant figure, or four? Sometimes people will draw a bar over the last significant digit, but if not, you must use your own best judgement.

If one uses scientific notation, finding the number of significant figures is much easier. Above, we saw that

0.000000356 = 3.56 x 10 ^-7 .

Well, we can immediately see that all those leading zeroes don't count; in scientific notation, they don't even show up! And if you had a trailing zero, it would be written like this:

0.00000035600 = 3.5600 x 10 ^-7 .

So using scientific notation, you have the very simple rule: The number of digits in the first part (before the " x 10^? ") is the number of significant figures. Easy, huh? That's one reason why scientists like scientific notation.