Things you should have learned in high school, but may have forgotten....

Scientific Notation

When numbers are very large or very small, you should write them in a short-hand form. For example, let's say you have a number like 560,000,000 or 0.0000003. Would you want to write those numbers over and over again? How can you simplify them? First of all, there is the way that the power of 10 works - that you have the following relations -

101 = 10 (anything to the power of 1 equals itself)
102 = 100
103 = 1000
104 = 10000
and so forth.

You'll notice how the power on the 10 equals the number of zeros after the 1. How do you write 560,000,000? You might notice that this number is also equal to 5.6 x 100,000,000, which can be written as 5.6 x 108. This format is known as scientific notation. What about numbers smaller than 10? How do you write those? Here's how -

100 = 1 (any number to the power of 0 = 1)
10-1 = 0.1
10-2 = 0.01
10-3 = 0.001
and so forth.

These aren't as easy as the others. In this case, the power on the 10 could represent how many places the decimal point is moved to the left of the 1. Now back to the original question: how do you write 0.0000003 using this? You might note that 0.0000003 is equal to 3 x 0.0000001, which is 3. x 10-7. You'll notice that in both cases, the decimal point is placed after the first non-zero number. This is the normal way that these numbers are written and it is also useful to use this method when writing similar numbers.

Let's try some more. How would you write the following numbers?

  1. 0.00045
  2. 345000
  3. 0.066
  4. -0.000102
  5. -53000
Check your answers here.

The next thing that you might have to do with numbers in scientific notation is to multiply or divide them. Generally this is pretty easy with a calculator. One of the things people don't know about their calculators is that there is usually a built in key that allows you to represent numbers in scientific notation in your calculator. The key is usually labeled with EE or Exp - what it actually is called will depend upon the type of calculator you have.

Let's go through an example of how you put a number into your calculator. Let's say you want to do the following problem -

4.5 x 10-5 x 3.3 x 106.

Here's what you do -
1. Put in the front part of the first number, 4.5 in this case.
2. Press the EE or Exp key. This usually causes a "00" or a "x10" to show up.
3. Enter the power on the 10, in this case -5. You don't want to use the subtraction key; use a key labeled ▒ or +/-. Don't put a "10" in since your calculator already has taken care of that.
4. Now press the multiply key.
5. Enter the second number, first the 3.3.
6. Press the EE or Exp key.
7. Put the power that is operating on the 10 in, which is 6 in this case.
8. Now press the equals or enter key to finish off the calculation.

If everything went all right, you should have gotten 148.5 (or 1.485 x 102, 1.485E2, or 1.485e2). Any way, that's the answer you should have gotten. If you didn't you should review the steps.

Here are some examples to try out - try to get the correct answer in each case

  1. 9.9 x 10-5 x 4.5 x 10-8
  2. 1.02 x 104 / 3.3 x 10-9
  3. 4.0 x 10-2 x 1.0 x 108
  4. 8.2 x 109 x 5.3 x 108
  5. 9.8 x 1019 + 4.2 x 10-9
Check your answers here.

In case you don't have a calculator, you can still do the math, mainly the multiplication and division, by following these rules -


If you don't believe this, try the practice problems above without a calculator or without using the EE/EXP key. You should get the same answers. Actually, you will only want to do this with problems 1-4 since the last one is an addition.

WORD OF WARNING: If you use a calculator to work with numbers in scientific notation, your calculator may write them in a way that they are not normally written.  For example, the number 3.4 x 1022 could appear in your calculator like 3.4     22  or 3.4     22.  The "x 10" part is often excluded to save space.  If you were to do a calculation and your calculator gives an answer similar to that shown above (without the "x 10" part), make sure you write the number out correctly - don't forget to write out the "x 10" part.

Why?  There is a big difference between those numbers.  If your calculator displays 3.4 22, and you write on your answer sheet 3.4 22, you'll lose points, since 3.4 22 means 3.4 taken to the power of 22, not what it is supposed to mean (3.4 x 1022).  That's a big difference - don't be lazy; write out the number properly.
 

When do you use scientific notation? This is one of those questions that doesn't have a solid answer. If you have a number like -0.2, or 123, you really don't need to write them in scientific notation, since that would be a bit silly and make the number harder to read. In general it is best to use scientific notation if the number is in the millions or greater, or if it is smaller than 0.001. While these are only guidelines you should do whatever you are comfortable with.

Accuracy - Significant Figures

One thing that you'll run across if you use a calculator is that it is very literal in doing calculations. For example, if you divide 10 by 13 you'll get 0.076923076923... Do you really have to write all those numbers down? No, of course not. You should round the numbers off so that there are the correct number of significant figures (SF). These are the number of digits that are needed to give an accuracy that is appropriate for the problem. Here're a few rules to follow -

1. Digits other than zero are always significant.
2. Zeros between other SF are significant. For example, 4003 has four SF.
3. Zeros to the left of the first non-zero digit in a number are not significant; they merely indicate the position of the decimal point. For example, 0.033 has two SF, 0.000401 has three SF. These are also called leading zeros.
4. When a number ends in zeros and the zeros are to the right of the decimal point, they are significant. For example, 0.00330 has three SF - 3, 3, and the rightmost 0 in this case.
5. When a number ends in zeros that are to the left of the decimal point location, the zeros are not necessarily significant. For example, 440 certainly has at least two SF, but the 0 may or may not be significant. You'd probably have to know more about the number, particularly how it was determined. One way to remove the ambiguity is to include the decimal point in the number. If I wrote "440." then I would want to include the zero as a SF, so there are now three SF in the number. Writing just "440" doesn't make it clear as to whether the zero is significant or not. Including a decimal point will help. So "329000" has three clearly SF, but the rest are uncertain. The number "329000." has six SF.

Sometimes a zero is included after a decimal to show that there is a need for greater accuracy, so that 3.20 has three SF, while 3.2 only has two. For some reason the person who wrote 3.20 wanted greater accuracy when the number is used in further calculations, and the rounding rules for significant figures takes effect (we'll get to those later).

Here are some for you to try - determine the number of SF in each number

  1. 6.751
  2. 0.157
  3. 28.0
  4. 2500
  5. 0.070
  6. 30.07
  7. 0.0067
  8. 6.02 x 1023
Check your answers here.

Now to actually use SF.  If the numbers you are using are measurements of some sort, odds are they are not exactly accurate, and the level of accuracy is given by the number of SF. In the case of addition and subtraction, your final answer should have the same number of decimal places as the value with the least number of decimal places. If you were to take 9.221 - 7.01 your answer should be 2.21, not 2.211. It is best to determine the number of SF when you get to the end of all of your mathematical steps - so when you are ready to write down the final answer, double check to see how many SF you should write down.

For multiplication and division, the answer canĺt be more accurate than the least accurate part. If you were to multiply 3.209 by 2.2 your answer should have only two SF in it since that is the least number of SF in the values you were given. You should write the answer as 7.1, not as 7.0598, which is the number your calculator spits out.

A word of warning - if you are using a number that is not a measured quantity like a constant that has an exact value, you should not use it to decide the number of SF in your final answer.  Exact values or constants are considered to have an infinite number of SF.  For example to calculate the circumference of a circle, you use the formula 2  r, where =3.14 (or more SF) and r is the radius of the circle. How many SF will your answer have?  Should there be only 1 SF, since that is how many are in the 2?  No. The 2 is not a measurement but a constant, a part of the formula.  The same rule applies to the The number of SF would depend upon the number of SF in the r, so depending upon the value of r, the number of SF in your answer can vary.

Here are some examples to try - determine the answer for each using the proper number of SF.

  1. 3.4 + 0.00344
  2. 4.50 x 3.3005
  3. 9.01/7.88
  4. 4.510 x 1012 x 3.401 x 10-11
  5. 607.1 x 4.4
Check your answers here. You should ALWAYS follow the rules of SF when you do math problems, especially when you calculator spits out numbers like 7.38029347234. If an answer should have only three SF and you write out an answer like 7.38029347234, you will get points taken off. If you aren't sure how many SF to include, ask.

Rounding

Once you know how many significant figures are in an answer, you need to round the value that your calculator produced appropriately. Let's say you got an answer of 36.329119 in your calculation. What would be the answer if you are required to have 5 SF? 4 SF? Or fewer? Well here are the possible answers:

You may wonder why a value like 40 is correct if you could also have a value like 36.24. It isn't that more significant figures are always better than fewer - if they aren't supposed to be there, then they are providing a false level of accuracy. Correctly presenting not only the answer but also the level of accuracy provides the best possible answer. So make sure you round up your final answer. Also do not round after each step of a calculation, but only at the end. You just need to know how many SF will be needed in each step of the calculation and then round appropriately at the end.

Order of Calculations

Sometimes when you are given a math problem the way that the formula is presented may not explicitly provide each mathematical symbol necessary to calculate the answer correctly. For example, what if you had the following problem to work out?
6/2(1+2) = x
What value did you get for x? What value should you get for x? How about this one?
6 - 1 x 0 +2/2 = y
What did you get for y? What is the order for the calculation? Is it just left to right or are some functions more important than others?

The answer is that there is an order for doing calculations, and that is the following -

  1. Parentheses are first
  2. Exponents (powers)
  3. Multiplication - Division (equal standing, so whichever is first going from left to right)
  4. Addition - Subtraction (equal standing, so whichever is first going from left to right)
There is often the common convention of not using the "x" symbol to show multiplication when it is possible to have two values next to each other and not confuse them, like a parentheses between values as in the first formula above.

So for x you should get a value of 6/2(1+2) = 6/2(3)= 3x(3) = 9.
And for y you should get 6 - 1 x 0 + 2/2 = 6 - 0 + 1 = 7.

These questions were actually on Facebook and many people get them incorrect by not following the rules of order.

How about this formula?
2 x 52/10 = z
What did you get?
How about the following -
(2 x 5)2/10 = zz
What did you get? Are z and zz the same value? For z you should have gotten the following
2 x 52/10 = 2 x 25 / 10 = 50/10 = 5
While for zz you should have gotten
(2x5)2/10 = (10)2/10 = 100/10=10.

Units of measure

For the most part, metric units are used in astronomy. Here's a quick re-cap of some of the common ones you'll run into -

Mass

kilograms - kg
grams - gm

Distance/Length

meters - m
kilometers - km
centimeters - cm
millimeters - mm
┼ngstroms - ┼

The trickiest thing is trying to remember what each of these things are in case you have to convert from one unit to another. Here is a listing if you need to convert from one value to another.

1 kg = 1000 gm
1 gm = 0.001 kg

1 meter = 100 cm = 1000 mm = 0.001 km = 1010
1 cm = 10 mm = 0.00001 km (10-5 km) = 0.01 m = 10 8
1 mm = 0.000001 km (10-6 km) = 0.001 m = 0.1 cm = 10 7
1 km = 1000 m = 100,000 cm = 1,000,000 mm = 1013
1 ┼ = 10-10 m = 10-8 cm = 10-7 mm = 10 -13 km
 

There are also some non-metric distances that are set up just for convenience. For example, there is the distance between the Earth and the Sun, which is defined as 1 A. U. (Astronomical Unit). This is useful for measuring distances within the solar system. There is also the distance of a light-year - the distance light travels in one year. Parsec is another distance which is often used interchangeably with light-year. These are often used to indicate the distances between stars. For even greater distances there are kiloparsecs and kilo light-years (1 kiloparsec = 1000 parsecs, 1 kilo light-year = 1000 light-years) and for very great distances there is the mega parsec (a million parsecs) and mega light-years (a million light-years). The actual values for these distances and other common units of measure can be found in the table of constants.