Things you should have learned in high school, but may have
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)
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.
10-1 = 0.1
10-2 = 0.01
10-3 = 0.001
and so forth.
Let's try some more. How would you write the following numbers?
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
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
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
Check your answers here.
- 9.9 x 10-5 x 4.5 x 10-8
- 1.02 x 104 / 3.3 x 10-9
- 4.0 x 10-2 x 1.0 x 108
- 8.2 x 109 x 5.3 x 108
- 9.8 x 1019 + 4.2 x 10-9
In case you don't have a calculator, you can still do the math,
mainly the multiplication and division, by following these rules -
- Multiplication: multiply the two front numbers, and then add the powers.
- Division: divide the two front numbers, and then subtract the powers.
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
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
Here are some for you to try - determine the number of SF in each
Check your answers here.
- 6.02 x 1023
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
Here are some examples to try - determine the answer for each using
the proper number of SF.
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.
- 3.4 + 0.00344
- 4.50 x 3.3005
- 4.510 x 1012 x 3.401 x 10-11
- 607.1 x 4.4
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.
- 5 SF = 36.239
- 4 SF = 36.24
- 3 SF = 36.2
- 2 SF = 36.
- 1 SF = 40
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 -
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.
- Parentheses are first
- Exponents (powers)
- Multiplication - Division (equal standing, so whichever is first going from left to right)
- Addition - Subtraction (equal standing, so whichever is first going from left to right)
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 -
kilograms - kg
grams - gm
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.