Study this lesson and work through hw 0 on your own.  This assignment is not graded and will not be handed in.  Work on the items as a check of your own mathematics skills.  If you are unsure of a problem, write down what you know and get started on it.  Then, e-mail the class list for assistance.

Algebra Review:

[This section is meant as a refresher, not a complete course.--KS]

In this section we'll review the basic rules of algebra, including using exponents and logarithms  Also, we'll take a look at binary numbers.  At the end of this section you should be able to apply concepts to manipulate algebraic equations and with practice do so with confidence and accuracy.

There are three rules for manipulating algebraic equations.  Knowing these three rules will allow you move terms around and solve for unknown items in an equation.

Commutative Law: 
The order in which you multiply or add values doesn't matter much.  This rule does not apply to subtraction and division.

a + b is the same as b + a
ab is the same as ba

Note that you can rearrange equations that have this pattern and be confident that you still have the same value as before.  A group of values may be considered as a single term. 

(a - b) + (cd) is the same as (dc) + (a - b)

Distributive Law: 
With addition, multiplication, subtraction and division you may redistribute the multiplier (the term outside the parenthesis).

a(b + c) is the same as ab + ac
a(b - c) is the same as ab - ac
(b - c)/a is the same as b/a - c/a
(b + c)/a is the same as b/a + c/a

This is important because it helps you isolate unknown values.  This law works in reverse, of course.

Associative Law: 
Building on the commutative law, we may group addition and multiplication in a different order and still have an equivalent equation.

(a + b) + c is the same as a + (b + c)
(ab)c is the same as a(bc)

Be very careful, subtraction and division do not allow such freedom.

Example: 
Simplify the following expression.

(a + b + c)b    bcc - abc
------------ X -----------
  bc - ba      (c - b)(cb)

Use the distributive law to "undistribute" the denominator (bottom) of the left side expression and the numerator (top) of the right side expression.

(a + b + c)b    b(cc - ac)
------------ X -----------
  b(c - a)     (c - b)(cb)

Notice that on the left side you now have 'b' divided by 'b' which is equal to 1.  Another way to say this is that they cancel.  This is also true on the right side, where the b's cancel out again.

Use the distributive law in the same manner on the right side to separate out the c's on top and bottom.

(a + b + c)   c(c - a)
----------- X --------
   c - a      (c - b)c

Now the c's on the right side will cancel.

(a + b + c)   c - a
----------- X -----
   c - a      c - b

Note that you have c - a in the numerator on the right and the denominator on the left.  These cancel out and you are left which a much simpler algebraic term.

a + b + c
---------
  c - b

You've done all you can do at this point.  Let's go on to a new subject.

Exponents:
What manipulations can be made with exponents?  First remember that exponents are a shorthand way to represent multiplication.  Who would want to write out sixteen y's in a row.  It's easier to use the exponent 16 on a base of y.  We can use superscripts for exponents and we can show exponents with the caret mark, ^. I will use either one or the other of these to indicate exponents.

2^4 (same as 2⁴) is shorthand for 2 x 2 x 2 x 2

y to the sixteenth power can be written as

   y¹⁶     or  y^16

Now how could your rewrite the following?

3^-x  (3^x * 3^y)  (3 * 4)^x  6^xy  x⁰  y¹

We need some equivalent ways to write these.  Then we can manipulate our equations with the rules.  Take a look at the following.  They show equivalent ways to write the expression.  Note that you can change from one to the other, either way, and be sure that you have not changed the values of the terms in the equation.

         1
x^-a   = ---
         x^a

x^ax^b  = x^a+b

(xy)^a  = x^ay^a 
 
(x^a)^b  = (x^b)^a  = x^ab

Let's prove that last one by putting in some numbers.  If the last one is true, then (2^3)^2 is the same as (2^2)^3 and the same as 2^(2 x 3)

(2^3)^2   is  (2 x 2 x 2)(2 x 2 x 2)  equals 2^6
(2^2)^3   is  (2 x 2)(2 x 2)(2 x 2)   equals 2^6
2^(2 x 3) is 2^6

OK, they are all the same.

Example:
Simplify the following.

        -3 2  2
     ( x  y  )
--------------------
     -2   4     2  2
[(xy)  )(y )(xy)  ]

Distribute the outside exponents to the inside terms.  Do this for both the numerator and the denominator.

        (-3 x 2)  (2 x 2)
       x         y
------------------------------
   (-2 x 2)  (4 x 2)   (2 x 2)
(xy)        y       (xy)

Hey, wait.  It's getting bigger, not smaller.  Don't let that bother you, just keep simplifying.  Multiply the exponents together to simplify.

      -6  4
     x   y
---------------
    -4  8     4
(xy)   y  (xy)

Distribute the exponents inside the parenthesis again.

       -6  4
      x   y
----------------
 -4  -4  8  4  4
x   y   y  x  y

Group the exponents together by their base and combine.  The exponents add together.

         -6  4
        x   y
-----------------------
 (-4 + 4)  (-4 + 8 + 4)
x         y

Keep simplifying ...

 -6  4
x   y
-------------
 0  8
x  y

Note that from our rules, x^0 is 1.  Also, we can move the y^8 into the numerator if we change the sign of the exponent.

 -6 (4 - 8) 
x  y         

            -6 -4
Which is   x  y  ,  our simplified answer.

Logarithms:
Basically, logarithms are the reverse of an exponent operation.  Let's say you calculate 6^4 on your calculator.  You get 1,296.  The equation looks like this:

y = 6⁴ = 6 x 6 x 6 x 6
y = 1,296

Now consider the following problem:

1296 = 6^x

We already know that x is 4, because we've used this example above.  How do you solve for x, though?  You need logarithms to do this.  Here is the general rule:

If   b^x  =  y, then

x = log_b ( y )

Yes, that is a subscript.  The b in the subscript means 'base b'.  In other words, b was the base of the exponent, b^x.  You can write it as a subscript or you can use an underscore like this, log_b(y).  I will use either the underbar or the subscript.

We don't use base b very often, but we do use a base of 10.  For example, 10^3, 10^5, and so on.

Your calculator has a "log" button on it.  This button probably assumes that you want to take the log base 10.  Check your calculator carefully to see if it will let you enter the base.  If not, then it is assuming base 10.

So, use your calculator to solve the following:

10^x  = 10,000,000

Enter 10,000,000 into your calculator and then hit the log key.

x = log₁₀( 10,000,000 )

If an unknown value is an exponent, then you have to use a logarithm to isolate the unknown value.  When people use the word 'log' and do not write the base subscript, they mean base 10.  We'll do this from this point on.  For example, y = 234 log x  means that y is equal to 243 times the base 10 logarithm of x.

You'll need some rules to manipulate logarithms.  These are very important rules, learn them by heart.

log_n( n ) = 1    example:  log₁₀( 10 ) = 1
log_n( 1 ) = 0    example:  log₂( 1 ) = 0

You should memorize these even if you don't understand why they are true.  Example:  log base 10 of 10 is 1,  log base 3 of 3 is 1, log base 2 of 2 is 1, and so on.  Also, log base 'anything' of 1 is 0. 

Want to take a closer look?  Well, let's do it anyway.  Look at the first one.  Use the definition you learned at the top of this section.

If  b^x  = y   then,  log_b( y ) = x

If  n¹  = n   then,  log_n( n ) = 1

You could do the same for the second one to prove that log_n (1) = 0.  Hopefully, your memory will tell you that anything to the zero power is 1, which is the opposite operation.

Here are three more very important rules:

log(xy) = log(x) + log(y)

log(x/y) = log(x) - log(y)

log(x^n) = n log(x)

You need one more rule if your calculator only has a log base 10 button.  You have to have some way to take log_b(x) when b isn't 10.  Write this one down so you don't forget.

           log_10(y)
log_n(y) = ---------
           log_10(n)

We can't do the part on the left side with our calculator, but we can solve the right-hand side with a calculator.

Examples:
Solve the following.

x = log_3(81)

Write out the base 10 version of this, then use your calculator.

                log_10(81)
x = log_3(81) = ---------- = 4
                log_10(3)

Simplify the following.

              2 6
           32x y
A = log_2(---------)
                 2
           (16xy)

Use the three rules above to split this up into individual terms.

A = log_2(32) + log_2(y^6) -

               log_2(16^2) - log_2(y^2)

Bring the exponents out front and use your calculator to simplify.  Use the rule from the example just above if your calculator only does log_10.

A = 5 + 6log_2(y) - 8 - 2log_2(y)
A = 4log_2(y) - 3

Now that's a lot simpler that what we started with.

Scientific Notation:
Scientific notation is used to write very large and very small numbers.  Use powers of 10 multiplied by a small decimal number to represent those difficult-to-write numbers.

You can count decimal places to determine the exponent.  Counting to the right of the decimal point means a negative exponent.

Example:
Would you want to write out the following or enter them in your calculator?

0.0000000000000000000000000000000000000384
38,393,084,000,000,000,000,000,000,000,000

Of course not.  Count the decimal places and make a power of ten with that exponent.  Write out the scientific notation.  Our first example number has 38 places, counting to the 3.  We are counting to the right of the decimal place so our exponent is negative. Place the decimal point just after the 3 and multiply by 10^-38.

3.84 x 10^-38

We'll place the decimal point just after the 3 in the second example number in a similar fashion.  This time we have 31 places to move in order to get there.

3.8393084 x 10³¹

What are the following in scientific notation?

322.5              1000 x 1000 x 321

0.01103            136,000,000,000,000,000.

32 ³                 10¹⁰(10^-3)(8²)

Answers:

3.225 x 10^2       3.21 x 10^8

1.103 x 10^-2      1.36 x 10^17

3.2768 x 10^4      6.4 x 10^8

Metric Prefixes:
We all know that 1000 meters is the same as one kilometer.  Therefore we all know, perhaps just intuitively, that kilo means 1,000, or 10^3.  However, there are many other prefixes that are beneficial.  Consider the following table.

Prefix         Symbol         Multiplication
                              Factor
-------------  -------------  --------------

P              peta           10^15
T              tera           10^12
G              giga           10^9
M              mega           10^6
k              kilo           10^3

d              deci           10^-1
c              centi          10^-2
m              mili           10^-3
u              micro          10^-6
n              nano           10^-9
p              pico           10^-12
f              femto          10^-15

Meters, grams, and seconds are the most common measurement units to take these prefixes.  You may also encounter, watts, amps, tones, and others.

Example:
What are the following written in a simpler manner using metric prefixes.

1000 meters              0.313 volts
299,000,000 watts        0.000125 seconds
0.000000008 meters       3 x 10^8 m/s
50,000,000 tones         1 mile

Answers:

1 km                     313 mV or 3.13 dV
299 MW                   125 us
8 nm                     300 Mm/s or 0.3 Gm/s
50 megatons              1.62 km

Note that upper and lower case prefixes have different meanings.  Also note that (3.15)^6 does not equal 3.15 x 10^6.  Try this out on your calculator and confirm the difference.

Binary Number System:
For some of you, this may be a new topic.  For others, this will be a review from as far back as junior high school.  Either way, it is useful to review and study the binary number system. 

Our decimal counting system is base 10.  We have 10 different values that can be placed in what we call the "one's place" when counting; 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  Once we get to 9 we run out of numbers and have to add another place, the 10's place.  Again, we cycle through the ten different numbers in the one's place; 10, 11, 12, 13, 14 and so on.  Yet again, we run out of digits for the one's place and we increment the value in the 10's place. 

Counting in other numbering systems, such as base 4 or base 2, functions in the same way.  You have  a limited number of unique digits.  When you use them all you have to add a new place to the left so that you may begin again.  Take a look at counting in base 10, base 4 and base 2.

Base 10     Base 4      Base 2
-------     -------     -------
    0           0           0
    1           1           1
    2           2          10
    3           3          11
    4          10         100
    5          11         101
    6          12         110
    7          13         111
    8          20        1000
    9          21        1001
   10          22        1010
   11          23        1011
   12          30        1100
   13          31        1101
  ...         ...         ...    etc.

The lower the base the more rapidly we add places to the number when counting.  You can see that base 2 adds places very quickly.

Now that you can count in base 2, binary, answer this:  How do you convert from binary to base 10 and back?

Just as base 10 has the 1's, 10's, 100's and 1000's places, base 2 has the 1's, 2's, 4's, 8's, 16,'s and so on places.  In base 2 and base 10 the values of each place can be written as exponents.

3156 in base 10 is:

1000's place      3000
 100's place       100
  10's place        50
   1's place         6
----------------------
Total             3156

101101 in base 2 is:

32's place      100000
16's place           0
 8's place        1000
 4's place         100
 2's place           0
 1's place           1
-----------------------
Total           101101

To convert to base 10 from base 2 use the following table which shows the value in base 10 for each 1 in a base 2 number.

     12  11  10  9  8  7  6  5  4  3  2  1  0
... 2   2   2   2  2  2  2  2  2  2  2  2  2

Example:

Base 2 number
    1   0   1   0  0  0  1  1  0  1  1  0  1
Add up the base 10 values for each 1.
     12     10           6   5    3   2    0
    2  +   2  +         2 + 2 +  2 + 2 +  2
Answer in base 10
    4096 + 1024 + 64 + 32 + 8 + 4 + 1 = 5229

Our example above, 101101 would equal in base 10:

= 2^5 + 2^3 + 2^2 + 2^0
= 45

Now, how do you convert  a number such as 116 into a binary number?  In this case you have to subtract the decimal value of each place one at a time until your remaining value reaches zero.  Mark the place you use under the table with a 1.

Our table and the positions we use below are:
     10  9  8  7  6  5  4  3  2  1  0
... 2   2  2  2  2  2  2  2  2  2  2

                 1  1  1     1

Subtraction:
    116
   - 64    (2^6)
   ----
     52
   - 32    (2^5)
   ----
     20
   - 16    (2^4)
   ----
      4
   -  4    (2^2)
   ----
      0

Our final base 2 number:  1110100

Example question:
Answer in base 2 and in base 10.  What  is the largest five digit binary number you can have?

Answer:  The largest number in base 2 that has five digits is 1 1 1 1 1.  This works out to be 31 in base 10.

Why use binary?
Computers use binary at the lowest levels.  Electricity can be 'on' or 'off' and this state can be represented by a binary digit.  On can be represented by a 1, off by a 0.  Magnetic media can have two polarities, north and south for example.  This too can be represented by 1's and 0's.  Understanding binary will allow you to understand the lowest layers of computer operations and we'll be investigating those layers fairly soon.

Used with permission of the Author; Copyright (C) Kevin A. Shaffer 1998 - 2022, all rights reserved.