Algebra

Algebra is great fun - you get to solve puzzles!

With computer games you play by running, jumping or finding secret things. Well, with Algebra you play with letters, numbers and symbols, and you also get to find secret things!

And once you learn some of the "tricks", it becomes a fun challenge to work out how to use your skills in solving each "puzzle".

Introduction to Algebra

Algebra is great fun - you get to solve puzzles!

A Puzzle

What is the missing number?

-

2

=

4

OK, the answer is 6, right? Because 6 − 2 = 4. Easy stuff.

Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we write:

x

-

2

=

4

It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable.

And when we solve it we write:

x

=

6

Why Use a Letter?

	Because:
	it is easier to write "x" than drawing empty boxes (and easier to say "x" than "the empty box").
	if there are several empty boxes (several "unknowns") we can use a different letter for each one.

So x is simply better than having an empty box. We aren't trying to make words with it!

And it doesn't have to be x, it could be y or w ... or any letter or symbol we like.

How to Solve

Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6".

But instead of saying "obviously x=6", use this neat step-by-step approach:

• Work out what to remove to get "x = ..."
• Remove it by doing the opposite (adding is the opposite of subtracting)
• Do that to both sides

Here is an example:

We want to remove the "-2"	To remove it, do the opposite, in this case add 2:	Do it to both sides:	Which is ...	Solved!

Why did we add 2 to both sides?

To "keep the balance"...

In Balance

Add 2 to Left Side

Out of Balance!

Add 2 to Right Side Also

In Balance Again

Just remember this:

To keep the balance, what we do to one side of the "=" we should also do to the other side!

Another Puzzle

Solve this one:

x

+

5

=

12

Start with:	x + 5 = 12
What we are aiming for is an answer like "x = ...", and the plus 5 is in the way of that! We can cancel out the plus 5 by doing a subtract 5 (because 5−5=0)
So, let us have a go at subtracting 5 from both sides:	x+5 −5 = 12 −5
A little arithmetic (5−5 = 0 and 12−5 = 7) becomes:	x+0 = 7
Which is just:	x = 7
	Solved!
(Quick Check: 7+5=12)

Have a Try Yourself

Now practice on this Simple Algebra Worksheet and then check your answers on the page after. Try to use the steps we have shown you here, rather than just guessing!

Name:____________________

Math is Fun Worksheet "print your own worksheets at http://smartmathsolution.blogspot.com"

Date:____________________

Solve the following

1: x+9 = 11

2: x-10 = 0

3: x+7 = 17

4: 2+x = 8

5: 9+x = 16

6: x+5 = 13

7: -9+x = -2

8: x+7 = 9

9: x-9 = -7

10: -10+x = 0

Introduction to Algebra - Multiplication

A Puzzle

What is the missing number?

×

4

=

8

The answer is 2, right? Because 2 × 4 = 8.

Well, in Algebra we don't use blank boxes, we use a letter. So we might write:

x

×

4

=

8

But the "x" looks like the "×" ... that can be very confusing ... so in Algebra we don't use the multiply symbol (×) between numbers and letters:

We put the number next to the letter to mean multiply:

4x

=

8

In English we say "four x equals eight", meaning that 4 x's make 8.

And the answer is written:

x

=

2

How to Solve

Instead of saying "obviously x=2", use this neat step-by-step approach:

• Work out what to remove to get "x = ..."
• Remove it by doing the opposite
• Do that to both sides

And what is the opposite of multiplying? Dividing!

Have a look at this example:

We want to remove the "4"	To remove it, do the opposite, in this case divide by 4:	Do it to both sides:	Which is ...	Solved!

Why did we divide by 4 on both sides?

Because of the need for balance ...

In Balance

Divide Left by 4

Out of Balance!

Divide Right by 4 Also

In Balance Again

Just remember ...

To keep the balance, what we do to one side of the "=" we should also do to the other side!

Another Puzzle

Solve this one:

x

/

3

=

5

Start with:	x/3 = 5
What we are aiming for is an answer like "x = ...", and the divide by 3 is in the way of that! If we multiply by 3 we can cancel out the divide by 3 (because 3/3=1)
So, let us have a go at multiplying by 3 on both sides:	x/3 ×3 = 5 ×3
A little arithmetic (3/3 = 1 and 5×3 = 15) becomes:	1x = 15
Which is just:	x = 15
	Solved!
(Quick Check: 15/3 = 5)

Have a Try Yourself

Now practice on this Algebra Multiplication Worksheet and then check your answers on the page after. Try to use the steps we have shown you here, rather than just guessing!

More Complicated Example

How do we solve this?

x

/

3

+

2

=

5

It might look hard, but not if we solve it in stages.

First let us get rid of the "+2":

Start with:	x/3 + 2 = 5
To remove the plus 2 use minus 2 (because 2-2=0)	x/3 + 2 -2 = 5 -2
A little arithmetic (2-2 = 0 and 5-2 = 3) becomes:	x/3 + 0 = 3
Which is just:	x/3 = 3

Now, get rid of the "/3":

Start with:	x/3 = 3
If we multiply by 3 we can cancel out the divide by 3:	x/3 ×3 = 3 ×3
A little arithmetic (3/3 = 1 and 3×3 = 9) becomes:	1x = 9
Which is just:	x = 9
	Solved!
(Quick Check: 9/3 + 2 = 3+2 = 5)

When you get more experienced:

When you get more experienced, you can solve it like this:

Start with:	x/3 + 2 = 5
Subtract 2 from both sides:	x/3 + 2 -2 = 5 -2
Simplify:	x/3 = 3
Multiply by 3 on both sides:	x/3 ×3 = 3 ×3
Simplify:	x = 9

Or even like this:

Start with:	x/3 + 2 = 5
Subtract 2:	x/3 = 3
Multiply by 3:	x = 9

Real World Example

Example: Sam bought 3 boxes of chocolates online.
Postage was $9 and the total cost was $45.
How much was each box?

Let's use x for the price of each box.

3 times x plus $9 is $45:

3x + 9 = 45

Let's solve!

Start with:	3x + 9 = 45
Subtract 9 from both sides:	3x + 9 − 9 = 45 − 9
Simplify:	3x = 36
Divide by 3:	3x /3 = 36 /3
Simplify:	x = 12

So each box was $12

Advanced: we can also do the "divide by 3" first (but we must do it to all terms):

Start with:	3x + 9 = 45
Divide by 3:	3x/3 + 9/3 = 45/3
Simplify:	x + 3 = 15
Subtract 3 from both sides:	x + 3 − 3 = 15 − 3
Simplify:	x = 12

Order of Operations - BODMAS

Operations

"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.

But, when you see something like...

7 + (6 × 5² + 3)

... what part should you calculate first?

Start at the left and go to the right?
Or go from right to left?

Calculate them in the wrong order, and you will get a wrong answer !

So, long ago people agreed to follow rules when doing calculations, and they are:

Order of Operations

Do things in Brackets First. Example:

		6 × (5 + 3)	=	6 × 8	=	48
		6 × (5 + 3)	=	30 + 3	=	33	(wrong)

Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract. Example:

		5 × 22	=	5 × 4	=	20
		5 × 22	=	102	=	100	(wrong)

Multiply or Divide before you Add or Subtract. Example:

		2 + 5 × 3	=	2 + 15	=	17
		2 + 5 × 3	=	7 × 3	=	21	(wrong)

Otherwise just go left to right. Example:

		30 ÷ 5 × 3	=	6 × 3	=	18
		30 ÷ 5 × 3	=	30 ÷ 15	=	2	(wrong)

How Do I Remember It All ... ? BODMAS !

B	Brackets first
O	Orders (ie Powers and Square Roots, etc.)
DM	Division and Multiplication (left-to-right)
AS	Addition and Subtraction (left-to-right)

Divide and Multiply rank equally (and go left to right).

Add and Subtract rank equally (and go left to right)

After you have done "B" and "O", just go from left to right doing any "D" or "M" as you find them. Then go from left to right doing any "A" or "S" as you find them.

Note: the only strange name is "Orders". "Exponents" is used in Canada, and so you might prefer "BEDMAS". There is also "Indices" which makes it "BIDMAS". In the US they say "Parentheses" instead of Brackets, so it is "PEMDAS"

Examples

Example: How do you work out 3 + 6 × 2 ?

Multiplication before Addition:

First 6 × 2 = 12, then 3 + 12 = 15

Example: How do you work out (3 + 6) × 2 ?

Brackets first:

First (3 + 6) = 9, then 9 × 2 = 18

Example: How do you work out 12 / 6 × 3 / 2 ?

Multiplication and Division rank equally, so just go left to right:

First 12 / 6 = 2, then 2 × 3 = 6, then 6 / 2 = 3

Exponents of Exponents ...

What about this example?

4³²

Exponents are special: they go right-to-left. So we caclulate it this way:

Start with:		432
32 = 3×3:		49
49 = 4×4×4×4×4×4×4×4×4:		262144

Oh, yes, and what about 7 + (6 × 5² + 3) ?

7 + (6 × 52 + 3)
7 + (6 × 25 + 3)	Start inside Brackets, and then use "Orders" First
7 + (150 + 3)	Then Multiply
7 + (153)	Then Add
7 + 153	Brackets completed, last operation is add
160	DONE !

Algebra - Substitution

"Substitute" means to put in the place of another.

Substitution

In Algebra "Substitution" means putting numbers where the letters are:

	If you have:	x − 2
	And you know that x=6 ...
	... then you can "substitute" 6 for x:	6 − 2 = 4

Example: If x=5 then what is 10/x + 4 ?

Put "5" where "x" is:

10/5 + 4 = 2 + 4 = 6

Example: If x=3 and y=4, then what is x² + xy ?

Put "3" where "x" is, and "4" where "y" is:

3² + 3×4 = 3×3 + 12 = 21

Example: If x=3 (but you don't know "y"), then what is x² + xy ?

Put "3" where "x" is:

3² + 3y = 9 + 3y

(that is as far as you can get)

As that last example showed, you may not always get a number for an answer, sometimes just a simpler formula.

Negative Numbers

When substituting negative numbers, put () around them so you get the calculations right.

Example: If x = −2, then what is 1 − x + x² ?

Put "(−2)" where "x" is:

1 − (−2) + (−2)² = 1 + 2 + 4 = 7

In that last example:

• the − (−2) became +2
• the (−2)² became +4

because of these special rules:

	Rule	Adding or Subtracting		Multiplying or Dividing
	Two like signs become a positive sign	3+(+2) = 3 + 2 = 5		3 × 2 = 6
		6−(−3) = 6 + 3 = 9		(−3) × (−2) = 6
	Two unlike signs become a negative sign	7+(−2) = 7 − 2 = 5		3 × (−2) = −6
		8−(+2) = 8 − 2 = 6		(−3) × 2 = −6

Equations and Formulas

What is an Equation?

An equation says that two things are equal. It will have an equals sign "=" like this:

x

+

2

=

6

That equations says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement "this equals that"

What is a Formula?

A formula is a special type of equation that shows the relationship between different variables.

A variable is a symbol like x or V that stands in for a number we don't know yet.

Example: The formula for finding the volume of a box is:

V = lwh

V stands for volume, l for length, w for width, and h for height.

When l=10, w=5, and h=4, then:

V = 10 × 5 × 4 = 200

A formula will have more than one variable.

These are all equations, but only some are formulas:

x = 2y - 7	Formula (relating x and y)
a2 + b2 = c2	Formula (relating a, b and c)
x/2 + 7 = 0	Not a Formula (just an equation)

Without the Equals

Sometimes a formula is written without the "=":

Example: The formula for the volume of a box is:

lwh

But in a way the "=" is still there, because we can write V = lwh if we want to.

Subject of a Formula

The "subject" of a formula is the single variable (usually on the left of the "=") that everything else is equal to.

Example: in the formula

s = ut + ½ at²

"s" is the subject of the formula

Changing the Subject

One of the very powerful things that Algebra can do is to "rearrange" a formula so that another variable is the subject.

Rearrange the volume of a box formula (V = lwh) so that the width is the subject:

Start with:	V = lwh
divide both sides by h:	V/h = lw
divide both sides by l:	V/(hl) = w
swap sides:	w = V/(hl)

So now when you want a box with a volume of 12m³, a length of 2m, and a height of 2m, you can calculate its width:

w = V/(hl)

w = 12m³ / (2m × 2m) = 12/4 = 3m

Exponents

The exponent of a number says how many times to use the number in a multiplication.

In 8² the "2" says to use 8 twice in a multiplication,
so 8² = 8 × 8 = 64

In words: 8² could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"

Exponents are also called Powers or Indices.

Some more examples:

Example: 5³ = 5 × 5 × 5 = 125

• In words: 5³ could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

Example: 2⁴ = 2 × 2 × 2 × 2 = 16

• In words: 2⁴ could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"

Exponents make it easier to write and use many multiplications

Example: 9⁶ is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9

In General

So in general:

an tells you to multiply a by itself, so there are n of those a's:

Other Way of Writing It

Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.

Example: 2^4 is the same as 2⁴

• 2^4 = 2 × 2 × 2 × 2 = 16

Negative Exponents

Negative? What could be the opposite of multiplying?

Dividing!

A negative exponent means how many times to divide one by the number.

Example: 8^-1 = 1 ÷ 8 = 0.125

You can have many divides:

Example: 5^-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008

But that can be done an easier way:

5^-3 could also be calculated like:

1 ÷ (5 × 5 × 5) = 1/5³ = 1/125 = 0.008

In General

That last example showed an easier way to handle negative exponents: Calculate the positive exponent (an) Then take the Reciprocal (i.e. 1/an)

More Examples:

Negative Exponent		Reciprocal of Positive Exponent		Answer
4-2	=	1 / 42	=	1/16 = 0.0625
10-3	=	1 / 103	=	1/1,000 = 0.001
(-2)-3	=	1 / (-2)3	=	1/(-8) = -0.125

What if the Exponent is 1, or 0?

1		If the exponent is 1, then you just have the number itself (example 91 = 9)
0		If the exponent is 0, then you get 1 (example 90 = 1)
		But what about 00 ? It could be either 1 or 0, and so people say it is "indeterminate".

It All Makes Sense

My favorite method is to start with "1" and then multiply or divide as many times as the exponent says, then you will get the right answer, for example:

Example: Powers of 5
	.. etc..
52	1 × 5 × 5	25
51	1 × 5	5
50	1	1
5-1	1 ÷ 5	0.2
5-2	1 ÷ 5 ÷ 5	0.04
	.. etc..

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern.

Be Careful About Grouping

To avoid confusion, use parentheses () in cases like this:

With () :	(-2)2 = (-2) × (-2) = 4
Without () :	-22 = -(22) = - (2 × 2) = -4

With () :	(ab)2 = ab × ab
Without () :	ab2 = a × (b)2 = a × b × b

  

Reciprocal In Algebra

Turn it upside down!

Reciprocal of a Number To get the reciprocal of a number, just divide 1 by the number.

Examples:

Number	Reciprocal	As a Decimal
2	1/2	= 0.5
8	1/8	= 0.125
1,000	1/1,000	= 0.001

Reciprocal of a Variable

If you are given a variable "x", its reciprocal is "1/x". Likewise, if you are given something more complicated like "x/y" then the reciprocal is just "y/x". In other words turn it upside down.

Example: What is the Reciprocal of x/(x-1) ?

Answer: take ^x/_(x-1) and flip it upside down: ^(x-1)/_x

Flipping a Flip

If you take the reciprocal of a reciprocal you end up back where you started!

Example:

The reciprocal of ax/y is y/ax

The reciprocal of y/ax is ax/y (back again)

Example: What is:

	1
	1/w

Answer: w

Notation

The reciprocal of "x" is shown as:

1/x

or

x-1

Squares and Square Roots

First learn about Squares, then Square Roots are easy.

How to Square A Number

To square a number, just multiply it by itself ...

Example: What is 3 squared?

3 Squared

=

= 3 × 3 = 9

"Squared" is often written as a little 2 like this:

This says "4 Squared equals 16"
(the little 2 says the number appears twice in multiplying)

Squares From 1² to 6²

1 Squared	=	12	=	1 × 1	=	1
2 Squared	=	22	=	2 × 2	=	4
3 Squared	=	32	=	3 × 3	=	9
4 Squared	=	42	=	4 × 4	=	16
5 Squared	=	52	=	5 × 5	=	25
6 Squared	=	62	=	6 × 6	=	36

The squares are also on the Multiplication Table:

Negative Numbers

We can also square negative numbers.

Example: What happens when we square (−5) ?

Answer:

(−5) × (−5) = 25

(because a negative times a negative gives a positive)

That was interesting!

When we square a negative number we get a positive result.

Just the same as squaring a positive number:

(For more detail read Squares and Square Roots in Algebra)

Square Roots

A square root goes the other way:

3 squared is 9, so a square root of 9 is 3

A square root of a number is ...

... a value that can be multiplied by itself to give the original number.

A square root of 9 is ...

... 3, because when 3 is multiplied by itself we get 9.

It is like asking:

What can we multiply by itself to get this?

To help you remember think of the root of a tree: "I know the tree, but what is the root that made it?" In this case the tree is "9", and the root is "3".

Here are some more squares and square roots:

4		16
5		25
6		36

Decimal Numbers

It also works for decimal numbers.

• What is the square root of 8?
• What is the square root of 9?
• What is the square root of 10?
• What is 1 squared?
• What is 1.1 squared?
• What is 2.6 squared?

Negatives

We found out before that we can square negative numbers:

Example: (−3) squared

(−3) × (−3) = 9

And of course 3 × 3 = 9 also.

So the square root of 9 could be −3 or +3

Example: What are the square roots of 25?

(−5) × (−5) = 25

5 × 5 = 25

So the square roots of 25 are −5 and +5

The Square Root Symbol

This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards. It is called the radical, and always makes mathematics look important!

We use it like this:

and we say "square root of 9 equals 3"

Example: What is √25?

Well, we just happen to know that 25 = 5 × 5, so when we multiply 5 by itself (5 × 5) we will get 25.

So the answer is:

√25 = 5

But wait a minute! Can't the square root also be −5? Because (−5) × (−5) = 25 too.

• Well the square root of 25 could be −5 or +5.
• But when we use the radical symbol √ we only give the positive (or zero) result.

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares

The Perfect Squares (also called "Square Numbers") are the squares of the whole numbers:

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	etc
Perfect Squares:	0	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225	...

Try to remember at least the first 10 of those.

Calculating Square Roots

It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.

Example: what is √10?

Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.

• Let's try 3.5: 3.5 × 3.5 = 12.25
• Let's try 3.2: 3.2 × 3.2 = 10.24
• Let's try 3.1: 3.1 × 3.1 = 9.61
• ...

Getting closer to 10, but it will take a long time to get a good answer!

At this point, I get out my calculator and it says: 3.1622776601683793319988935444327 But the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation !

Note: numbers like that are called Irrational Numbers, if you want to know more.

The Easiest Way to Calculate a Square Root

Use your calculator's square root button!

And also use your common sense to make sure you have the right answer.

A Fun Way to Calculate a Square Root

There is a fun method for calculating a square root that gets more and more accurate each time around:

	a) start with a guess (let's guess 4 is the square root of 10)
	b) divide by the guess (10/4 = 2.5) c) add that to the guess (4 + 2.5 = 6.5) d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25) e) now, set that as the new guess, and start at b) again

• Our first attempt got us from 4 to 3.25
• Going again (b to e) gets us: 3.163
• Going again (b to e) gets us: 3.1623

And so, after 3 times around the answer is 3.1623, which is pretty good, because:

3.1623 x 3.1623 = 10.00014

Now ... why don't you try calculating the square root of 2 this way?

How to Guess

What if we have to guess the square root for a difficult number such as "82,163" ... ?

In that case we could think "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 is a good start.

Square Root Day

The 4th of April 2016 is a Square Root Day, because the date looks like 4/4/16

The next after that is the 5th of May 2025 (5/5/25)

Factoring in Algebra

Factors

Numbers have factors:

And expressions (like x²+4x+3) also have factors:

Factoring

Factoring (called "Factorising" in the UK) is the process of finding the factors:

Factoring: Finding what to multiply together to get an expression.

It is like "splitting" an expression into a multiplication of simpler expressions.

Example: factor 2y+6

Both 2y and 6 have a common factor of 2:

• 2y is 2 × y
• 6 is 2 × 3

So you can factor the whole expression into:

2y+6 = 2(y+3)

So 2y+6 has been "factored into" 2 and y+3

Factoring is also the opposite of Expanding:

Common Factor

In the previous example we saw that 2y and 6 had a common factor of 2

But to do the job properly make sure you have the highest common factor, including any variables

Example: factor 3y²+12y

Firstly, 3 and 12 have a common factor of 3.

So you could have:

3y²+12y = 3(y²+4y)

But we can do better!

3y² and 12y also share the variable y.

Together that makes 3y:

• 3y² is 3y × y
• 12y is 3y × 4

So you can factor the whole expression into:

3y²+12y = 3y(y+4)

Check: 3y(y+4) = 3y × y + 3y × 4 = 3y²+12y

More Complicated Factoring

Factoring Can Be Hard !

The examples have been simple so far, but factoring can be very tricky.

Because you have to figure what got multiplied to produce the expression you are given!

It can be like trying to find out what ingredients went into a cake to make it so delicious. It is sometimes not obvious at all!

Experience Helps

But the more experience you get, the easier it becomes.

Example: Factor 4x² - 9

Hmmm... I can't see any common factors.

But if you know your Special Binomial Products you might see it as the "difference of squares":

Because 4x² is (2x)², and 9 is (3)²,

so we have:

4x² - 9 = (2x)² - (3)²

And that can be produced by the difference of squares formula:

(a+b)(a-b) = a² - b²

Where a is 2x, and b is 3.

So let us try doing that:

(2x+3)(2x-3) = (2x)² - (3)² = 4x² - 9

Yes!

So the factors of 4x² - 9 are (2x+3) and (2x-3):

Answer: 4x² - 9 = (2x+3)(2x-3)

How can you learn to do that? By getting lots of practice, and knowing "Identities"!

Remember these Identities

Here is a list of common "Identities" (including the "difference of squares" used above).

It is worth remembering these, as they can make factoring easier.

a2 - b2	=	(a+b)(a-b)
a2 + 2ab + b2	=	(a+b)(a+b)
a2 - 2ab + b2	=	(a-b)(a-b)
a3 + b3	=	(a+b)(a2-ab+b2)
a3 - b3	=	(a-b)(a2+ab+b2)
a3+3a2b+3ab2+b3	=	(a+b)3
a3-3a2b+3ab2-b3	=	(a-b)3

There are many more like those, but those are the simplest ones.

Advice

The factored form is usually best.

When trying to factor, follow these steps:

"Factor out" any common terms
See if it fits any of the identities, plus any more you may know
Keep going till you can't factor any more

You can also use computers! There are Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and many more that are good at factoring.

More Examples

I said that experience helps, so here are more examples to help you on the way:

Example: w⁴ - 16

An exponent of 4? Maybe we could try an exponent of 2:

w⁴ - 16 = (w²)² - 4²

Yes, it is the difference of squares

w⁴ - 16 = (w² + 4)(w² - 4)

And "(w² - 4)" is another difference of squares

w⁴ - 16 = (w² + 4)(w + 2)(w - 2)

That is as far as I can go (unless I use imaginary numbers)

Example: 3u⁴ - 24uv³

Remove common factor "3u":

3u⁴ - 24uv³ = 3u(u³ - 8v³)

Then a difference of cubes:

3u⁴ - 24uv³ = 3u(u³ - (2v)³)

= 3u(u-2v)(u²+2uv+4v²)

That is as far as I can go.

Example: z³ - z² - 9z + 9

Try factoring the first two and second two separately:

z²(z-1) - 9(z-1)

Wow, (z-1) is on both, so let us use that:

(z²-9)(z-1)

And z²-9 is a difference of squares

(z-3)(z+3)(z-1)

That is as far as I can go.

Introduction to Logarithms

In its simplest form, a logarithm answers the question:

How many of one number do we multiply to get another number?

Example: How many 2s do we multiply to get 8?

Answer: 2 × 2 × 2 = 8, so we needed to multiply 3 of the 2s to get 8

So the logarithm is 3

How to Write it

We write "the number of 2s we need to multiply to get 8 is 3" as:

log₂(8) = 3

So these two things are the same:

The number we are multiplying is called the "base", so we can say:

• "the logarithm of 8 with base 2 is 3"
• or "log base 2 of 8 is 3"
• or "the base-2 log of 8 is 3"

Notice we are dealing with three numbers:

• the base: the number we are multiplying (a "2" in the example above)
• how many times to use it in a multiplication (3 times, which is the logarithm)
• The number we want to get (an "8")

More Examples

Example: What is log₅(625) ... ?

We are asking "how many 5s need to be multiplied together to get 625?"

5 × 5 × 5 × 5 = 625, so we need 4 of the 5s

Answer: log₅(625) = 4

Example: What is log₂(64) ... ?

We are asking "how many 2s need to be multiplied together to get 64?"

2 × 2 × 2 × 2 × 2 × 2 = 64, so we need 6 of the 2s

Answer: log₂(64) = 6

Exponents

Exponents and Logarithms are related, let's find out how ...

The exponent says how many times to use the number in a multiplication. In this example: 23 = 2 × 2 × 2 = 8 (2 is used 3 times in a multiplication to get 8)

So a logarithm answers a question like this:

In this way:

The logarithm tells us what the exponent is!

In that example the "base" is 2 and the "exponent" is 3:

So the logarithm answers the question:

What exponent do we need 
(for one number to become another number) ?

The general case is:

Example: What is log₁₀(100) ... ?

10² = 100

So an exponent of 2 is needed to make 10 into 100, and:

log₁₀(100) = 2

Example: What is log₃(81) ... ?

3⁴ = 81

So an exponent of 4 is needed to make 3 into 81, and:

log₃(81) = 4

Common Logarithms: Base 10

Sometimes a logarithm is written without a base, like this:

log(100)

This usually means that the base is really 10.

It is called a "common logarithm". Engineers love to use it.

On a calculator it is the "log" button.

It is how many times we need to use 10 in a multiplication, to get our desired number.

Example: log(1000) = log₁₀(1000) = 3

Natural Logarithms: Base "e"

Another base that is often used is e (Euler's Number) which is about 2.71828.

This is called a "natural logarithm". Mathematicians use this one a lot.

On a calculator it is the "ln" button.

It is how many times we need to use "e" in a multiplication, to get our desired number.

Example: ln(7.389) = log_e(7.389) ≈ 2

Because 2.71828² ≈ 7.389

But Sometimes There Is Confusion ... !

Mathematicians use "log" (instead of "ln") to mean the natural logarithm. This can lead to confusion:

Example	Engineer Thinks	Mathematician Thinks
log(50)	log10(50)	loge(50)	confusion
ln(50)	loge(50)	loge(50)	no confusion
log10(50)	log10(50)	log10(50)	no confusion

So, be careful when you read "log" that you know what base they mean!

Logarithms Can Have Decimals

All of our examples have used whole number logarithms (like 2 or 3), but logarithms can have decimal values like 2.5, or 6.081, etc.

Example: what is log₁₀(26) ... ?

Get your calculator, type in 26 and press log Answer is: 1.41497...

The logarithm is saying that 10^1.41497... = 26
(10 with an exponent of 1.41497... equals 26)

This is what it looks like on a graph: See how nice and smooth the line is.

Read Logarithms Can Have Decimals to find out more.

Negative Logarithms

−	Negative? But logarithms deal with multiplying. What could be the opposite of multiplying? Dividing!

A negative logarithm means how many times to divide by the number.

We could have just one divide:

Example: What is log₈(0.125) ... ?

Well, 1 ÷ 8 = 0.125,

So log₈(0.125) = −1

Or many divides:

Example: What is log₅(0.008) ... ?

1 ÷ 5 ÷ 5 ÷ 5 = 5⁻³,

So log₅(0.008) = −3

It All Makes Sense

Multiplying and Dividing are all part of the same simple pattern.

Let us look at some Base-10 logarithms as an example:

	Number	How Many 10s	Base-10 Logarithm
	.. etc..
	1000	1 × 10 × 10 × 10	log10(1000)	= 3
	100	1 × 10 × 10	log10(100)	= 2
	10	1 × 10	log10(10)	= 1
	1	1	log10(1)	= 0
	0.1	1 ÷ 10	log10(0.1)	= −1
	0.01	1 ÷ 10 ÷ 10	log10(0.01)	= −2
	0.001	1 ÷ 10 ÷ 10 ÷ 10	log10(0.001)	= −3
	.. etc..

Looking at that table, see how positive, zero or negative logarithms are really part of the same (fairly simple) pattern.

The Word

"Logarithm" is a word made up by Scottish mathematician John Napier (1550-1617), from the Greek word logos meaning "proportion, ratio or word" andarithmos meaning "number", ... which together makes "ratio-number" !