Monday, July 18, 2016

A Puzzle

What is the missing number?

−

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:

−

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:

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!

See this in action at the Algebra Balance Animation.

What is an Equation

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

x

+

2

=

6

That equation 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"

Parts of an Equation

So people can talk about equations, there are names for different parts (better than saying "that thingy there"!)

Here we have an equation that says 4x − 7 equals 5, and all its parts:

A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.

A number on its own is called a Constant.

A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient)

Sometimes a letter stands in for the number:

Example: ax² + bx + c

x is a variable
a and b are coefficients
c is a constant

An Operator is a symbol (such as +, ×, etc) that shows an operation (ie we want to do something with the values).

A Term is either a single number or a variable, or numbers and variables multiplied together.

An Expression is a group of terms (the terms are separated by + or − signs)

So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"

Exponents

The exponent (such as the 2 in x²) says how many times to use the value in a multiplication.

Examples:

8² = 8 × 8 = 64

y³ = y × y × y

y²z = y × y × z

Exponents make it easier to write and use many multiplications

Example: y⁴z² is easier than y × y × y × y × z × z, or even yyyyzz

Polynomial

Example of a Polynomial: 3x² + x - 2

A polynomial can have constants, variables and the exponents 0,1,2,3,...

But it never has division by a variable.

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

Like Terms

Like Terms are terms whose variables (and their exponents such as the 2 in x²) are the same.

In other words, terms that are "like" each other. (Note: the coefficients can be different)

Example:

(1/3)xy²

-2xy²

6xy²

Are all like terms because the variables are all xy²

Sunday, June 12, 2016

Name

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x²).

It is also called an "Equation of Degree 2" (because of the "2" on the x)

Standard Form

The Standard Form of a Quadratic Equation looks like this:

a, b and c are known values. a can't be 0.

"x" is the variable or unknown (we don't know it yet).

Here are some more examples:

2x² + 5x + 3 = 0		In this one a=2, b=5 and c=3

x² − 3x = 0		This one is a little more tricky: Where is a? Well a=1, and we don't usually write "1x²" b = -3 And where is c? Well c=0, so is not shown.
5x − 3 = 0		Oops! This one is not a quadratic equation: it is missing x² (in other words a=0, which means it can't be quadratic)

Hidden Quadratic Equations!

So the "Standard Form" of a Quadratic Equation is

ax² + bx + c = 0

But sometimes a quadratic equation doesn't look like that! For example:

In disguise	→	In Standard Form	a, b and c
x² = 3x − 1	Move all terms to left hand side	x² − 3x + 1 = 0	a=1, b=−3, c=1
2(w² − 2w) = 5	Expand (undo the brackets), and move 5 to left	2w² − 4w − 5 = 0	a=2, b=−4, c=−5
z(z−1) = 3	Expand, and move 3 to left	z² − z − 3 = 0	a=1, b=−1, c=−3

Have a Play With It

Play with the "Quadratic Equation Explorer" so you can see:

the graph it makes, and
the solutions (called "roots").

How To Solve It?

The "solutions" to the Quadratic Equation are where it is equal to zero.

There are usually 2 solutions (as shown in the graph above).

They are also called "roots", or sometimes "zeros"

There are 3 ways to find the solutions:

1. We can Factor the Quadratic (find what to multiply to make the Quadratic Equation)

2. We can Complete the Square, or

3. We can use the special Quadratic Formula:

Just plug in the values of a, b and c, and do the calculations.

We will look at this method in more detail now.

About the Quadratic Formula

Plus/Minus

First of all what is that plus/minus thing that looks like ± ?

The ± means there are TWO answers:

Here is why we can get two answers:

But sometimes we don't get two real answers, and the "Discriminant" shows why ...

Discriminant

Do you see b² − 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:

when b² - 4ac is positive, we get two Real solutions

when it is zero we get just ONE real solution (both answers are the same)

when it is negative we get two Complex solutions

Complex solutions? Let's talk about them after we see how to use the formula.

Using the Quadratic Formula

Just put the values of a, b and c into the Quadratic Formula, and do the calculations.

Example: Solve 5x² + 6x + 1 = 0

Coefficients are:		a = 5, b = 6, c = 1

Quadratic Formula:		x = −b ± √(b²− 4ac)2a

Put in a, b and c:		x = −6 ± √(6²− 4×5×1)2×5

Solve:		x = −6 ± √(36− 20)10
		x = −6 ± √(16)10
		x = −6 ± 410
		x = −0.2 or −1

Answer: x = −0.2 or x = −1

And we see them on this graph.

Check -0.2:	5×(−0.2)² + 6×(−0.2) + 1 = 5×(0.04) + 6×(−0.2) + 1 = 0.2 − 1.2 + 1 = 0
Check -1:	5×(−1)² + 6×(−1) + 1 = 5×(1) + 6×(−1) + 1 = 5 − 6 + 1 = 0

Remembering The Formula

I don't know of an easy way to remember the Quadratic Formula, but a kind reader suggested singing it to "Pop Goes the Weasel":

♫	"x equals minus b	♫	"All around the mulberry bush
♫	*plus or minus the square root*	♫	The monkey chased the weasel
	*of b-squared minus four a c*		The monkey thought 'twas all in fun
	*all over two a"*		Pop! goes the weasel"

Try singing it a few times and it will get stuck in your head!

Or you can remember this story:

x = −b ± √(b²− 4ac)2a

"A negative boy was thinking yes or no about going to a party,
at the party he talked to a square boy but not to the 4 awesome chicks.
It was all over at 2 am."

Complex Solutions?

When the Discriminant (the value b² − 4ac) is negative we get Complex solutions ... what does that mean?

It means our answer will include Imaginary Numbers. Wow!

Example: Solve 5x² + 2x + 1 = 0

Coefficients are:		a = 5, b = 2, c = 1

Note that the Discriminant is negative:		b² − 4ac = 2² − 4×5×1 = -16

Use the Quadratic Formula:		x = −2 ± √(−16)10

The square root of -16 is 4i (i is √-1, read Imaginary Numbers to find out more)

So:		x = −2 ± 4i10

Answer: x = −0.2 ± 0.4i

The graph does not cross the x-axis. That is why we ended up with complex numbers.

In some ways it is easier: we don't need more calculation, just leave it as −0.2 ± 0.4i.

Summary

Quadratic Equation in Standard Form: ax² + bx + c = 0

Quadratic Equations can be factored

Quadratic Formula: x = −b ± √(b²− 4ac)2a

When the Discriminant (b²−4ac) is:

positive, there are 2 real solutions
zero, there is one real solution
negative, there are 2 complex solutions

Saturday, June 11, 2016

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)	log_e(50)	confusion
ln(50)	log_e(50)	log_e(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" !