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Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. For K-12 kids, teachers and parents.

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Addition, subtraction, multiplication and Division. Substitution, power laws, expanding, factorising, simplifying and solving linear and quadratics

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When we add or subtract the same thing from both sides the equation stays in balance.

Watch the equation below and try to get "x = (something)" using the +x, −x, +1, −1 buttons.

Things to Try

Each "equation" will have a solution like "x=2" (or maybe "x=3", etc). Try to see all the different ways this answer can be hidden.

First, try to solve the equation by adding and subtracting until you get an answer like x = (value)

Then add and subtract some more until you get x - (value) = 0

Then try to get a solution like - (value) = -x

Then just add and subtract different amounts to see all the possibilities.

Now press "New Equation" and try out some more ... !

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Have a play with this and you will learn about keeping the balance in Algebra

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A Puzzle

What is the missing number?

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:

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:

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

And the answer is written:

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 reverse

Do that to both sides

And what is the reverse of multiplying? Dividing!

Have a look at this example:

To remove it, do
the reverse, in
this case divide by 4

Solved!
Why did we divide by 4 on both sides?

Because of the need for balance ...

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:
What we want is an answer like "x = ...", but 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)

More Complicated Example

How do we solve this?

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

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

A little arithmetic (2−2 = 0 and 5−2 = 3) becomes: x/3 + 0 = 3
Now, get rid of the "/3":
A little arithmetic (3/3 = 1 and 3 × 3 = 9) becomes: 1x = 9

(Quick Check: 9/3 + 2 = 3+2 = 5)

When you get more experienced:

When you get more experienced, you can solve it like this:
faster like this:
Subtract 2 from both sides: x/3 = 3

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!

So each box was $12

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

Same answer!

Have a Try Yourself

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

1729, 1730, 1731, 1732, 1733, 1734, 3139, 3140, 3141, 3142

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What is the missing number? 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: But the "x" looks like the "×" ... that can be very confusing ...

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Substitution

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

When we have:

And we know that x=6 ...

... then we can "substitute" 6 for x:

Example: When x=2, what is 10/x + 4 ?

Put "2" where "x" is:

Example: When x=5, what is x + x/2 ?

Put "5" where "x" is:

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

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

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

Put "3" where "x" is:

(that is as far as we can get)

As that last example showed, we 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.

Put "(−2)" where "x" is:

In that last example:

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

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In Algebra "Substitution" means putting numbers where the letters are: When we have: And we know that x=6 ... ... then we can "substitute" 6 for x: Put "2" where "x" is: Put "5" where "x" is: Put "3" where "x" is, and "4" where "y" is: Put "3" where "x" is: (that is as far as we can get) As that last example showed, we may not always get a number for an answer, sometimes just a simpler formula.

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Exponents are also called Powers or Indices

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

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

Try it yourself:

So an Exponent saves us writing out lots of multiplies!

a⁷ = a × a × a × a × a × a × a = aaaaaaa

Notice how we wrote the letters together to mean multiply? We will do that a lot here.

The Key to the Laws

Writing all the letters down is the key to understanding the Laws

Which shows that x²x³ = x⁵, but more on that later!

So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.

All you need to know ...

The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:

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

A negative exponent means divide, because the opposite of multiplying is dividing

If you understand those, then you understand exponents!

And all the laws below are based on those ideas.

Laws of Exponents

Here are the Laws (explanations follow):

Law

Example

x ¹ = x

6 ¹ = 6

x ⁰ = 1

7 ⁰ = 1

x⁻¹ = 1/x

4⁻¹ = 1/4

x mx n = x m+n

x ²x ³ = x ²⁺³ = x ⁵

x m/x n = x m-n

x ⁶/x ² = x ⁶⁻² = x ⁴

(x m) n = x mn

(x ²) ³ = x 2×3 = x ⁶

(xy) n = x ny n

(xy) ³ = x ³y ³

(x/y) n = x n/y n

(x/y) ² = x ² / y ²

x-n = 1/x n

x⁻³ = 1/x ³

And the law about Fractional Exponents:

Laws Explained

The first three laws above (, and ) are just part of the natural sequence of exponents. Have a look at this:

Example: Powers of 5

.. etc..

.. etc..

Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).

With x mx n, how many times do we end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.

Like the previous example, how many times do we end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because we are dividing), for a total of "m-n" times.

(Remember that x/ x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)

This law can also show you why x⁰=1 :

First you multiply "m" times. Then you have to do that "n" times, for a total of m×n times.

Example: (x³)⁴ = (xxx)⁴ = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x¹²

To show how this one works, just think of re-arranging all the "x"s and "y"s as in this example:

Similar to the previous example, just re-arrange the "x"s and "y"s

The law that

OK, this one is a little more complicated!

I suggest you read Fractional Exponents first, or this may not make sense.

Anyway, the important idea is that:

And so a fractional exponent like is really saying to do a cube (3) and a square root (1/2), in any order.

Just remember from fractions that m/n = m × (1/n):

Example:

The order does not matter, so it also works for m/n = (1/n) × m:

Example:

Exponents of Exponents ...

What about this example?

We do the exponent at the top first, so we calculate it this way:

And That Is It!
If you find it hard to remember all these rules, then remember this:
you can work them out when you understand the
three ideas near the top of this page

Oh, One More Thing ... What if x = 0?

The Strange Case of 0⁰

There are different arguments for the correct value of 0 ⁰

0 ⁰ could be 1, or possibly 0, so some people say it is really "indeterminate":

x ⁰ = 1, so ...

0 ⁰ = 1

0 n = 0, so ...

0 ⁰ = 0

When in doubt ...

0 ⁰ = "indeterminate"

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Exponents are also called Powers or Indices The exponent of a number says how many times to use the number in a multiplication. In words: 8 2 could be called "8 to the second power", "8 to the power 2" or simply "8 squared" Try it yourself: So an Exponent saves us writing out lots of multiplies!

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When We Multiply:

Example

×

two positives make a positive:

3 × 2 = 6

×

two negatives make a positive:

(−3) × (−2) = 6

×

a negative and a positive
make a negative:

(−3) × 2 = −6

×

a positive and a negative
make a negative:

3 × (−2) = −6

Yes indeed, two negatives make a positive, and we will explain why, with examples!

Signs

Let's talk about signs.

"+" is the positive sign, "−" is the negative sign.

When a number has no sign it usually means that it is positive.

And we can put () around the numbers to avoid confusion.

Example: 3 × −2 can be written as (+3) × (−2)

Two Signs: The Rules

"Two like signs make a positive sign,
two unlike signs make a negative sign"

Example: (−2) × (+5)

The signs are − and + (a negative sign and a positive sign), so they are unlike signs (they are different to each other)

So the result must be negative:

(−2) × (+5) = −10

Example: (−4) × (−3)

The signs are − and − (they are both negative signs), so they are like signs (like each other)

So the result must be positive:

(−4) × (−3) = +12

Why does multiplying two negative numbers make a positive?

Well, first there is the "common sense" explanation:

When I say "Eat!" I am encouraging you to eat (positive)

But when I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you don't need to read any more.

Direction

It is all about direction. Remember the Number Line?

Well here we have Baby Steven taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 2 steps x 3 = 6 steps forward:

Now, Baby Steven can also step backwards (he is a clever little guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this three times:

Once again Steven's Dad puts him back at the start, but facing the other way. Steven takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times:

Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times:

So, by walking backwards, while facing in the negative direction, he moves in the positive direction.

Try it yourself! Try walking forwards and backwards, then again but facing the other direction.

Play With It

But maybe you would like to see it in action? Use the sliders below:

numbers/images/number-line-mult.js

More Examples

Example: Money

Sam gives you three $10 notes:

+3 × +10 = you gain $30

Sam gives you three $10 debts:

+3 × −10 = you lose $30

Sam takes three $10 notes from you:

−3 × +10 = you lose $30

Sam takes three $10 debts from you:

−3 × −10 = you gain $30

Example: A Video of People Running

People Run Forward, Video Normal:

All normal, people running forwards: +1 × +1 = +1

People Run Forward, but Video in Reverse:

Looks like people running backwards: +1 × −1 = −1

People Run Backward, Video Normal:

You see people running backwards: −1 × +1 = −1

People Run Backward, but Video in Reverse:

Looks like people running forwards: −1 × −1 = +1

Example: Tank Levels Rising/Falling

The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank 3 days ago?

We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change is:

−3 × −1,000 = +3,000

The full calculation is:

30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000

So 3 days ago there were 33,000 liters of water in the tank.

Multiplication Table

Here is another way of looking at it.

First have a play with this (explanations below):

numbers/images/mult-grid.js?min=-5&max=5

Start with the multiplication table (just up to 4×4 will do):

Now see what happens when we head into negatives!

Let's go backwards through zero:

Look at the "4" column: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.

Look over that table again, make sure you are comfortable with how it works, because ...

... now we go further to the left, through zero:

We can follow along a row (or column) and the values change consistently:

Follow the "4" row: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.

Follow the "-4" row: it goes 16, 12, 8, 4, 0, -4, -8, -12, -16. Getting 4 smaller each time.

etc...

So it all follows a consistent pattern!

Also see this at Real Multiplication Table.

What About Multiplying 3 or More Numbers Together?

Multiply two at a time and follow the rules.

Example: What is (−2) × (−3) × (−4) ?

First multiply (−2) × (−3). Two like signs make a positive sign, so:

(−2) × (−3) = +6

Next multiply +6 × (−4). Two unlike signs make a negative sign, so:

+6 × (−4) = −24
Result: (−2) × (−3) × (−4) = −24
330, 1615, 1616, 1617, 3447, 3448,331, 1618, 3170, 3171

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Yes indeed, two negatives make a positive, and we will explain why, with examples ... Lets talk about signs. ... is the positive sign, is the negative sign.

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"Expanding" means removing the ( ) ... but we have to do it the right way!

( ) are called "parentheses" or " brackets"

Whatever is inside the ( ) needs to be treated as a "package".

So when multiplying: multiply by everything inside the "package".

Example: Expand 3 × (5+2)

Answer:

It is now expanded.

We can also complete the calculation:

In Algebra

In Algebra putting two things next to each other usually means to multiply.

Here is an example of expanding, using variables a, b and c instead of numbers:

And here is another example involving some numbers. Notice the "·" between the 3 and 6 to mean multiply, so 3·6 = 18:

Multiplying negatives has special rules: a negative times a positive gives a negative, but multiplying two negatives gives a positive:

In that case −3 · -5 = +15 (a positive answer), but here is an example where the second part is negative:

So the second term ended up negative because 2x · −a = −2ax, (it is also neater to write "−2ax" rather than "−2xa").

That was also interesting because of x being squared (x ²)

Lastly, we have an example with three terms inside:

The same rule applies: multiply by everything inside the ().

And here is a hint: when a multiplication is obvious (like a · 2) do it straight away, but when it needs more thought (like a · −b) leave it for the next line.

Many Times Many

How do we do something like this?

(x + 2y)(3x − 4y)

Read Multiplying Polynomials to find out!

Conclusion

Multiply by everything inside the ()

Do it in two stages:

Write down the multiplications

Then do the multiplications

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"Expanding" means removing the ( ) ... but we have to do it the right way! ( ) are called "parentheses" or " brackets" Whatever is inside the ( ) needs to be treated as a "package". So when multiplying: multiply by everything inside the "package". Answer: It is now expanded.

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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:

So we 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 we need the highest common factor, including any variables

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

So we could have:

But we can do better!

Together that makes 3y:

So we can factor the whole expression into:

3y ²+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 we have to figure what got multiplied to produce the expression we are given!

It is like trying to find which ingredients
went into a cake to make it so delicious.
It can be hard to figure out!

Experience Helps

With more experience factoring becomes easier.

Example: Factor 4x² − 9

Hmmm... there don't seem to be any common factors.

But knowing the Special Binomial Products gives us a clue called the "difference of squares":

So we have:

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

So let us try doing that:

Yes!

Answer: 4x ² − 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.

=

(a+b)(a−b)

=

(a+b) (a+b)

=

(a−b) (a−b)

=

=

a ³+3a ²b+3ab ²+b ³

=

(a+b) ³

a ³−3a ²b+3ab ²−b ³

=

(a−b) ³

There are many more like those, but those are the most useful ones.

Advice

The factored form is usually best.

When trying to factor, follow these steps:

There are also Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and others that can do factoring.

More Examples

Experience does help, 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:

Yes, it is the difference of squares

And "(w ² − 4)" is another difference of squares

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

Remove common factor "3u":

Then a difference of cubes:

That is as far as I can go.

Try factoring the first two and second two separately:

z ²(z−1) − 9(z−1)

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

And is a difference of squares

(z−3)(z+3)(z−1)

That is as far as I can go.

Now get some more experience:

338, 339, 2047, 2048, 2049, 178, 2050, 3177, 3178, 3179

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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The equation of a straight line is usually written this way:

(or "y = mx + c" in the UK see below)

What does it stand for?

y = how far up

x = how far along

m = Slope or Gradient (how steep the line is)

How do you find "m" and "b"?

b is easy: just see where the line crosses the Y axis.

m (the Slope) needs some calculation:

m = Change in Y Change in X

Knowing this we can work out the equation of a straight line:

Example 1

m = 2 1 = 2

b = 1 (value of y when x=0)

So: y = 2x + 1

With that equation you can now ...

... choose any value for x and find the matching value for y

For example, when x is 1:

Check for yourself that x=1 and y=3 is actually on the line.

Or we could choose another value for x, such as 7:

Positive or Negative Slope?

Going from left-to-right, the cyclist has to Push on a P ositive Slope:

Example 2

m = −3 1 = −3

b = 0

This gives us y = −3x + 0

We do not need the zero!

So: y = −3x

Example 3: Vertical Line

What is the equation for a vertical line?
The slope is undefined ... and where does it cross the Y-Axis?

In fact, this is a special case, and you use a different equation, not " y=...", but instead you use " x=...".

x = 1.5

Every point on the line has x coordinate 1.5,
that is why its equation is x = 1.5

Rise and Run

Sometimes the words "rise" and "run" are used.

Rise is how far up

Run is how far along

And so the slope "m" is:

You might find that easier to remember.

Other Forms

We have been looking at the "slope-intercept" form. The equation of a straight line can be written in many other ways.

Another popular form is the Point-Slope Equation of a Straight Line.

Footnote

Country Note:

Different Countries teach different "notation" (as sent to me by kind readers):

In the UK, Australia (also), Bahamas, Bangladesh, Belgium, Brunei, Bulgaria, Cyprus, Egypt, Germany, Ghana, India, Indonesia, Ireland, Jamaica, Kenya, Kuwait, Malaysia, Malawi, Malta, Nepal, New Zealand, Nigeria, Oman, Pakistan, Peru, Singapore, Solomon Islands, South Africa, Sri Lanka, Turkey, UAE, Zambia and Zimbabwe

y = mx + c

In Afghanistan, Albania, Algeria, Brazil, China, Czech Republic, Denmark, Ethiopia, France, Lebanon, Netherlands, Kosovo, Kyrgyzstan, Norway, Poland, Romania, South Korea, Surinam, Spain, Tunisia and Viet Nam:

y = ax + b

... but it all means the same thing, just different letters.

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The equation of a straight line is usually written this way: (or "y = mx + c" in the UK see below) y = how far up x = how far along m = Slope or Gradient (how steep the line is) b is easy: just see where the line crosses the Y axis.

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Keep the sequence, Finding the equation

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What is a Sequence?

A Sequence is a list of things (usually numbers) that are in order.

Infinite or Finite

When the sequence goes on forever it is called an infinite sequence,
otherwise it is a finite sequence

Examples:

{1, 2, 3, 4, ...} is a very simple sequence (and it is an infinite sequence)

{20, 25, 30, 35, ...} is also an infinite sequence

{1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a finite sequence)

{4, 3, 2, 1} is 4 to 1 backwards

{1, 2, 4, 8, 16, 32, ...} is an infinite sequence where every term doubles

{a, b, c, d, e} is the sequence of the first 5 letters alphabetically

{f, r, e, d} is the sequence of letters in the name "fred"

{0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s (yes they are in order, it is an alternating order in this case)

In Order

When we say the terms are "in order", we are free to define what order that is! They could go forwards, backwards ... or they could alternate ... or any type of order we want!

Like a Set

A Sequence is like a Set, except:

the terms are in order (with Sets the order does not matter)

the same value can appear many times (only once in Sets)

Example: {0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s.

The set is just {0,1}

Notation

Sequences also use the same notation as sets:
list each element, separated by a comma,
and then put curly brackets around the whole thing.

{3, 5, 7, ...}

The curly brackets { } are sometimes called "set brackets" or "braces".

A Rule

A Sequence usually has a Rule, which is a way to find the value of each term.

Example: the sequence {3, 5, 7, 9, ...} starts at 3 and jumps 2 every time:

As a Formula

Saying " starts at 3 and jumps 2 every time" is fine, but it doesn't help us calculate the:

So, we want a formula with "n" in it (where n is any term number).

So, What Can A Rule For {3, 5, 7, 9, ...} Be?

Firstly, we can see the sequence goes up 2 every time, so we can guess that a Rule is something like "2 times n" (where "n" is the term number). Let's test it out:

Test Rule: 2n

That nearly worked ... but it is too low by 1 every time, so let us try changing it to:

Test Rule: 2n+1
That Works!
So instead of saying "starts at 3 and jumps 2 every time" we write this:
2n+1
Now we can calculate, for example, the 100th term:

2 × 100 + 1 = 201

Many Rules

But mathematics is so powerful we can find more than one Rule that works for any sequence.

Example: the sequence {3, 5, 7, 9, ...}

We have just shown a Rule for {3, 5, 7, 9, ...} is: 2n+1

And so we get: {3, 5, 7, 9, 11, 13, ...}

But can we find another rule?

How about "odd numbers without a 1 in them":

And we get: {3, 5, 7, 9, 23, 25, ...}
A completely different sequence!
And we could find more rules that match {3, 5, 7, 9, ...}. Really we could.

So it is best to say "A Rule" rather than "The Rule" (unless we know it is the right Rule).

Notation

To make it easier to use rules, we often use this special style:

xn is the term

n is the term number

Example: to mention the "5th term" we write:

So a rule for {3, 5, 7, 9, ...} can be written as an equation like this:

And to calculate the 10th term we can write:
Can you calculate x₅₀ (the 50th term) doing this?
Here is another example:

Example: Calculate the first 4 terms of this sequence:

Calculations:

Answer:

{a n} = { -1, 1/4, -1/27, 1/256, ... }

Special Sequences

Now let's look at some special sequences, and their rules.

Arithmetic Sequences

In an Arithmetic Sequence the difference between one term and the next is a constant.

In other words, we just add some value each time ... on to infinity.

Example:

In General we can write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }

where:

a is the first term, and

d is the difference between the terms (called the "common difference")

And we can make the rule:

(We use "n-1" because d is not used in the 1st term).

Geometric Sequences

In a Geometric Sequence each term is found by multiplying the previous term by a constant.

Example:

This sequence has a factor of 2 between each number.
Its Rule is

In General we can write a geometric sequence like this:

where:

a is the first term, and

r is the factor between the terms (called the "common ratio")

Note: r should not be 0.

When r=0, we get the sequence {a,0,0,...} which is not geometric

And the rule is:

(We use "n-1" because is the 1st term)

Triangular Numbers

The Triangular Number Sequence is generated from a pattern of dots which form a triangle:

By adding another row of dots and counting all the dots we can find the next number of the sequence.

But it is easier to use this Rule:

Example:

Square Numbers

The next number is made by squaring where it is in the pattern.

Rule is xn = n²

Cube Numbers

The next number is made by cubing where it is in the pattern.

Rule is xn = n³

Fibonacci Sequence

The next number is found by adding the two numbers before it together:

The 2 is found by adding the two numbers before it (1+1)

The 21 is found by adding the two numbers before it (8+13)

etc...

Rule is xn = xn-1 + xn-2

That rule is interesting because it depends on the values of the previous two terms.

Rules like that are called recursive formulas.

The Fibonacci Sequence is numbered from 0 onwards like this:

Example: term "6" is calculated like this:
x₆ = x₆₋₁ + x₆₋₂ = x₅ + x₄ = 5 + 3 = 8
Series and Partial Sums

Now you know about sequences, the next thing to learn about is how to sum them up. Read our page on Partial Sums.

When we sum up just part of a sequence it is called a Partial Sum.

But a sum of an infinite sequence it is called a "Series" (it sounds like another name for sequence, but it is actually a sum). See Infinite Series.

Example: Odd numbers

Sequence: {1, 3, 5, 7, ...}

Series: 1 + 3 + 5 + 7 + ...

Partial Sum of first 3 terms: 1 + 3 + 5

595, 1242, 596, 1243, 597, 3009, 3010, 1244, 3011, 3012

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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To find a missing number in a Sequence, first we must have a Rule

Sequence

A Sequence is a set of things (usually numbers) that are in order.

Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for a more in-depth discussion.

Finding Missing Numbers

To find a missing number, first find a Rule behind the Sequence.

Sometimes we can just look at the numbers and see a pattern:

Example: 1, 4, 9, 16, ?

Rule: xn = n²

Sequence: 1, 4, 9, 16, 25, 36, 49, ...

Did you see how we wrote that rule using "x" and "n" ?

And we can calculate term 3 using:

We can use a Rule to find any term. For example, the 25th term can be found by "plugging in" 25 wherever n is.

How about another example:

Example: 3, 5, 8, 13, 21, ?

After 3 and 5 all the rest are the sum of the two numbers before,

That is 3 + 5 = 8, 5 + 8 = 13 etc, which is part of the Fibonacci Sequence:

3, 5, 8, 13, 21, 34, 55, 89, ...

Which has this Rule:

Rule: xn = xn-1 + xn-2

Now what does mean? It means "the previous term" as term number n-1 is 1 less than term number n.

And means the term before that one.

Let's try that Rule for the 6th term:

So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:

Many Rules

One of the troubles with finding "the next number" in a sequence is that mathematics is so powerful we can find more than one Rule that works.

What is the next number in the sequence 1, 2, 4, 7, ?

Here are three solutions (there can be more!):

Solution 1: Add 1, then add 2, 3, 4, ...
Rule: xn = n(n-1)/2 + 1
Sequence: 1, 2, 4, 7, 11, 16, 22, ...

(That rule looks a bit complicated, but it works)

Solution 2: After 1 and 2, add the two previous numbers, plus 1:
Rule: xn = xn-1 + xn-2 + 1
Sequence: 1, 2, 4, 7, 12, 20, 33, ...

Solution 3: After 1, 2 and 4, add the three previous numbers
Rule: xn = xn-1 + xn-2 + xn-3
Sequence: 1, 2, 4, 7, 13, 24, 44, ...

So, we have three perfectly reasonable solutions, and they create totally different sequences.

Which is right? They are all right.

And there are other solutions ...

... it may be a list of the winners' numbers ... so the next number could be ... anything!

Simplest Rule

When in doubt choose the simplest rule that makes sense, but also mention that there are other solutions.

Finding Differences

Sometimes it helps to find the differences between each pair of numbers ... this can often reveal an underlying pattern.

Here is a simple case:

The differences are always 2, so we can guess that "2n" is part of the answer.

Let us try 2n:

The last row shows that we are always wrong by 5, so just add 5 and we are done:

OK, we could have worked out "2n+5" by just playing around with the numbers a bit, but we want a systematic way to do it, for when the sequences get more complicated.

Second Differences

In the sequence {1, 2, 4, 7, 11, 16, 22, ...} we need to find the differences ...

... and then find the differences of those (called second differences), like this:

The second differences in this case are 1.

In our case the difference is 1, so let us try just :

Wrong by 1 now, so let us add 1:

We did it!

So by "trial-and-error" we discovered a rule that works:

Sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, ...

Other Types of Sequences

Read Sequences and Series to learn about:

And there are also:

And many more!

In truth there are too many types of sequences to mention here, but if there is a special one you would like me to add just let me know.

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To find a missing number in a Sequence, first we must have a Rule A Sequence is a set of things (usually numbers) that are in order. Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for a more in-depth discussion.

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Pythagoras (no angles given). Trigonometry (1 angle known)

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Pythagoras

Over 2000 years ago there was an amazing discovery about triangles:
When a triangle has a right angle (90°) ...... and squares are made on each of the three sides, ...... then the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

Note:

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Let's check if the areas are the same:

3 ² + 4 ² = 5 ²

Calculating this becomes:

9 + 16 = 25
It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a ² + b ² = c ²

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

Read Builder's Mathematics to see practical uses for this.

Also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

Example: What is the diagonal distance across a square of size 1?

It works the other way around, too: when the three sides of a triangle make , then the triangle is right angled.

Example: Does this triangle have a Right Angle?

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

Draw a right angled triangle on the paper, leaving plenty of space.

Draw a square along the hypotenuse (the longest side)

Draw the same sized square on the other side of the hypotenuse

Draw lines as shown on the animation, like this:

Cut out the shapes

Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

becomes

We also have a proof by adding up the areas.

Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361

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Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... ... then the biggest square has the exact same area as the other two squares put together!

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Want to learn Trigonometry? Here is a quick summary.
Follow the links for more, or go to Trigonometry Index

Trigonometry ... is all about triangles.

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

Right-Angled Triangle

The triangle of most interest is the right-angled triangle. The right angle is shown by the little box in the corner:

Another angle is often labeled θ, and the three sides are then called:

Adjacent: adjacent (next to) the angle θ

Opposite: opposite the angle θ

and the longest side is the Hypotenuse

Why a Right-Angled Triangle?

Why is this triangle so important?

Imagine we can measure along and up but want to know the direct distance and angle:

Trigonometry can find that missing angle and distance.

Or maybe we have a distance and angle and need to "plot the dot" along and up:

Questions like these are common in engineering, computer animation and more.

And trigonometry gives the answers!

Sine, Cosine and Tangent

The main functions in trigonometry are Sine, Cosine and Tangent

They are simply one side of a right-angled triangle divided by another.

For any angle "θ":
(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan.)
Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

The triangle could be larger, smaller or turned around, but that angle will always have that ratio.

Calculators have sin, cos and tan to help us, so let's see how to use them:

Example: How Tall is The Tree?

We can't reach the top of the tree, so we walk away and measure an angle (using a protractor) and distance (using a laser):

Sine is the ratio of Opposite / Hypotenuse:

Get a calculator, type in "45", then the "sin" key:

sin(45°) = 0.7071...

What does the 0.7071... mean? It is the ratio of the side lengths, so the Opposite is about 0.7071 times as long as the Hypotenuse.

We can now put 0.7071... in place of sin(45°):

And we also know the hypotenuse is 20:

To solve, first multiply both sides by 20:

20 × 0.7071... = Opposite

Finally:

Opposite = 14.14m (to 2 decimals)

Example: How Tall is The Tree?

The tree is 14.14m tall

Try Sin Cos and Tan

Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.

../algebra/images/circle-triangle.js

Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be positive or negative by the rules of Cartesian coordinates, so the sine, cosine and tangent change between positive and negative also.

So trigonometry is also about circles!

Unit Circle

What you just played with is the Unit Circle.

It is a circle with a radius of 1 with its center at 0.

Because the radius is 1, we can directly measure sine, cosine and tangent.

Here we see the sine function being made by the unit circle:
Note: you can see the nice graphs made by sine, cosine and tangent.
Degrees and Radians

Angles can be in Degrees or Radians. Here are some examples:

Angle

Degrees

Radians

Right Angle

90°

π/2

__ Straight Angle

180°

π

Full Rotation

360°

2π

Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation (see Amplitude, Period, Phase Shift and Frequency).

When we want to calculate the function for an angle larger than a full rotation of 360° (2π radians) we subtract as many full rotations as needed to bring it back below 360° (2π radians):

Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° − 360° = 10°

cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

And when the angle is less than zero, just add full rotations.

Example: what is the sine of −3 radians?

−3 is less than 0 so let us add 2π radians

−3 + 2π = −3 + 6.283... = 3.283... radians

sin(−3) = sin(3.283...) = −0.141 (to 3 decimal places)

Solving Triangles

Trigonometry is also useful for general triangles, not just right-angled ones .

It helps us in Solving Triangles. "Solving" means finding missing sides and angles.

We can also find missing side lengths. The general rule is:

When we know any 3 of the sides or angles we can find the other 3
(except for the three angles case)

See Solving Triangles for more details.

Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

Trigonometric and Triangle Identities

And as you get better at Trigonometry you can learn these:

Enjoy becoming a triangle (and circle) expert!

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Want to learn Trigonometry? Here is a quick summary. Follow the links for more, or go to Trigonometry Index Trigonometry ... is all about triangles. Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more! The triangle of most interest is the right-angled triangle.

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Perimeter, Areas, Volume and unit conversion

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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Area of Circle, Triangle, Square, Rectangle, Parallelogram, Trapezium, Ellipse and Sector

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Area is the size of a surface!
Learn more about Area, or try the Area Calculator.

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Solid Geometry is the geometry of three-dimensional space,
the kind of space we live in.

Three Dimensions

It is called three-dimensional,
or 3D, because there are
three dimensions: width, depth and height.

Simple Shapes

Let us start with some of the simplest shapes:

Common 3D Shapes
Properties

Solids have properties (special things about them), such as:

Polyhedra and Non-Polyhedra

There are two main types of solids, "Polyhedra", and "Non-Polyhedra":

Polyhedra (they must have flat faces):

Non-Polyhedra (when any surface is not flat):

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Solid Geometry is the geometry of three-dimensional space, the kind of space we live in. It is called three-dimensional, or 3D, because there are three dimensions: width, depth and height.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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We can help you solve an equation of the form "ax² + bx + c = 0"
Just enter the values of a, b and c below:

algebra/images/quadratic-solver.js

Is it Quadratic?

The name comes from "quad" meaning square, as the variable is squared (in other words x²).

These are all quadratic equations in disguise:

How Does this Work?

The solution(s) to a quadratic equation can be calculated using the Quadratic Formula:

The "±" means we need to do a plus AND a minus, so there are normally TWO solutions !

The blue part ( b² - 4ac) is called the "discriminant", because it can "discriminate" between the possible types of answer:

when it is positive, we get two real solutions,

when it is zero we get just ONE solution,

when it is negative we get complex solutions.

Learn more at Quadratic Equations

Note: you can still access the old version here.

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We can help you solve an equation of the form "ax2 + bx + c = 0" Just enter the values of a, b and c below: algebra/images/quadratic-solver.js The name comes from "quad" meaning square, as the variable is squared (in other words x2 ).