An Introduction to US Standard Units

Also known as "English Units" or "US Customary Units"

Wow, I just flew in from planet Micron.  It was a long flight, but well worth it to get to spend time with you!
My name is Maggie in your language (but you couldn't pronounce my real name!)
When I first arrived I couldn't understand how you measure things, but my friend Tom taught me all about measurement, and I am going to share with you everything he taught me.
The first thing Tom told me was that you can measure things using two different systems: US Standard Units and Metric.
Today is my day to learn US Standard Units !

Liquids

Since it was such a long flight, the first thing I could use is something cold to drink.   But I want to be sure how much to ask for! So I can get a drink that is not too big or too small.

The first thing I need to know when asking for a drink is the types of units used to measure liquids, which are:

• Fluid Ounces
• Cups
• Pints
• Quarts
• Gallons

Fluid Ounces (oz) are small. About how much fits into a small medicine cup ... but that isn't enough for someone who is thirsty!

Then Tom showed me a small carton, and told me it held 8 fluid ounces, which is also called 1 cup. But I wanted more!

So Tom showed me a pint, which is equal to 2 cups. That seemed about right for someone who was very thirsty!

(Tom also told me that I can measure things using measuring cups)

To measure many cups of liquid all put together you can use quarts. A quart (qt) is the same thing as 4 cups or 2 pints.

If you still need more liquid you may want to switch to using gallons. A gallon (gal) is the same as 16 cups or 8 pints or 4 quarts.  It is the largest liquid measurement. (Oh wow! A quart is a quarter of a gallon!)

So now I know that 1 ounce is too small for me, but 1 gallon is too much. I think I will ask for a pint of juice!
Final thoughts about measuring liquids: 

1 Gallon = 4 quarts = 8 pints = 16 cups= 128 fluid ounces

Mass (Weight)

Next I wanted to eat some chocolate ... so I should learn about mass. You often call it "weight", but it is only because of the gravity on your planet that items have weight! 
Tom says I need to know:

• Ounces
• Pounds
• Tons

Ounces Again! One thing that really confused me is that when I asked for a drink I could use ounces, but ounces are also used for mass ... ... the same word can be used in two different measuring systems!  How amazing is that? But they are really different.

Tom says:

If you mean an ounce of fluid say "fluid ounce" ("fl oz")
Otherwise ounce usually means mass.

So we are not talking about fluids! I already had a drink. I need to know about mass.

The smallest unit of mass is ounces (oz).  A slice of bread is about one ounce.  It is very light.

But when you add up the ounces you get a new label for mass:

If you have 16 ounces, it can also be called a pound (lb).  Typically, this is the unit that you use to measure your own weight.  1 pound = 16 ounces Pounds are used to measure lots of things from people to food to animals.  Tom says he weighs 90 pounds.

But if something weighs many pounds we use yet another label

2,000 pounds is also one ton.  1 ton = 2,000 pounds That is really heavy!  Trucks, ships and heavy equipment are measured using tons instead of pounds. An elephant has a mass of about 8 tons!

So now I know that 1 ounce of chocolate is too small for me, but 1 ton is way too much. I think I will ask for a pound of chocolate!
Final thoughts about mass:

1 pound = 16 ounces
1 Ton = 2,000 pounds = 32,000 ounces

Length

The last kind of measurement we will explore is length.  This is important for lots of different reasons.  Measurement of length helps you to know how far you have traveled, how far you have left to travel, how tall you are and many other things.

I need to know about:

• Inches
• Feet
• Yards
• Miles

Small units of length are called inches.  The last joint of your finger or thumb is about 1 inch (depending on how big your fingers are!).

Lots of things are measured in inches from rainfall to paper length.
Measuring in inches gives us a way for everyone to understand the size of something.

When we have 12 inches together, it is known as a foot. 

1 foot = 12 inches

A long time ago, people used their feet to measure things. But everyone has different sized feet so it did not work very well.
Using 12 inches put together to make one foot lets everyone have an accurate picture of what exactly a "foot" of length is.

When 3 feet are together, this is called a yard.  (This isn't the same thing as a lawn, though they are both referred to as a "yard"!)

1 yard = 3 feet

The length of this guitar is about 1 yard.

When you put together 1,760 yards, you have a mile.  1 mile = 1,760 yards = 5,280 feet Miles are long distances and are mostly used to measure the distance between places which are far away from each other.  Most people refer to miles when they are driving, biking or jogging.

Final thoughts about measuring length:

1 foot = 12 inches

1 yard = 3 feet = 36 inches

1 mile = 1,760 yards = 5,280 feet = 63,360 inches

Temperature

I was feeling a bit hot, so I asked Tom how to measure temperature.
So he showed me a thermometer. But I saw 2 sets of numbers!
Tom explained that a thermometer measures in degrees (°) of either Celsius or Fahrenheit.
"Why two scales?", I asked.
Tom said that some people like one scale and some like the other, and that I should learn both!
He then gave me an example: when water freezes the thermometer shows:

• 0 degrees Celsius on the left side,
• but on the right side it shows 32 degrees Fahrenheit.

Two numbers for the same thing!
He gave me more examples.

• A hot sunny day with a temperature of 30 degrees Celsius is 86 degrees in Fahrenheit.
• Water boils at 100 degrees Celsius or 212 degrees Fahrenheit.
• And you can bake cookies in your oven at a temperature of 180 degrees Celsius, which is 356 degrees Fahrenheit.

I decided to get my own thermometer, so I can learn all about this.

Bye for Now!

I hope you enjoyed learning all about measurement.  Now I must return home.  Keep measuring until I see you again!!!!!!!!!

The Evolution of Numbers

I want to take you on an adventure ...

... an adventure through the world of numbers.

Let us start at the beginning:

Q: What is the simplest idea of a number?

A: Something to count with!

The Counting Numbers

We can use numbers to count: 1, 2, 3, 4, etc
Humans have been using numbers to count with for thousands of years. It is a very natural thing to do.

• You can have "3 friends",
• a field can have "6 cows"
• and so on.

So we have:

Counting Numbers: {1, 2, 3, ...}

And the "Counting Numbers" satisfied people for a long time.

Zero

The idea of zero, though natural to us now, was not natural to early humans ... if there is nothing to count, how can you count it?

Example: you can count dogs, but you can't count an empty space:

Two Dogs		Zero Dogs? Zero Cats?

An empty patch of grass is just an empty patch of grass!

Placeholder

But about 3,000 years ago people needed to tell the difference between numbers like 4 and 40. Without the zero they look the same!
So they used a "placeholder", a space or special symbol, to show "there are no digits here"

5 2	So "5 2" meant "502" (5 hundreds, nothing for the tens, and 2 units)

The idea of zero had begun, but it wasn't for another thousand years or so that people started thinking of it as an actual number.
But now we can think

"I had 3 oranges, then I ate the 3 oranges, now I have zero oranges...!"

The Whole Numbers

So, let us add zero to the counting numbers to make a new set of numbers.
But we need a new name, and that name is "Whole Numbers":

Whole Numbers: {0, 1, 2, 3, ...}

The Natural Numbers

You may also hear the term "Natural Numbers" ... which can mean:

• the "Counting Numbers": {1, 2, 3, ...}
• or the "Whole Numbers": {0, 1, 2, 3, ...}

depending on the subject. I guess they disagree on whether zero is "natural" or not.

Negative Numbers

But the history of mathematics is all about people asking questions, and seeking the answers!
One of the good questions to ask is

"if you can go one way, can you go the opposite way?"

We can count forwards: 1, 2, 3, 4, ...

... but what if we count backwards: 3, 2, 1, 0, ... what happens next?

The answer is: you get negative numbers:

Now we can go forwards and backwards as far as we want

But how can a number be "negative"?

By simply being less than zero.

A simple example is temperature. We define zero degrees Celsius (0° C) to be when water freezes ... but if we get colder we need negative temperatures. So -20° C is 20° below Zero.

Negative Cows?

And in theory you can have a negative cow!
Think about this ...If you had just sold two bulls, but can only find one to hand over to the new owner... you actually have minus one bull ... you are in debt one bull!
So negative numbers exist, and we're going to need a new set of numbers to include them ...

Integers

If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers

Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}

The Integers include zero, the counting numbers, and the negative of the counting numbers, to make a list of numbers that stretch in either direction indefinitely.

Fractions

If you have one orange and want to share it with someone, you need to cut it in half. You have just invented a new type of number!

You took a number (1) and divided by another number (2) to come up with half (1/2)

The same thing happens when we have four biscuits (4) and want to share them among three people (3) ... they get (4/3) biscuits each.
A new type of number, and a new name:

Rational Numbers

Any number that can be written as a fraction is called a Rational Number.
So, if "p" and "q" are integers (remember we talked about integers), then p/q is a rational number.

Example: If p is 3 and q is 2, then:

p/q = 3/2 = 1.5 is a rational number

The only time this doesn't work is when q is zero, because dividing by zero is undefined.

Rational Numbers: {p/q : p and q are integers, q is not zero}

So half (½) is a rational number.
And 2 is a rational number also, because you could write it as 2/1
So, Rational Numbers include:

• all the integers
• and all fractions.

Even a number like 13.3168980325 is a Rational Number.

13.3168980325 = 133,168,980,325 / 10,000,000,000

That seems to include all possible numbers, right?

But There Is More

People didn't stop asking the questions ...and here is one that caused a lot of fuss during the time of Pythagoras:

If you draw a square (of size "1"), what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950...(etc)
But it is not a number like 3, or five-thirds, or anything like that ...

... in fact you cannot answer that question using a ratio of two integers

square root of 2 ≠ p/q

... and so it is not a rational number (read more here)

Wow! There are numbers that are NOT rational numbers! What do we call them?

What is "Not Rational" ...? Irrational !

Irrational Numbers

So, the square root of 2 (√2) is an irrational number. It is called irrational because it is not rational (can't be made using a simple ratio of integers). It isn't crazy or anything, just not rational.
And we know there are many more irrational numbers. Pi (π) is a famous one.

Useful

So irrational numbers are useful. You need them to

• find the diagonal distance across some squares,
• to work out lots of calculations with circles (using π),
• and more,

So we really should include them.
And so, we introduce a new set of numbers ...

Real Numbers

That's right, another name!
Real Numbers include:

• the rational numbers, and
• the irrational numbers

Real Numbers: {x : x is a rational or an irrational number}

In fact a Real Number can be thought of as any point anywhere on the number line:

© 2015 MathsIsFun.com v0.77

This only shows a few decimal places (it is just a simple computer)
but Real Numbers can have lots more decimal places!

Any point Anywhere on the number line, that is surely enough numbers!
But there is one more number which has turned out to be very useful. And once again, it came from a question.

Imagine ...

The question is:

"is there a square root of minus one?"

In other words, what can you multiply by itself to get -1?
Think about this: if you multiply any number by itself you can't get a negative result:

• 1×1 = 1,
• and also (-1)×(-1) = 1 (because a negative times a negative gives a positive)

So what number, when multiplied by itself, results in -1?
This is normally not possible, but ...

"if you can imagine it, then you can play with it"

So, ...

Imaginary Numbers

... let us just imagine that the square root of minus one exists. We can even give it a special symbol: the letter i

And we can use it to answer questions:

Example: what is the square root of -9 ?
Answer: √(-9) = √(9 × -1) = √(9) × √(-1) = 3 × √(-1) = 3i

OK, the answer still involves i, but it gives a sensible and consistent answer.
And i has this interesting property that if you square it (i×i) you get -1 which is back to being a Real Number. In fact that is the correct definition:

Imaginary Number: A number whose square is a negative Real Number.

And i (the square root of -1) times any Real Number is an Imaginary Number. So these are all Imaginary Numbers:

• 3i
• -6i
• 0.05i
• πi

There are also many applications for Imaginary Numbers, for example in the fields of electricity and electronics.

Real vs Imaginary Numbers

Imaginary Numbers were originally laughed at, and so got the name "imaginary". And Real Numbers got their name to distinguish them from the Imaginary Numbers.
So the names are just a historical thing. Real Numbers aren't "in the Real World" (in fact, try to find exactly half of something in the real world!) and Imaginary Numbers aren't "just in the Imagination" ... they are both valid and useful types of Numbers!
In fact they are often used together ...

"what if you put a Real Number and an Imaginary Number together?"

Complex Numbers

Yes, if you put a Real Number and an Imaginary Number together you get a new type of number called a Complex Number and here are some examples:

• 3 + 2i
• 27.2 - 11.05i

A Complex Number has a Real Part and an Imaginary Part, but either one could be zero

So a Real Number is also a Complex Number (with an imaginary part of 0):

• 4 is a Complex Number (because it is 4 + 0i)

and likewise an Imaginary Number is also a Complex Number (with a real part of 0):

• 7i is a Complex Number (because it is 0 + 7i)

So the Complex Numbers include all Real Numbers and all Imaginary Numbers, and all combinations of them.

And that's it!

That's all of the most important number types in mathematics.

From the Counting Numbers through to the Complex Numbers.

There are other types of numbers, because mathematics is a broad subject, but that should do you for now.

Summary

Here they are again:

Type of Number	Quick Description
Counting Numbers	{1, 2, 3, ...}
Whole Numbers	{0, 1, 2, 3, ...}
Integers	{..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational Numbers	p/q : p and q are integers, q is not zero
Irrational Numbers	Not Rational
Real Numbers	Rationals and Irrationals
Imaginary Numbers	Squaring them gives a negative Real Number
Complex Numbers	Combinations of Real and Imaginary Numbers

End Notes

History

The history of mathematics is very broad, with different cultures (Greeks, Romans, Arabic, Chinese, Indians and European) following different paths, and many claims for "we thought of it first!", but the general order of discovery I discussed here gives a good idea of it.

Questions

And isn't it amazing how many times that asking a question, like

• "what happens if we count backwards through zero", or
• "what is the exact distance across the diagonal of the square"

first led to disagreement (and even ridicule!), but eventually to amazing breakthroughs in understanding.
I wonder what interesting questions are being asked now?

Over to You!

Here are two questions you can ask when you learn something new:

Can it go the other way?

• Positive numbers lead to negative numbers
• Squares lead to square roots
• etc

Can I use this with something else I know?

• If fractions are numbers, can they be added, subtracted, etc?
• Can I take the square root of a complex number? (can you?)
• etc

And one day your questions may lead to a new discovery!

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 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

Example: factor 3y²+12y

Firstly, 3 and 12 have a common factor of 3.
So we 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 we 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 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":

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 most useful 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

There are also Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and many more that are good at 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:

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 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!

See this in action at the Algebra Balance Animation.

Another Puzzle

Solve this one:

x

+

5

=

12

What we want is an answer like "x = ...",
but the +5 is in the way of that!
We can cancel out the +5 with a −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)

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)
Variables without a number have a coefficient of 1 (x is really 1x)
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

Question 1

Algebra (Grade 5, Easy)

Help

Solve x + 2 = 6

A

x = 3

B

x = 4

C

x = 6

D

x = 8

Don't waste this chance to learn

Question 2

Algebra (Grade 5, Easy)

Help

Solve x + 5 = 11

A

x = 16

B

x = 10

C

x = 8

D

x = 6

Use pen and paper to work out the answer