This is about the math you can love. This is math you can actually use.

Live and breathe this stuff and your life can change.

This is the math which is every human's birthright and one of mankind'smost powerful celebrations.

Here are a some principles I've used in writing this:

  1. I don't prove anything. Proofs are usually difficult and boring--needed by the working mathematician, confusing to most others. Most of these ideas are thousands of years old, anyway.
  2. I've included the stuff I love. Knowing this stuff enhances my power and understanding. These ideas have touched my heart.
  3. I include the math which is so simple and obvious that it actually gets used by the people who know it.

In reading this, don't skip any of the words. That way, you won't miss anything because you thought you knew it.

If something doesn't make sense, reread it.

If it still doesn't make sense, put the book down until the next day.

If it still doesn't make sense, find one of the ten thousand books written on that subject which explains it better, or get one of your friends to explain it--over and over till you get it.

Knowing the math in this book is part of your birthright as a human. Don't let anyone cheat you out of it.


Math begins with the point. A point is a location in space. We represent them with dots, arrows, or crossed lines.

We usually need two lines to define a point's location. In fact, an arrow is really just three crossed lines ending in a point.

We need two straight lines to determine a point. How do we get straight lines?

A straight line is the intersection of two plane surfaces. Two lines determine a plane surface and two plane surfaces determine a line.

One way to get a reasonably straight line in the real world is to stretch a string between two points.

Accurate straight lines in the real world are made by intersecting two accurate plane surfaces.

Accurate plane surfaces are made by grinding three surfaces together until they all fit each other. These are usually made of cast iron, glass, or granite.

Here's how that works ( for those really interested):


Three lines intersecting form a triangle. Triangles are the most basic and most useful figure. They are rigid. Once you fix two sides and their common angle, or two angles and their common side, or the lengths of the three sides, the triangle is formed and cannot change shape and remain a triangle.

If you have a square or rectangle of metal, and cut the middle material out, it becomes flimsy. A triangular piece with the middle material cut out can be as strong as the original piece.

Types of Triangles

The most useful triangles are the isoceles, right, and equilateral.

Isoceles means equal sided and refers to a triangle with two equal sides and angles.

Right means right as in upright. The right triangle has a right angle in it. The right angle is the angle we make with the ground. It's also the angle a string makes with the ground when it's suspended by a weight. There are two right angles in a straight line and the sum of all the angles in a triangle seems to be two right angles or the angle of a straight line. A right angle is 90 degrees. A straight line is 180 degrees.

Equilateral means equal sided. All sides and angles are equal. The angle is one third of 180 or 60 degrees.

An isoceles right triangle has angles of 90, 45 and 45 degrees.

One other weird word used with triangles is the word "hypotenuse". The hypotenuse is the longest side in a right triangle.

We can get another useful triangle from the equilateral by drawing a line from one corner to the middle of the other side, making two right triangles:

These two new triangles are called the 30-60-90 triangles.

Most of the work people have done with triangles has been done with the 45-45-90, the 30-60-90, and the equilateral.

The Pythagorean theorem

Draw a right triangle and form the squares on its sides:

The area in the square on the longest side (the "hypotenuse") is equal to the sum of the areas in the other two squares.

For example, take a right triangle with sides 3 , 4 , and 5.

This theorem is thousands of years old and serves as the basis for a lot of mathematics.

The right isoceles triangle with short sides of 1 has a hypotenuse of the square root of two or about 1.414.

The 30-60-90 triangle has sides of 1,2 and the square root of 3 or about 1.732

These results are very useful since the triangles are used so often.

Triangle Area

The formula for the area of a triangle is to multiply one-half the size of the base by the height.


Polygons are many sided figures ( from the Greek, pollos, many, and gonos, side). They are figures which have more sides than a triangle.

The first group are the quadrilaterals, or four sided figures.

The special ones are:

In the polygons with more sides, there is really only interest in the figure with all sides of equal length.

Any polygon can be broken up into triangles to find things like the area, the way to construct it, and so on, like this:

Always try to reduce your structure to its triangles, since this will make them simple to work with, though, like all such processes, it reduces the mystery.


A circle is all the points that can be drawn at the same distance from a single point called the center. They are usually drawn with a compass. In the real world, lathes are designed to make circles of great accuracy.

Circles are useful because they can be used to measure things, as in the famous ruler and compass constructions.


We are always surrounded by things. In my mind I can collect these things into groups or collections--all the shoes in my house, or all the split peas. Any collection can be identified by its size if I have a name for each size.

A number system is a way of naming each size collection. By having a system of numbers, we can make a name for any size we want.

If we have a number, and we add one to it, we must be able to name the new number.

In the decimal system, we have the numbers one through nine, the numbers ten through ninety, the name hundred, the name thousand, the name, million, the name billion, and the name trillion. This is a total of 23 names for numbers which in diffrent combinations can name every size of collection from one to much farther than I care to contemplate--say the number of atoms on every beach in the world, or the number of stars in the known universe.

This is how it works:

Counting is adding one to each number in succession while moving the collection one by one from one pile to another.

When you have counted to nine, you say ten and start over until you reach nineteen (I've ignored the quaint words eleven, twelve, etc. ) then you say twenty and start over until twenty-nine, and so on until you reach ninety-nine.

You then say one hundred and start over.

When you get to ten hundred, you say one thousand and start over.

When you get to one thousand thousand, you say one million.

When your children get to one thousand million ( in the United States), they say one billion.

When your children's children's ...(this goes on for four hundred generations)...children's children get to one thousand billion, they say one trillion.

We can always add one and name the new number. Our children's...children can always add one and name a new number.

We call this system of numbers the NATURAL NUMBERS.

We reverse an addition by subtracting.

Can we always subtract and name a new number? Not if the number is one.

To solve this we have a number called zero which stands for no things.

We call this system of numbers ( with the zero) the WHOLE NUMBERS.

What if we subtract from zero? When this has happened in my bank account before, I changed from black to red ink or put paraentheses,like (204.30), around the number to indicate that the collection that was in my pile is now in someone else's pile. Another way to indicate this is to put a minus sign before the number, like -204.30.

We call numbers with a minus sign NEGATIVE NUMBERS, those without are the POSITIVE NUMBERS.

We call the system of numbers with positive numbers and negative numbers the INTEGERS.

A faster way to count numbers is to group everything in equal clumps first:

Then we count all the clumps and multiply by the size of the clump.

To multiply, you memorize some tables and activities or use a calculator.

If a machine can do all the clumping for you, then this method of counting really flies. For example, a machine puts 12 eggs in each carton. To find the number of eggs, count the cartons and multiply by 12.

You can always multiply and get a nameble new number in the integers.

Here's how to handle the minuses:

A positive number ( no minus siqn) multiplied by a positive number yields a positive number.

A negative number (with a minus sign) multiplied by a positive number yields a negative number.

A negative number multiplied by a negative number yields a positive number.

Note that by these rules a number squared ( multiplied by itself) is always positive.

You reverse an addition by subtracting, you reverse a multiplication by dividing.

You can't always divide and name the number in the system of integers.

To get to nameble numbers again, you need FRACTIONS.

Fractions are unique names for numbers which cannot be divided. If I divide nine by three the result is three ( or, in fraction form, three oneths).If I divide three by nine the result is three nineths. If I divide twenty-two by two hundred sixty-five, I get twenty-two two hundred sixty-fifths.

We also have a special way of naming fractions in decimals: The decimal point. In the number 123456.789 , each number going left is ten times larger than its nieghbor. To the right of the decimal point, the fractions start. .789 is seven tenths plus 8 hundreths, plus 9 thousanths or seven hundred eighty-nine thousanths.

Fractions can be expressed as a top and a bottom part ( numerator and denominator, to be technical).

There are two tricks which tame fractions. Converting them to decimals, and changing the numbers.

To convert to decimals, first multiply the top of the fraction by a ten enough times to make it larger ( maybe even a lot larger ) than the bottom.

Next, divide the bottom into the large top number normally.

Finally, divide by ten the same number of times you multiplied before.

This is simple since multiplying by ten means moving the decimal point to the right, and dividing moving to the left.

The second trick comes from noticing that you can always multiply or divide the top and the bottom by the same number and the fraction is unchanged.

Calculators make fractions a lot easier to deal with.

Numbers expressed as fractions are called RATIONAL NUMBERS.

Multiplication allows us to deal with big numbers easily, but a special form of multiplication makes even bigger numbers easy. This is called raising to a power. The word power is in there because people feel powerful when they do this.

Multiplication is repeated addition. A number is added to itself a number of times.

Raising to a power is repeated multiplication. A number is multiplied by itself a number of times.

The little number 6 in 10 6 is called the exponent and expressing the number in this form is called the exponential form

We can also divide sucessively:

Here is the short version:

1,000,000 106
100,000 105
10,000 104
1000 103
100 102
10 101
1 100
.1 10-1
.01 10-2
.001 10-3
.0001 10-4
.00001 10-5
.000001 10-6

Scientific Notation

It is obvious that the easy way to express big powers of ten is as 10x.It would be nice to use this ability to express any number.

We can do this by breaking the number into two parts,a power of one part and a power of ten part.

In this scheme, 1 is 1 times 100, 2 is 2 times 100,10 is 1 times 101,20 is 2 times 101.

Here are some examples:

3,260,000 3.26 times 106 3.26 X 1063.26E6
326,000 3.26 times 105 3.26 X 1053.26E5
32,600 3.26 times 104 3.26 X 1043.26E4
3260 3.26 times 103 3.26 X 1033.26E3
326 3.26 times 102 3.26 X 1023.26E2
32.6 3.26 times 101 3.26 X 1013.26E1
3.26 3.26 times 100 3.26 X 1003.26E0
.326 3.26 times 10-1 3.26 X 10-13.26E-1
.0326 3.26 times 10-2 3.26 X 10-23.26E-2
.00326 3.26 times 10-3 3.26 X 10-33.26E-3
.000326 3.26 times 10-4 3.26 X 10-43.26E-4
.0000326 3.26 times 10-5 3.26 X 10-53.26E-5
.00000326 3.26 times 10-6 3.26 X 10-63.26E-6


102, or ten squared, is 100. Ten is the square root of one hundred.

103, or ten cubed, is 1000. Ten is the cube root of one thousand.

The easiest way to express a root is as a fractional exponent: the square root of 100 is 1001/2.

Roots are sometimes difficult to evaluate, requiring a calculator or table. Some can never be perfectly calculated and must be rounded off.

Here are some common roots (rounded off where needed):

1000,0001/2 1000
10,0001/2 100
1001/2 10
101/2 3.162...
91/2 3
41/2 2
31/2 1.732...
21/2 1.414...
11/2 1

The three dots after some numbers indicate that the number continues on forever.


Evaluating formulas

To evaluate a formula, say f=3x where x = 5, substitute the number for the letter,f=3(5), and multiply (where a number is right beside a letter, that means to multiply) to get 15.

To evaluate a formula, say f=3x2 where x = 5, substitute the number for the letter with its exponent, f=3(52). Do the calculation inside the parenthesis first-- 52=25--then multiply--3 times 25 = 75.

,p.To evaluate something really complex, like f=10((3+5)x2) where x=5, use the following order: