time

What is time?

Everyone knows what time is. We can practically feel it ticking away, marching on in the same direction with horrifying regularity. Time has enslaved the Western world and become our most precious commodity. Turn it over to the physicists however, and it begins to morph, twist and even crumble away. So what is time exactly?

To many people throughout history time would have been synonymous with the rhythms of nature; the passing of the seasons and the cycles of the celestial bodies. If this idea seems naive today, it's not only because modern clocks are infinitely more accurate time keepers than the celestial bodies ever were. It's also because we've come to think of time as something universal, something that would keep marching on even if all clocks, celestial or man-made, were to stop. The notion of an absolute time, one that's measurable and the same for all observers, was expressed most succinctly by Newton: "absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external.

Money

Spending It, Saving It,

and Growing It

The purpose of this chapter is to make you

rich. If that fails, then at the very least I

hope this chapter will give you some

understanding of the basic principles of

money management, the tools to make

sound financial decisions, and ideas on

how to avoid wasting your hard-earned

money. Not surprisingly, there is a good

amount of mathematics to learn along the

way, but when it comes to financial matters,

mathematics alone is not sufficient. As a

complement to your understanding of the numbers, to

be a good money manager you will need a dose of

common sense, a dash of skepticism, and plenty of selfdiscipline.

fraction

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.

A common or vulgar fraction, such as ,  ,and  consists of an integer numerator and a non-zero integer denominator—the numerator representing a number of equal parts and the denominator indicating how many of those parts make up a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts equal a whole. The picture to the right illustrates  or 3/4 of a cake.

Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and 10⁻² respectively, all of which are equivalent to 1/100). An integer such as the number 7 can be thought of as having an implied denominator of one: 7 equals 7/1.

Other uses for fractions are to represent ratios and to represent division.^[1] Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four).

In mathematics the set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational).

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.

A common or vulgar fraction, such as ,  ,and  consists of an integer numerator and a non-zero integer denominator—the numerator representing a number of equal parts and the denominator indicating how many of those parts make up a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts equal a whole. The picture to the right illustrates  or 3/4 of a cake.

Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and 10⁻² respectively, all of which are equivalent to 1/100). An integer such as the number 7 can be thought of as having an implied denominator of one: 7 equals 7/1.

Other uses for fractions are to represent ratios and to represent division.^[1] Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four).

In mathematics the set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational).

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.

A common or vulgar fraction, such as ,  ,and  consists of an integer numerator and a non-zero integer denominator—the numerator representing a number of equal parts and the denominator indicating how many of those parts make up a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts equal a whole. The picture to the right illustrates  or 3/4 of a cake.

Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and 10⁻² respectively, all of which are equivalent to 1/100). An integer such as the number 7 can be thought of as having an implied denominator of one: 7 equals 7/1.

Other uses for fractions are to represent ratios and to represent division.^[1] Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four).

In mathematics the set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational).

Whole Numbers and Integers

Whole Numbers

Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, … (and so on)

No Fractions!

Counting Numbers

Counting Numbers are Whole Numbers, but without the zero. Because you can't "count" zero. So they are 1, 2, 3, 4, 5, … (and so on).

Natural Numbers

"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.

Integers

Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed!

So, integers can be negative {-1, -2,-3, -4, -5, … }, positive {1, 2, 3, 4, 5, … }, or zero {0}

We can put that all together like this:

Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }

Confusing

Just to be confusing, some people say that whole numbers can also be negative, so that would make them exactly the same as integers. And sometimes people say that zero is NOT a whole number. So there you go, not everyone agrees on a simple thing!

My Standard

I must admit that sometimes I say "negative whole number", but usually I stick to:

Numbers	Name
{ 0, 1, 2, 3, 4, 5, … }	Whole Numbers
{ 1, 2, 3, 4, 5, … }	Counting Numbers
{ ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … }	Integers

But nobody disagrees on the definition of an integer, so when in doubt say "integer", and if you only want positive integers, say "positive integers". It is not only accurate, it makes you sound intelligent. Like this (note: zero is neither positive nor negative):

• Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }
• Negative Integers = { ..., -5, -4, -3, -2, -1 } 
• Positive Integers = { 1, 2, 3, 4, 5, ... }
• Non-Negative Integers = { 0, 1, 2, 3, 4, 5, ... } (includes zero, see?)