2.1 - The Types of Numbers
Let's survey the types of numbers. We will look at natural and whole numbers,
integers, rational numbers,
irrational numbers, real numbers,
imaginary numbers and complex numbers.
The Natural and Whole Numbers
We start with the natural numbers. These are the numbers
1, 2, 3, … (The … symbol means that the sequence goes on forever.)
They are used for counting. If we include zero then we get the
whole numbers, 0, 1, 2, 3, ….
The natural and whole numbers are usually considered to be exact
(e.g. there are 4 tires on a car, 8 legs on a spider). But sometimes they are approximate
(e.g. there were 1000 people in the crowd). Here they are, shown on the number line:
Now for each of the numbers 1, 2, 3, … let’s create its opposite or negative and put it on the opposite
side of the number line, like this:
We say that 1 and −1 are opposites, 2 and −2 are opposites, etc. and we also say that
−1 is the opposite of 1 and that 1 is the opposite of −1.
Negative numbers are used to describe debts as opposed to assets,
temperatures below zero as opposed to temperatures above zero,
heights below sea level as opposed to heights above sea level, and so on.
The set of numbers …, −3, −2, −1, 0, 1, 2, 3, …
(the whole numbers and their opposites) is called the integers.
The Rational Numbers
Next are the rational numbers. A rational number is any number that can be expressed
as the quotient of two integers.
We can use any of these fraction notations
to express rational numbers:
a is called the numerator and b is called the denominator. The denominator cannot be zero.
The notation means that we break something into b equal pieces and we have a of those pieces.
For example if we break a pie into 4 pieces and take 1 piece then we have 1/4 of the pie:
Rational numbers can also be written in decimal notation instead of fraction notation. For example:
1/4 = 0.25
The decimal notation 0.25 means literally “25/100” and 25/100 and 1/4 are equivalent fractions.
Notice that some rational numbers have no exact decimal equivalent. For example 1/3
is approximately equal to 33/100 but not exactly:
1/3 ≈ 0.33
(The symbol ≈ means “is approximately equal to”.)
Rational numbers are usually considered to be exact. For this reason
the Algebra Coach program will not convert fractions to decimals when it is running in exact mode.
Notice that since 3/1 = 3 and −5/1 = −5, the rational numbers include all the integers.
The Irrational Numbers
The irrational numbers are those that cannot be expressed as a ratio of two integers.
Examples are as well as the square roots of many other numbers,
and special numbers like e and π. It turns out that there are as many irrational numbers
as rational. Irrational numbers have no exact decimal equivalents. To write any irrational number
in decimal notation would require an infinite number of decimal digits.
Thus these are only approximations:
≈ 1.732, e ≈ 2.718 and π ≈ 3.14,
For this reason the Algebra Coach program will not convert irrational numbers to decimals
when it is operating in exact mode.
The Real Numbers
The rational numbers and the irrational numbers together make up the real numbers. The real numbers are
said to be dense. They include every single number that is on the number line.
The number line is useful for understanding the order of numbers. Smaller numbers are farther to the left
and larger numbers are farther to the right.
We use the symbol < to mean “is less than” and
the symbol > to mean “is greater than”. Here are some examples of the use of these symbols:
Real numbers often result from making measurements and measurements are always approximate.
For example, with one piece of equipment the length of an object might be measured to be 5.28
(in some units). This doesn’t mean the length is exactly 5.28. It just means that it is
closer to 5.28 than it is to 5.27 or 5.29.
With a more accurate (and usually much more expensive piece of equipment) the length
might be measured to be 5.283, which just means it is closer to 5.283 than it is to
5.282 or 5.284, and so on. If an expression contains an approximate number then that
whole expression is also approximate.
- 5 < 8 because 5 is to the left of 8.
- −5 < 2 because −5 is to the left of 2.
In fact any negative number is less than any positive number.
- 5 < 8 and 8 > 5 are two ways of stating the same fact.
- 3.14 < π < 3.15 I call this a “less than sandwich”.
It means that 3.14 < π and also π < 3.15. In other words the number
π is somewhere between 3.14 and 3.15.
Click here for more information on accuracy and significant figures.
The Imaginary Numbers and the Complex Numbers
If real numbers include every single number on the number line, then what other numbers could there be?
To answer that question, consider how we built up the number system so far:
We now want to take the square root of negative numbers but we will need a new type of number
to describe the result. We define the square root of a negative number
to be an imaginary number. How much further can this process of
creating new types of numbers go? The answer is one more step.
We can add a real number to an imaginary number. The result is called a complex number.
This is the end of the line because it turns out that every possible operation with every possible complex number
results only in other complex numbers. We say that complex numbers make the numbers complete.
- We started with the whole numbers (numbers like 3)
- We wanted opposites for these numbers so we created the integers (numbers like −3)
- We wanted to divide these numbers but needed the rationals to describe
some of the results (numbers like 3/4)
- We wanted to take the square root of these numbers but needed the irrationals to describe
some of the results (e.g. )
Where do imaginary and complex numbers go on the number line? The answer is they don’t.
This picture shows the complex plane. It contains the number line
(which is now called the real axis) and a new axis called the
imaginary axis, perpendicular to it. Real numbers lie on the real axis, imaginary numbers
lie on the imaginary axis and complex numbers generally lie off the real axis,
either above it or below it.
The Algebra Coach program can run in real mode or complex mode. In real mode it will not
carry out any operation that leads to a non-real number (such as taking the square root
of a negative number).
Click here for more information on complex numbers.
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