Number Types
Purplemath
What are the different types of numbers?
The different types of numbers are the counting numbers, the natural or whole numbers, the integers, the rationals and irrationals, the real numbers, the imaginary numbers, and the complex numbers.
What are the counting numbers?
The counting numbers are the numbers that you first learned as a little child, being the numbers that you used to count things; namely, the counting numbers are:
1, 2, 3, 4, 5, 6, …
The "…", or "ellipsis", means "and so on and so forth, forever".
What are the natural or whole numbers?
The natural or whole numbers are the counting numbers, together with zero; in other words:
0, 1, 2, 3, 4, 5, 6, …
Sometimes the counting numbers are called natural numbers. If your textbook or instructor uses slightly different terminology or definitions, make sure that you're clear on which answers will get you full credit; and be aware that a different textbook or instructor might use different terminology or definitions.
Mathematicians have a symbol that they use for the set of all natural numbers; namely, ℕ.
Historical note: The number zero came to Europe from India via north-African scholars. Zero, also called "the cipher", was originally viewed by European authorities with suspicion; they said that it, along with the other Arabic numerals, were conducive of fraud, and the use of zero was sometimes banned outright.
What are the integers?
The integers are zero, the counting numbers, and the negatives of the counting numbers; namely, the integers are:
…, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, …
Mathematicians have a symbol that they use for the set of all integers; namely, ℤ. The double-struck Z stands for "Zahlen", being the German word for "numbers".
What are the rational numbers?
The rational numbers are the fractions, where the numerators (top numbers) and denominators (bottom numbers) are integers. While counting numbers, natural/whole numbers, and integers all have the property of being one unit apart (so there is always a "next" number of these types), fractions can be any rational distance apart. For any given fraction, there is no "next" fraction that immediately follows it.
Mathematicians have a symbol that they use for the set of all rationals; namely, ℚ. The double-struck Q stands for "quotient", which is a helpful reminder that all fractions are divisions.
What are the irrational numbers?
The irrational numbers are the numbers which cannot be expressed as fractions. For instance, √2 cannot be expressed as a fraction. For any given irrational, there is no "next" number that immediately follows it.
There is no symbol for the set of irrationals.
What is the difference between rationals and irrationals when they're written as decimals?
Fractions (also called rational numbers) can be written as terminating decimals (that is, decimal numbers that ends after some finite number of decimal places) or as repeating decimals (that is, as decimal numbers that have some block of decimal places that repeates forever). Irrational numbers, on the other hand, have decimal forms that never end and never have a continually-repeating block of decimal places. Rationals are nice and neat; irrationals are very much not nice or neat, at least in decimal form.
For example, the fraction can be written in decimal form as 0.5, and can be written as 0.76. These two fractions are terminating decimals.
The fraction can be written in decimal form as 0.333333…, and can be written as 0.538461538461…. These two fractions are repeating decimals. In the first case, the repeated block is just 3; in the second case, the repeated block is 538461.
On the other hand, we have loads of other numbers whose decimal forms are non-repeating, non-terminating decimals; these number are non-rational (that is, they cannot be written as ratios of two integers); this is why they are called the "irrationals". Examples of irrationals in decimal form would be = 1.41421356…, or the number π = 3.14159.... (Yes, you may have used a fraction as an approximation of π, but that fraction did not equal π.)
The rationals and the irrationals are two totally separate number types; there is no overlap.
What are the real numbers?
The real numbers are the integers, the rationals, and the irrationals. As is the case with rational numbers, there is, for any given real number, no "next" real number that immediately follows it.
Mathematicians have a symbol that they use for the set of real numbers; namely, ℝ.
(Once you know the symbol for the real numbers, they can use symbols to indicate the set of irrational numbers. The irrationals are the reals, less the rationals; symbolically, the irrationals are ℝ ∖ ℚ.)
How are these number types related to each other?
Note that each new type of number is either the opposite of the previous type, or else it contains the previous type within it. The natural numbers are just the counting numbers with zero thrown in. The integers are just the natural numbers with the negatives of the counting numbers thrown in. And the fractions are just the integers with all their divisions thrown in. (Remember that you can turn any integer into a fraction by putting it over the number 1. For example, the integer 4 is also the fraction .)
The rationals and the irrationals are sort of opposite types of numbers: the rationals *can* be written as fractions and the irrationals can *not* be written as fractions. Putting these two sets of numbers together, we get the set of all real numbers.
Are there any numbers that aren't real numbers?
There are numbers that are not real numbers: they are called the imaginary numbers. (No, I'm not making that up.) They are the numbers that include the number i, which is defined as . Imaginaries allow you to take the square roots of negative numbers, and they are crucial to electrical engineering, physics, and other areas of science.
You can then add real numbers and imaginary numbers, and get the complex numbers, indicated by ℂ. Complex numbers can be written in different ways; one of them is as a + bi, where a and b are real numbers. If b = 0, then you've got a regular real number. If a = 0, then you've got a strictly imaginary number. The set of complex numbers is the reals, the imaginaries, and their sums.
If you've never seen numbers with i in them, then every number you've ever seen is a real number.
The most common question I hear regarding number types is something along the lines of "Is a real number irrational, or is an irrational number real, or neither... or both?" Unless you know about complexes, everything you've ever done has used real numbers. Unless the number has an "i" in it, it's a real number. And counting numbers, natural/whole numbers, rationals, and irrationals are all inside the set of real numbers.
Here are some typical number-type questions (assuming that you haven't yet learned about imaginaries and complexes):
- True or False: An integer is also a rational number.
Since any integer can be formatted as a fraction by putting it over 1, then this statement is true.
- True or False: A rational number is also an integer.
Not necessarily; the integer 4 is also the rational number but, for instance, the rational number is not also a integer. So this statement is false.
- True or False: A number is either a rational number or an irrational number, but not both.
True! In decimal form, a number is either non-terminating and non-repeating (so it's an irrational) or else it's not (so it's a rational); there is *no* overlap between these two number types!
Classify according to number type; some numbers may be of more than one type.
- 0.45
This is a terminating decimal, so it can be written as a fraction: . Since this fraction does not reduce to a whole number, then it's not an integer or a natural. And everything is a real, so the answer is: rational, real
- 3.14159265358979323846264338327950288419716939937510...
You probably recognize this as being π, though this may be more decimal places than you customarily use. The point, however, is that the decimal does not repeat and does not end, so π is an irrational. And everything (that you know about so far) is a real, so the answer is: irrational, real
- 3.14159
Don't let this fool you! Yes, you often use something like this as an approximation of π, but it isn't π! This is a rounded decimal approximation, and, since this approximation terminates, this is actually a rational, unlike π itself, which is irrational! The answer is: rational, real
- 10
Obviously, this is a counting number. That means it is also a whole number and an integer. Depending on the text and teacher (there is some inconsistency), this may also be counted as a rational, which technically-speaking it is. And of course it's also a real. The answer is: natural, whole, integer, rational (possibly), real
This is a fraction, so it's a rational. It's also a real, so the answer is: rational, real
This can also be written as , which is the same as the previous problem. The answer is: rational, real
Your first impulse may be to say that this is irrational, because it's a square root, but notice that this square root simplifies: , which is just an integer. The answer is: integer, rational, real
This number is stated a fraction, but notice that it reduces to −3, so this may also count as an integer. The answer is: integer (possibly), rational, real
Except for the section in your book where you have to classify numbers according to type, you really won't need to be terribly familiar with this hierarchy. It's more important to know what the terms mean when you hear them. For instance, if your teacher talks about "integers", you should know that the term refers to the counting numbers, their negatives, and zero.
(If you're wondering, yes, there are other classifications of numbers. Many other classifications.)
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