A real number is a number that can be found on the real number line. At first, this may seem a silly way to define a real number. But, when you look into it, you can see why this makes sense.
First off, any whole number or integer is a real number. Usually, we use whole number for our tick-marks on the real number line as in figure 1:
Figure 1: Number line from -3 to 3. |
Obviously, we can find any integer on the number line simply by extending the number line far enough. How about rational numbers? Take 1/2 for instance. It can be found halfway between zero and one as shown in figure 2.
Figure 2: Number line with the number 1/2 marked. |
Now look at irrational numbers. Take for instance. The value of is about 1.41. So it is on the number line. See figure 3 for this example.
Figure 3: Number line with marked. |
And how about the number for π? Yes! It's right there between 3 and 4 as shown in figure 4.
Figure 4: Number line with π marked. |
So what numbers are not on the real number line? The answer is complex numbers. The diagram below is the complex number plane. Real numbers can be found on the real number line which is the horizontal axis. Any other complex number is found on the complex plane, but not on the real number line. There are three numbers labeled on the plane in figure 5. The first is A( 2, 0 ). Since this is on the real number line it is a real number. The second is B( 0, √2 ). This is the number . It is not on the real number line. The third is C(1,1). This is the number 1 + i. It, too, is not on the real number line. So and 1 + i are not real numbers.
Figure 5: The complex plane. |
The set of real numbers can be divided into several subsets:
Subset name | Description | Examples |
---|---|---|
Natural numbers | A natural number is a positive whole number: ℕ = {1, 2, 3, 4, …}. | 1, 17, 2, 325, 985 |
Integers | An integer is a positive or negative whole number: ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}. | -3, 19, 215 |
Rational numbers | Any number that can be expressed as the ratio of two integers. {x = a / b | a, b ∈ ℤ}. | 3 = 1/3, -6/5, 22/7 |
Irrational numbers | Any real number that is not a rational number. | π, |
Algebraic number | Any number that is a root of a single variable, nonzero polynomial with rational coefficients. | 5, -3/2, , . |
Transcendental number | Any real number that is not an algebraic number. | π, e. |
Table 1: Types of real numbers. |
Property | Description |
---|---|
Associativity | The set of real numbers is associative with respect to addition, subtraction, multiplication and division. Example: a + (b + c) = (a + b) + c. |
Commutativity | The set of real numbers is commutative with respect to addition and multiplication. The set of real numbers is not commutative with respect to subtraction or division. Example: a + b = b + a. |
Additive identity | The additive identity for real numbers is 0. Example: a + 0 = 0 + a = a. |
Multiplicative identity | The multiplicative identity for real numbers is 1. Example: a · 1 = 1 · a = a. |
Closure | The set of real numbers is closed with respect to addition, subtraction, multiplication, and division. Example: if a and b are real numbers then a + b is also a real number. |
Discrete | The set of real numbers is a continuous (not discrete) set. |
Cardinality | The cardinality of the set of real numbers is ℵ_{0}. |
Trichotomy | The Trichotomy Property of Real Numbers states that for any two real numbers a and b, exactly one of the following is true: a < b, a = b or a > b. |
Table 2: Properties of the real numbers. |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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