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Real Numbers

Define and identify numbers that are real numbers.

Learning Objectives

After completing this section, you should be able to:

  1. Define and identify numbers that are real numbers.
  2. Identify subsets of the real numbers.
  3. Recognize properties of real numbers.

Defining and Identifying Real Numbers

Real numbers are the rational and irrational numbers combined. The real numbers represent the collection of all physical distances that exist, along with 0 and the negatives of those physical distances. For example, if you take a measure of three units, and divide that distance into eight (8) equal lengths, the distance you have formed is \(\frac{3}{8}\) units. Also, if you draw a right triangle (a triangle with one angle equal to 90 degrees) with one side length of 1, and the other side length of 3, the long side of the triangle will have length \(\sqrt{10}\) units, as shown in .

Of course, if we name something the real numbers, there must be numbers that aren't real. Otherwise, they’d just be called the numbers. One such not real number, one that cannot be a length, is \(\sqrt{-1}\). It is part of a collection of numbers called the complex numbers, it is denoted with the letter \(i\). As an extension, the square root of any negative number is not a real number, but instead a complex number.

To determine if a number is real, check to see if there are any negatives under a square root or any \(i's\). If there are any present, the number is not real.

Identifying Real Numbers

Try it.

Determine if each of the following are real numbers:

  1. \(\frac{4\sqrt{3}}{7}\)
  2. \(13.3381\)
  3. \(17\sqrt{-8}\)
Solution
  1. \(\frac{4\sqrt{3}}{7}\) is a real number, as there are no negatives under the square roots, nor is there any factor of \(i\).
  2. \(13.3381\) is a rational number, and so it is a real number.
  3. \(17\sqrt{-8}\) is not a real number, as there is a negative number under the square root.

Identifying Subsets of Real Numbers

The real numbers were built out of pieces, including integers, rational numbers, and irrational numbers. As such, the real numbers have named subsets, as shown in the table below.

Set NameSet SymbolSet Description
Natural Numbers\(\text{ℕ}\)The counting numbers
Whole NumbersThe counting numbers and 0
Integers\(\text{ℤ}\)The natural numbers, their negatives, and 0
Rational Numbers\(\text{ℚ}\)Fractions of integers
Irrational Numbers\(\text{ℙ}\)Numbers that cannot be written as a fraction of integers
Real Numbers\(\text{ℝ}\)The union of the rational and irrational numbers, all possible physical lengths, and their negatives

When we categorize numbers using these sets, we use the smallest set that they belong to. For instance, −7 is an integer, and a rational number, and a real number. The smallest set to which −7 belongs is integer, so we’d say it belongs to the integers.

We can also represent the relationships between the different sets of real numbers using set notation. All natural numbers are integers, but there are integers that are not natural numbers, so \(\text{ℕ}⊂\text{ℤ}\). Similarly, every integer is a rational number, but there are rational numbers that are not integers, so \(\text{ℤ}⊂\text{ℚ}\). The same is true of the rational numbers and the real numbers, so \(\text{ℚ}⊂\text{ℝ}\).

There is no agreed-upon symbol for the irrational numbers. If we represent the irrationals as the set \(A\), we should note that the following are true: \(\text{ℚ}\cup A=\text{ℝ}\) and \(\text{ℚ}\cap A=\emptyset\). Recall that this means the irrationals are the complement of the rational numbers in the universal set of real numbers.

Categorizing Numbers

Try it.

Identify all subsets of the real numbers to which the following real numbers belong:

  1. \(14\)
  2. \(-14.223\)
  3. \(\sqrt{17}\)
Solution
  1. \(14\) is a natural number, integer, and rational number.
  2. \(-14.223\) is a rational number.
  3. \(\sqrt{17}\) is an irrational number.
Categorizing Numbers within a Venn Diagram

Try it.

Place the following numbers correctly in the Venn diagram ().

\[\ -4\sqrt{2}\ -10\ \frac{37}{150}\ 41\ \frac{1}{20}\ 4\pi\]
Solution

Since \(-4\sqrt{2}\) is irrational, it belongs in the real numbers, but outside the rational numbers ().

Since −10 is an integer, it belongs in the integers but outside the natural numbers ().

Since \[\frac{37}{150}\] is a rational number, it belongs in the rational numbers but not in the integers ().

Since 41 is a natural number, it belongs in the natural numbers circle ().

Since \(\frac{1}{20}\) is a rational number, it belongs in the rational numbers but not in the integers ().

Since \(4\pi\) is irrational, it belongs in the real numbers, but outside the rational numbers ().

Condensed — the full section is in OpenStax Contemporary Mathematics.

Recognizing Properties of Real Numbers

The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers. Knowing these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with these properties is helpful in all settings where numbers are used and manipulated. For example, when multiplying \(4\times 13\times 25\), you could multiply the 4 and 25 first. If you know that product is 100, it makes the multiplication easier.

The table below is a partial list of properties of real numbers.

PropertyExampleIn Words
Distributive property
\(a\times (b+c)=a\times b+a\times c\)		\(5\times (3+4)=5\times 3+5\times 4\)Multiplication distributes across addition
Commutative property of addition
\(a+b=b+a\)		\(3+7=7+3\) Numbers can be added in any order
Commutative property of multiplication
\(a\times b=b\times a\)		\(10\times 4=4\times 10\)Numbers can be multiplied in any order
Associative property of addition
\(a+(b+c)=(a+b)+c\)		\(4+(3+8)=(4+3)+8\)Doesn't matter which pair of numbers is added first
Associative property of multiplication
\(a\times (b\times c)=(a\times b)\times c\)		\(2\times (5\times 7)=(2\times 5)\times 7\)Doesn't matter which pair of numbers is multiplied first
Additive identity property
\(a+0=a\)		\(17+0=17\)Any number plus 0 is the number
Multiplicative identity property
\(a\times 1=a\)		\(21\times 1=21\)Any number times one is the number
Additive inverse property
\(a+(-a)=0\)		\(14+(-14)=0\)Every number plus its negative is 0
Multiplicative inverse property
\(a\times (\frac{1}{a})=1\), provided \(a\ne 0\)		\(3\times (\frac{1}{3})=1\)Every non-zero number times its reciprocal is 1

The names of the properties are suggestive. The commutative properties, for example, suggest commuting, or moving. Associative properties suggest which items are associated with others, or if order matters in the computation. The distributive property addresses how a number is distributed across parentheses.

Identifying Properties of Real Numbers

Try it.

In each of the following, identify which property of the real numbers is being applied.

  1. \(4+(8+13)=(4+8)+13\)
  2. \(34\times (\frac{1}{34})=1\)
  3. \(14+27=27+14\)
Solution
  1. Here, the pair of numbers that is added first is switched. This is the associative property of addition.
  2. Here, a number is multiplied by its reciprocal, resulting in 1. This is the multiplicative inverse property.
  3. Here, the order in which numbers are added is switched. This is the commutative property of addition.

Using these properties to perform arithmetic quickly relies on spotting easy numbers to work with. Look for numbers that add to a multiple of 10, or multiply to a multiple of 10 or 100.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

  • Real numbers is the collection of all rational and irrational numbers. Conceptually, it is the collection of all values that can be represented on a number line, or, as a length along with sign.
  • The subsets of the real numbers include the natural numbers, integers, rational numbers and irrational numbers. The natural numbers are a subset of the integers, which is a subset of the rational numbers. The rational and irrational numbers are disjoint sets.
  • The real numbers, due to order of operation rules and that performing arithmetic operations on real number always results in a real number, have arithmetic properties that apply in all cases. There include the distributive property, the commutative property, and the associative property. Also, every real number has an additive inverse and, except for zero (0), have a multiplicative inverse.

Formulas

  • \(a\times (b+c)=a\times b+a\times c\)
  • \(a+b=b+a\)
  • \(a\times b=b\times a\)
  • \(a+(b+c)=(a+b)+c\)
  • \(a\times (b\times c)=(a\times b)\times c\)
  • \(a+0=a\)
  • \(a\times 1=a\)
  • \(a+(-a)=0\)
  • \(a\times (\frac{1}{a})=1\)

Videos

  • Arthur Benjamin TED talk, Faster than a Calculator
  • Identifying Sets of Real Numbers
  • Properties of the Real Numbers #1
  • Properties of the Real Numbers #2
  • Properties of the Real Numbers #3

Practice (5)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Determine if each of the following are real numbers:

    1. \(\frac{4\sqrt{3}}{7}\)
    2. \(13.3381\)
    3. \(17\sqrt{-8}\)
    Zbulo përgjigjen
    1. \(\frac{4\sqrt{3}}{7}\) is a real number, as there are no negatives under the square roots, nor is there any factor of \(i\).
    2. \(13.3381\) is a rational number, and so it is a real number.
    3. \(17\sqrt{-8}\) is not a real number, as there is a negative number under the square root.

  2. Identify all subsets of the real numbers to which the following real numbers belong:

    1. \(14\)
    2. \(-14.223\)
    3. \(\sqrt{17}\)
    Zbulo përgjigjen
    1. \(14\) is a natural number, integer, and rational number.
    2. \(-14.223\) is a rational number.
    3. \(\sqrt{17}\) is an irrational number.

  3. Place the following numbers correctly in the Venn diagram ().

    \[\ -4\sqrt{2}\ -10\ \frac{37}{150}\ 41\ \frac{1}{20}\ 4\pi\]
    Zbulo përgjigjen

    Since \(-4\sqrt{2}\) is irrational, it belongs in the real numbers, but outside the rational numbers ().

    Since −10 is an integer, it belongs in the integers but outside the natural numbers ().

    Since \[\frac{37}{150}\] is a rational number, it belongs in the rational numbers but not in the integers ().

    Since 41 is a natural number, it belongs in the natural numbers circle ().

    Since \(\frac{1}{20}\) is a rational number, it belongs in the rational numbers but not in the integers ().

    Since \(4\pi\) is irrational, it belongs in the real numbers, but outside the rational numbers ().

  4. In each of the following, identify which property of the real numbers is being applied.

    1. \(4+(8+13)=(4+8)+13\)
    2. \(34\times (\frac{1}{34})=1\)
    3. \(14+27=27+14\)
    Zbulo përgjigjen
    1. Here, the pair of numbers that is added first is switched. This is the associative property of addition.
    2. Here, a number is multiplied by its reciprocal, resulting in 1. This is the multiplicative inverse property.
    3. Here, the order in which numbers are added is switched. This is the commutative property of addition.

  5. Use properties of the real numbers and mental math to calculate the following:

    1. \(2\times 13\times 50\)
    2. \(13+84+27\)
    3. \(9\times 16\times 11\)
    Zbulo përgjigjen
    1. Notice that \(2\times 50=100\), so that becomes the multiplication to do first. Use the commutative property of multiplication to change the order of the numbers being multiplied.
      \[2\times 13\times 50=2\times 50\times 13=100\times 13=1,300\]
    2. Notice that \(13+27=40\), so that becomes the addition to do first. Use the commutative property of addition to change the order in which the numbers are added.
      \[13+84+27=13+27+84=40+84=124\]
    3. Notice that \(9\times 11=99\). Using that, the problem can be changed to \(99\times 16\). That, however, doesn't look easy at all. But \(99=(100-1)\). Using the distributive property, we rewrite and expand this as \(99\times 16=(100-1)\times 16=100\times 16-1\times 16=1,600-16\). The last step is subtraction, so the final answer is 1,584. So, multiplying by 99 is the same as multiplying by 100, and then subtracting the other number once.

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
i
imaginary unit
i² = −1.
\neq
not equal
The two sides are different.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
a \bmod n
remainder
What is left after dividing a by n.

How to: Real Numbers

  1. Define and identify numbers that are real numbers.
  2. Identify subsets of the real numbers.
  3. Recognize properties of real numbers.
  5. Here, the pair of numbers that is added first is switched. This is the associative property of addition.
  6. Here, a number is multiplied by its reciprocal, resulting in 1. This is the multiplicative inverse property.
  7. Here, the order in which numbers are added is switched. This is the commutative property of addition.
  8. Notice that

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller–Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

Provo timen.

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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