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== 3.6#h(0.6em)Real Numbers


  #figure(figph[An illustration shows a man thinking about the equation 4 times 31 times 25 equals 4 times 25 times 31 equals 100 times 31 equals 3,100.], alt: "An illustration shows a man thinking about the equation 4 times 31 times 25 equals 4 times 25 times 31 equals 100 times 31 equals 3,100.", caption: [Quick mental math involves using the known properties of real numbers.])


  
    
=== Learning Objectives


    After completing this section, you should be able to:


    + Define and identify numbers that are real numbers.
+ Identify subsets of the real numbers.
+ Recognize properties of real numbers.


  
  Have you ever been impressed by the speed at which someone can do math in their head? Most of us at one time or another have witnessed a person speed through mental math, an impressive feat that often bests calculators. One such person is Neelkantha Bhanu Prakash. As of September 20, 2020, he is considered the world’s fastest human calculator. He currently holds four world records. How does someone do that, though? Have they memorized lots of arithmetic facts? Are they simply brilliant?


  The answer isn't simple so much as it is about knowledge. Real numbers behave in some very regular ways, following rules that can be learned. In this section, those rules are explored.


  Watch the video of Arthur Benjamin’s TED Talk to learn about another mathematician with remarkable mental abilities.


  #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[
#link("https://openstax.org/r/Arthur_Benjamin_TED_talk,_Faster_than_a_Calculator")[Arthur Benjamin TED Talk, Faster Than a Calculator]
]


  
    
=== Defining and Identifying Real Numbers


    #strong[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 #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] units. Also, if you draw a #strong[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 #math.equation(block: false, alt: "the square root of 10")[$sqrt(10)$] units, as shown.


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    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 #math.equation(block: false, alt: "the square root of minus 1")[$sqrt(− 1)$]. It is part of a collection of numbers called the complex numbers, it is denoted with the letter #math.equation(block: false, alt: "i")[$i$]. 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 #math.equation(block: false, alt: "i ' s")[$i ' upright(s)$]. If there are any present, the number is not real.


    #examplebox("Example 1")[Identifying Real Numbers][
Determine if each of the following are real numbers:


        + #math.equation(block: false, alt: "the fraction 4 the square root of 3 over 7")[$frac(4 sqrt(3), 7)$]
+ #math.equation(block: false, alt: "13.3381")[$13.3381$]
+ #math.equation(block: false, alt: "17 the square root of minus 8")[$17 sqrt(− 8)$]


      
      #solutionbox[
+ #math.equation(block: false, alt: "the fraction 4 the square root of 3 over 7")[$frac(4 sqrt(3), 7)$] is a real number, as there are no negatives under the square roots, nor is there any factor of #math.equation(block: false, alt: "i")[$i$].
+ #math.equation(block: false, alt: "13.3381")[$13.3381$] is a rational number, and so it is a real number.
+ #math.equation(block: false, alt: "17 the square root of minus 8")[$17 sqrt(− 8)$] 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.


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  [Natural Numbers], [#math.equation(block: false, alt: "ℕ")[$"ℕ"$]], [The counting numbers],
  [Whole Numbers], [], [The counting numbers and 0],
  [Integers], [#math.equation(block: false, alt: "ℤ")[$"ℤ"$]], [The natural numbers, their negatives, and 0],
  [Rational Numbers], [#math.equation(block: false, alt: "ℚ")[$"ℚ"$]], [Fractions of integers],
  [Irrational Numbers], [#math.equation(block: false, alt: "ℙ")[$"ℙ"$]], [Numbers that cannot be written as a fraction of integers],
  [Real Numbers], [#math.equation(block: false, alt: "ℝ")[$"ℝ"$]], [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 #math.equation(block: false, alt: "ℕ ⊂ ℤ")[$"ℕ" ⊂ "ℤ"$]. Similarly, every integer is a rational number, but there are rational numbers that are not integers, so #math.equation(block: false, alt: "ℤ ⊂ ℚ")[$"ℤ" ⊂ "ℚ"$]. The same is true of the rational numbers and the real numbers, so #math.equation(block: false, alt: "ℚ ⊂ ℝ")[$"ℚ" ⊂ "ℝ"$].


    There is no agreed-upon symbol for the irrational numbers. If we represent the irrationals as the set #math.equation(block: false, alt: "A")[$A$], we should note that the following are true: #math.equation(block: false, alt: "ℚ union A equals ℝ")[$"ℚ" ∪ A = "ℝ"$] and #math.equation(block: false, alt: "ℚ intersection A equals the empty set")[$"ℚ" ∩ A = ∅$]. Recall that this means the irrationals are the complement of the rational numbers in the universal set of real numbers.


    #examplebox("Example 2")[Categorizing Numbers][
Identify all subsets of the real numbers to which the following real numbers belong:


        + #math.equation(block: false, alt: "14")[$14$]
+ #math.equation(block: false, alt: "minus 14.223")[$− 14.223$]
+ #math.equation(block: false, alt: "the square root of 17")[$sqrt(17)$]


      
      #solutionbox[
+ #math.equation(block: false, alt: "14")[$14$] is a natural number, integer, and rational number.
+ #math.equation(block: false, alt: "minus 14.223")[$− 14.223$] is a rational number.
+ #math.equation(block: false, alt: "the square root of 17")[$sqrt(17)$] is an irrational number.
]
]


    
    #examplebox("Example 3")[Categorizing Numbers within a Venn Diagram][
Place the following numbers correctly in the Venn diagram .


        #math.equation(block: true, alt: "minus 4 the square root of 2 minus 10 the fraction 37 over 150 41 the fraction 1 over 20 4 π")[$#h(1em) − 4 sqrt(2) #h(1em) − 10 #h(1em) frac(37, 150) #h(1em) 41 #h(1em) frac(1, 20) #h(1em) 4 π$]
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      #solutionbox[
Since #math.equation(block: false, alt: "minus 4 the square root of 2")[$− 4 sqrt(2)$] is irrational, it belongs in the real numbers, but outside the rational numbers .


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


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        Since #math.equation(block: false, alt: "the fraction 37 over 150")[$frac(37, 150)$] is a rational number, it belongs in the rational numbers but not in the integers .


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


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        Since #math.equation(block: false, alt: "the fraction 1 over 20")[$frac(1, 20)$] is a rational number, it belongs in the rational numbers but not in the integers .


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)


        Since #math.equation(block: false, alt: "4 π")[$4 π$] is irrational, it belongs in the real numbers, but outside the rational numbers .


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]
]


    
    #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[
#link("https://openstax.org/r/Sets_of_Real_Numbers")[Identifying Sets of Real Numbers]
]


  
  
    
=== Recognizing Properties of Real Numbers


    The real numbers behave in very regular ways. These behaviors are called the #strong[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 #math.equation(block: false, alt: "4 times 13 times 25")[$4 × 13 × 25$], 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.


    #figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Property], [Example], [In Words]),
  [Distributive property  #linebreak() #math.equation(block: false, alt: "a times open parenthesis b plus c close parenthesis equals a times b plus a times c")[$a × ( b + c ) = a × b + a × c$]], [#math.equation(block: false, alt: "5 times open parenthesis 3 plus 4 close parenthesis equals 5 times 3 plus 5 times 4")[$5 × ( 3 + 4 ) = 5 × 3 + 5 × 4$]], [Multiplication distributes across addition],
  [Commutative property of addition  #linebreak() #math.equation(block: false, alt: "a plus b equals b plus a")[$a + b = b + a$]], [#math.equation(block: false, alt: "3 plus 7 equals 7 plus 3")[$3 + 7 = 7 + 3$]], [Numbers can be added in any order],
  [Commutative property of multiplication  #linebreak() #math.equation(block: false, alt: "a times b equals b times a")[$a × b = b × a$]], [#math.equation(block: false, alt: "10 times 4 equals 4 times 10")[$10 × 4 = 4 × 10$]], [Numbers can be multiplied in any order],
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  [Associative property of multiplication  #linebreak() #math.equation(block: false, alt: "a times open parenthesis b times c close parenthesis equals open parenthesis a times b close parenthesis times c")[$a × ( b × c ) = ( a × b ) × c$]], [#math.equation(block: false, alt: "2 times open parenthesis 5 times 7 close parenthesis equals open parenthesis 2 times 5 close parenthesis times 7")[$2 × ( 5 × 7 ) = ( 2 × 5 ) × 7$]], [Doesn't matter which pair of numbers is multiplied first],
  [Additive identity property  #linebreak() #math.equation(block: false, alt: "a plus 0 equals a")[$a + 0 = a$]], [#math.equation(block: false, alt: "17 plus 0 equals 17")[$17 + 0 = 17$]], [Any number plus 0 is the number],
  [Multiplicative identity property  #linebreak() #math.equation(block: false, alt: "a times 1 equals a")[$a × 1 = a$]], [#math.equation(block: false, alt: "21 times 1 equals 21")[$21 × 1 = 21$]], [Any number times one is the number],
  [Additive inverse property  #linebreak() #math.equation(block: false, alt: "a plus open parenthesis minus a close parenthesis equals 0")[$a + ( − a ) = 0$]], [#math.equation(block: false, alt: "14 plus open parenthesis minus 14 close parenthesis equals 0")[$14 + ( − 14 ) = 0$]], [Every number plus its negative is 0],
  [Multiplicative inverse property  #linebreak() #math.equation(block: false, alt: "a times open parenthesis the fraction 1 over a close parenthesis equals 1")[$a × ( frac(1, a) ) = 1$], provided #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$]], [#math.equation(block: false, alt: "3 times open parenthesis the fraction 1 over 3 close parenthesis equals 1")[$3 × ( frac(1, 3) ) = 1$]], [Every non-zero number times its reciprocal is 1],
))


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


    #examplebox("Example 4")[Identifying Properties of Real Numbers][
In each of the following, identify which property of the real numbers is being applied.


        + #math.equation(block: false, alt: "4 plus open parenthesis 8 plus 13 close parenthesis equals open parenthesis 4 plus 8 close parenthesis plus 13")[$4 + ( 8 + 13 ) = ( 4 + 8 ) + 13$]
+ #math.equation(block: false, alt: "34 times open parenthesis the fraction 1 over 34 close parenthesis equals 1")[$34 × ( frac(1, 34) ) = 1$]
+ #math.equation(block: false, alt: "14 plus 27 equals 27 plus 14")[$14 + 27 = 27 + 14$]


      
      #solutionbox[
+ Here, the pair of numbers that is added first is switched. This is the associative property of addition.
+ Here, a number is multiplied by its reciprocal, resulting in 1. This is the multiplicative inverse property.
+ 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.


    #examplebox("Example 5")[Using Properties of Real Numbers in Calculations][
Use properties of the real numbers and mental math to calculate the following:


        + #math.equation(block: false, alt: "2 times 13 times 50")[$2 × 13 × 50$]
+ #math.equation(block: false, alt: "13 plus 84 plus 27")[$13 + 84 + 27$]
+ #math.equation(block: false, alt: "9 times 16 times 11")[$9 × 16 × 11$]


      
      #solutionbox[
+ Notice that #math.equation(block: false, alt: "2 times 50 equals 100")[$2 × 50 = 100$], so that becomes the multiplication to do first. Use the commutative property of multiplication to change the order of the numbers being multiplied. #linebreak() 
    #math.equation(block: true, alt: "2 times 13 times 50 equals 2 times 50 times 13 equals 100 times 13 equals 1,300")[$2 × 13 × 50 = 2 × 50 × 13 = 100 × 13 = 1,300$]
+ Notice that #math.equation(block: false, alt: "13 plus 27 equals 40")[$13 + 27 = 40$], so that becomes the addition to do first. Use the commutative property of addition to change the order in which the numbers are added. #linebreak() 
    #math.equation(block: true, alt: "13 plus 84 plus 27 equals 13 plus 27 plus 84 equals 40 plus 84 equals 124")[$13 + 84 + 27 = 13 + 27 + 84 = 40 + 84 = 124$]
+ Notice that #math.equation(block: false, alt: "9 times 11 equals 99")[$9 × 11 = 99$]. Using that, the problem can be changed to #math.equation(block: false, alt: "99 times 16")[$99 × 16$]. That, however, doesn't look easy at all. But #math.equation(block: false, alt: "99 equals open parenthesis 100 minus 1 close parenthesis")[$99 = ( 100 − 1 )$]. Using the distributive property, we rewrite and expand this as #math.equation(block: false, alt: "99 times 16 equals open parenthesis 100 minus 1 close parenthesis times 16 equals 100 times 16 minus 1 times 16 equals 1,600 minus 16")[$99 × 16 = ( 100 − 1 ) × 16 = 100 × 16 − 1 × 16 = 1,600 − 16$]. 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.
]
]


    
  
  
    
=== Videos


    - #link("https://openstax.org/r/Properties_of_the_Real_Numbers_1")[Properties of the Real Numbers 1]
- #link("https://openstax.org/r/Properties_of_the_Real_Numbers_2")[Properties of the Real Numbers 2]
- #link("https://openstax.org/r/Properties_of_the_Real_Numbers_3")[Properties of the Real Numbers 3]


  
  
  
  
    
=== Key Terms


    - complex number
- imaginary number
- real number


  
  
    
=== 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


    - #math.equation(block: false, alt: "a times open parenthesis b plus c close parenthesis equals a times b plus a times c")[$a × ( b + c ) = a × b + a × c$]
- #math.equation(block: false, alt: "a plus b equals b plus a")[$a + b = b + a$]
- #math.equation(block: false, alt: "a times b equals b times a")[$a × b = b × a$]
- #math.equation(block: false, alt: "a plus open parenthesis b plus c close parenthesis equals open parenthesis a plus b close parenthesis plus c")[$a + ( b + c ) = ( a + b ) + c$]
- #math.equation(block: false, alt: "a times open parenthesis b times c close parenthesis equals open parenthesis a times b close parenthesis times c")[$a × ( b × c ) = ( a × b ) × c$]
- #math.equation(block: false, alt: "a plus 0 equals a")[$a + 0 = a$]
- #math.equation(block: false, alt: "a times 1 equals a")[$a × 1 = a$]
- #math.equation(block: false, alt: "a plus open parenthesis minus a close parenthesis equals 0")[$a + ( − a ) = 0$]
- #math.equation(block: false, alt: "a times open parenthesis the fraction 1 over a close parenthesis equals 1")[$a × ( frac(1, a) ) = 1$]


  
  
    
=== Videos


    - #link("https://openstax.org/r/Arthur_Benjamin_TED_talk,_Faster_than_a_Calculator")[Arthur Benjamin TED talk, Faster than a Calculator]
- #link("https://openstax.org/r/Sets_of_Real_Numbers")[Identifying Sets of Real Numbers]
- #link("https://openstax.org/r/Properties_of_the_Real_Numbers_1")[Properties of the Real Numbers \#1]
- #link("https://openstax.org/r/Properties_of_the_Real_Numbers_2")[Properties of the Real Numbers \#2]
- #link("https://openstax.org/r/Properties_of_the_Real_Numbers_3")[Properties of the Real Numbers \#3]