### Learning Objectives

construct a number line and graph clues on it. Use a number heat to recognize the order of real numbers. Recognize the the contrary of a genuine number. Identify the absolute worth of a genuine number.

## Definitions

A setAny repertoire of objects. Is a arsenal of objects, commonly grouped within braces , wherein each thing is called an elementAn object within a set.. Because that example, red, green, blue is a set of colors. A subsetA set consisting of facets that belong to a given set. Is a collection consisting of aspects that belong come a given set. For example, green, blue is a subset that the color collection above. A set with no facets is dubbed the empty setA subset v no elements, denoted ∅ or . And also has its own special notation, or ∅.

You are watching: All real numbers that are less than 5

When researching mathematics, we emphasis on special sets that numbers. The collection of natural (or counting) numbersThe collection of counting numbers 1, 2, 3, 4, 5, …., denoted N, is The three durations (…) is referred to as an ellipsis and indicates that the numbers continue without bound. The collection of entirety numbersThe set of organic numbers an unified with zero 0, 1, 2, 3, 4, 5, …., denoted W , is the set of herbal numbers combined with zero. The set of integersThe set of positive and an unfavorable whole numbers merged with zero …, −3, −2, −1, 0, 1, 2, 3, …., denoted Z, is composed of both optimistic and an unfavorable whole numbers, and also zero. Notice that the to adjust of natural and also whole numbers space both subsets of the collection of integers.

Rational numbersNumbers that the form ab, wherein a and b space integers and also b is nonzero., denoted Q, are characterized as any number of the form ab, whereby a and b room integers and also b is nonzero. Decimals the repeat or terminate are rational. Because that example, The set of integers is a subset that the collection of reasonable numbers due to the fact that every integer have the right to be expressed together a proportion of the integer and 1. In other words, any type of integer have the right to be composed over 1 and can be thought about a rational number. For example, Irrational numbersNumbers the cannot be written as a ratio of 2 integers. Are characterized as any kind of number that cannot be composed as a ratio of 2 integers. Nonterminating decimals that execute not repeat space irrational. Because that example,

The collection of actual numbersThe collection of every rational and irrational numbers., denoted R, is defined as the collection of all rational numbers an unified with the collection of every irrational numbers. Therefore, every the numbers identified so far are subsets that the set of real numbers. In summary,

## Number Line

A genuine number lineA heat that enables us to visually represent genuine numbers by link them v points ~ above the line., or merely number line, enables us come visually screen real number by associating them with distinct points top top a line. The real number linked with a point is referred to as a coordinateThe genuine number associated with a point on a number line.. A allude on the actual number line that is associated with a name: coordinates is referred to as its graphA point on the number line associated with a coordinate..

To construct a number line, draw a horizontal line with arrows top top both end to show that it proceeds without bound. Next, select any suggest to stand for the number zero; this point is referred to as the originThe suggest on the number line that represtents zero..

Mark off regular lengths top top both sides of the origin and label every tick note to define the scale. Positive real numbers lie to the best of the beginning and an adverse real number lie come the left. The number zero (0) is neither optimistic nor negative. Typically, each mite represents one unit.

As depicted below, the scale need not constantly be one unit. In the an initial number line, each tick note represents 2 units. In the second, every tick note represents 17.

The graph that each genuine number is displayed as a period at the appropriate suggest on the number line. A partial graph that the collection of integers Z follows:

Example 1: Graph the following collection of genuine numbers: −1, −13, 0, 53.

Solution: Graph the numbers on a number line v a range where each tick note represents 13 unit.

## Ordering real Numbers

When comparing actual numbers top top a number line, the larger number will always lie to the best of the smaller one. It is clear the 15 is higher than 5, however it may not it is in so clean to see that −1 is higher than −5 till we graph every number top top a number line.

We use signs to help us efficiently connect relationships in between numbers ~ above the number line. The symbols offered to explain an equality relationshipExpress equality with the price =. If two quantities are not equal, usage the symbol ≠. Between numbers follow:

These symbols space used and also interpreted in the following manner:

We next specify symbols that signify an order relationship in between real numbers.

These symbols enable us come compare two numbers. For example,

Since the graph the −120 is come the left that the graph that –10 ~ above the number line, that number is much less than −10. We can write an equivalent statement as follows:

Similarly, because the graph the zero is to the appropriate of the graph of any an adverse number ~ above the number line, zero is higher than any an adverse number.

The symbols are used to denote strict inequalitiesExpress bespeak relationships utilizing the price for “greater than.”, and the symbols ≤ and ≥ are used to represent inclusive inequalitiesUse the price ≤ to express amounts that room “less 보다 or equal to” and ≥ for quantities that room “greater than or same to” each other.. In part situations, more than one symbol deserve to be effectively applied. For example, the adhering to two statements space both true:

In addition, the “or same to” component of one inclusive inequality permits us to properly write the following:

The logical use of words “or” requires that only one of the problems need be true: the “less than” or the “equal to.”

Example 2: fill in the blank with : −2 ____ −12.

Solution: use > due to the fact that the graph that −2 is come the best of the graph the −12 on a number line. Therefore, −2 > −12, which reads “negative 2 is higher than an unfavorable twelve.”

In this text, we will certainly often point out the indistinguishable notation supplied to express mathematical quantities electronically making use of the traditional symbols obtainable on a keyboard. We begin with the tantamount textual notation because that inequalities:

Many calculators, computer system algebra systems, and also programming languages usage this notation.

## Opposites

The oppositeReal numbers who graphs are on opposite sides of the beginning with the very same distance to the origin. Of any real number a is −a. Opposite genuine numbers room the very same distance from the beginning on a number line, but their graphs lie on opposite political parties of the origin and also the numbers have opposite signs.

For example, us say the the the contrary of 10 is −10.

Next, think about the the contrary of a an adverse number. Provided the essence −7, the creature the same distance indigenous the origin and with the opposite sign is +7, or simply 7.

Therefore, we say that the the opposite of −7 is −(−7) = 7. This idea leader to what is often referred to together the double-negative propertyThe the contrary of a an adverse number is positive: −(−a) = a.. For any real number a,

Example 3: What is the contrary of −34?

Solution: here we use the double-negative property.

Example 4: Simplify: −(−(4)).

Solution: begin with the innermost bracket by recognize the opposite of +4.

Example 5: Simplify: −(−(−2)).

Solution: apply the double-negative property starting with the innermost parentheses.

### Tip

If over there is one even number of consecutive an adverse signs, then the an outcome is positive. If over there is an odd number of consecutive an unfavorable signs, then the result is negative.

Try this! Simplify: −(−(−(5))).

### Video Solution

(click to check out video)

## Absolute Value

The pure valueThe absolute worth of a number is the distance from the graph the the number come zero top top a number line. Of a genuine number a, denoted |a|, is defined as the distance between zero (the origin) and also the graph that that genuine number top top the number line. Since it is a distance, that is always positive. For example,

Both 4 and also −4 are 4 units native the origin, as portrayed below:

Example 6: Simplify:

a.|−12|

b.|12|

Solution: Both −12 and also 12 space twelve units from the origin on a number line. Therefore,

Also, it is worth noting that

The pure value can be expressed textually making use of the notation abs(a). We frequently encounter an adverse absolute values, such together −|3| or −abs(3). An alert that the an unfavorable sign is in front of the absolute value symbol. In this case, occupational the pure value very first and then find the the opposite of the result.

Try no to confuse this with the double-negative property, which says that −(−7)=+7.

Example 7: Simplify: −|−(−7)|.

Solution: First, find the the opposite of −7 inside the pure value. Then uncover the opposite of the result.

At this point, we can determine what real numbers have actually a specific absolute value. For example,

Think that a actual number whose street to the origin is 5 units. There are two solutions: the street to the best of the origin and also the distance to the left of the origin, namely, ±5. The prize (±) is check out “plus or minus” and indicates the there are two answers, one positive and one negative.

Here we wish to discover a value for i m sorry the distance to the origin is negative. Since an unfavorable distance is no defined, this equation has actually no solution. If an equation has actually no solution, us say the solution is the north set: Ø.

### Key Takeaways

any real number have the right to be connected with a suggest on a line. Produce a number heat by an initial identifying the origin and also marking off a scale ideal for the offered problem. Negative numbers lie to the left that the origin and positive numbers lie come the right. Smaller numbers constantly lie come the left of bigger numbers ~ above the number line. The contrary of a confident number is an adverse and opposing of a an unfavorable number is positive. The absolute worth of any real number is constantly positive because it is characterized to it is in the distance from zero (the origin) on a number line. The absolute value of zero is zero.

### Topic Exercises

Part A: actual Numbers

Use set notation to perform the explained elements.

1.The hrs on a clock.

2.The days of the week.

3.The first ten whole numbers.

4.The very first ten herbal numbers.

5.The first five positive also integers.

6.The an initial five optimistic odd integers.

Determine even if it is the complying with real numbers space integers, rational, or irrational.

7.12

8.−3

9.4.5

10.−5

11.0.36¯

12.0.3¯

13.1.001000100001…

14.1.001¯

15.e=2.71828…

16.7=2.645751…

17.−7

18.3.14

19.227

20.1.33

21.0

22.8,675,309

True or false.

23.All integers space rational numbers.

24.All integers are entirety numbers.

25.All rational number are totality numbers.

26.Some irrational numbers space rational.

27.All terminating decimal numbers are rational.

28.All irrational numbers room real.

Part B: real Number Line

Choose an proper scale and graph the adhering to sets of real numbers top top a number line.

29.−3, 0 3

30.−2, 2, 4, 6, 8, 10

31.−2, −13, 23, 53

32.−52, −12, 0, 12 , 2

33.−57, 0, 27 , 1

34. –5, –2, –1, 0

35. −3, −2, 0, 2, 5

36.−2.5, −1.5, 0, 1, 2.5

37.0, 0.3, 0.6, 0.9, 1.2

38.−10, 30, 50

39.−6, 0, 3, 9, 12

40.−15, −9, 0, 9, 15

Part C: Ordering actual Numbers

Fill in the empty with .

41.−7 ___ 0

42.30 ___ 2

43.10 ___−10

44.−150 ___−75

45.−0.5 ___−1.5

46.0___ 0

47.−500 ___ 200

48.−1 ___−200

49.−10 ___−10

50.−40 ___−41

True or false.

51.5≠7

52.4=5

53.1≠1

54.−5>−10

55.4≤4

56.−12≥0

57.−10=−10

58.3>3

59.−1000−20

60.0=0

61.List 3 integers less than −5.

62.List three integers higher than −10.

63.List three rational numbers less than zero.

64.List three rational numbers better than zero.

65.List three integers in between −20 and −5.

66.List three rational numbers between 0 and also 1.

Translate each statement right into an English sentence.

67.1020

68.−50≤−10

69.−4≠0

70.30≥−1

71.0=0

72.e≈2.718

Translate the following into a mathematical statement.

73.Negative 7 is less than zero.

74.Twenty-four is not equal come ten.

75.Zero is higher than or equal to an adverse one.

76.Four is better than or same to an unfavorable twenty-one.

77.Negative two is same to negative two.

78.Negative 2 thousand is less than negative one thousand.

Part D: Opposites

Simplify.

79.−(−9)

80.−(−35)

81.−(10)

82.−(3)

83.−(5)

84.−(34)

85.−(−1)

86.−(−(−1))

87.−(−(1))

88.−(−(−3))

89.−(−(−(−11)))

90.What is opposing of −12

91.What is the opposite of π ?

92.What is the opposite −0.01?

93.Is the opposite of −12 smaller or larger than −11?

94.Is the contrary of 7 smaller or bigger than −6?

Fill in the blank with .

95.−7 ___−(−8)

96.6 ___−(6)

97.13 ___−(−12)

98.−(−5) ___−(−2)

99.−100 ___−(−(−50))

100.44 ___−(−44)

Part E: pure Value

Simplify.

101.|20|

102.|−20|

103.|−33|

104.|−0.75|

105.|−25|

106.|38|

107.|0|

108.|1|

109.−|12|

110.−|−20|

111.−|20|

112.−|−8|

113.−|7|

114.−|−316|

115.−(−|89|)

116.|−(−2)|

117.−|−(−3)|

118.−(−|5|)

119.−(−|−45|)

120.−|−(−21)|

121.abs(6)

122.abs(−7)

123.−abs(5)

124.−abs(−19)

125.− (−abs(9))

126.−abs(−(−12))

Determine the unknown.

127.| ? |=9

128.| ? |=15

129.| ? |=0

130.| ? |=1

131.| ? |=−8

132.| ? |=−20

133.|?|−10=−2

134.|?|+5=14

Fill in the empty with = , or >.

135.|−2| ____ 0

136.|−7| ____ |−10|

137.−10 ____−|−2|

138.|−6| ____ |−(−6)|

139.−|3| ____ |−(−5)|

140.0 ____−|−(−4)|

Part F: discussion Board Topics

141.Research and discuss the background of the number zero.

142.Research and discuss the assorted numbering equipment throughout history.

See more: Driving Distance Between Tampa Florida And Naples Florida ) From Tampa

143.Research and also discuss the an interpretation and history of π .

144.Research the background of irrational numbers. Who is attributed with proving that the square source of 2 is irrational and also what happened to him?