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A horizontal line with arrows on the end. The center has a hash mark labeled 0. On the right side is a bracket, labeled Positive numbers. On the left side is a bracket, labeled Negative numbers.

The number 0 is neither positive nor negative.

Subsets of real numbers

The set of real numbers has many subsets. Some of the subsets that are of interest in the study of algebra are listed below along with their notations and graphs.

Natural numbers, counting numbers

The natural or counting numbers ( N ): 1, 2, 3, 4, . . . Read “and so on.”

A number line containing dots on the hash marks for numbers one through seven.

Whole numbers

The whole numbers ( W ): 0, 1, 2, 3, 4, . . .

A number line containing dots on the hash marks for numbers zero through seven.

Notice that every natural number is a whole number.

Integers

The integers ( Z ): . . . -3, -2, -1, 0, 1, 2, 3, . . .

A number line containing dots on the hash marks for numbers -4 through 4.

Notice that every whole number is an integer.

Rational numbers (fractions)

The rational numbers ( Q ): Rational numbers are sometimes called fractions . They are numbers that can be written as the quotient of two integers. They have decimal representations that either terminate or do not terminate but contain a repeating block of digits. Some examples are

- 3 4 = - 0.75 Terminating       8 11 27 = 8.407407407 . . . Nonterminating, but repeating

Some rational numbers are graphed below.

A number line containing hash marks for numbers -3 through 4. There are dots for negative three and one-eighths, negative one-half, two-fifths, two divided by one, and three and one-half.

Notice that every integer is a rational number .

Notice that there are still a great many points on the number line that have not yet been assigned a type of number. We will not examine these other types of numbers in this text. They are examined in detail in algebra. An example of these numbers is the number π , whose decimal representation does not terminate nor contain a repeating block of digits. An approximation for π is 3.14.

Sample set a

Is every whole number a natural number?

No. The number 0 is a whole number but it is not a natural number.

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Is there an integer that is not a natural number?

Yes. Some examples are 0, -1, -2, -3, and -4.

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Is there an integer that is a whole number?

Yes. In fact, every whole number is an integer.

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Practice set a

Is every natural number a whole number?

yes

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Is every whole number an integer?

yes

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Is every integer a real number?

yes

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Is there an integer that is a whole number?

yes

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Is there an integer that is not a natural number?

yes

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Ordering real numbers

Ordering real numbers

A real number b size 12{b} {} is said to be greater than a real number a size 12{a} {} , denoted b > a size 12{b>a} {} , if b size 12{b} {} is to the right of a size 12{a} {} on the number line. Thus, as we would expect, 5 > 2 size 12{5>2} {} since 5 is to the right of 2 on the number line. Also, - 2 > - 5 size 12{"- 2 ">"-5"} {} since -2 is to the right of -5 on the number line.

A number line containing hash marks for numbers -5 through 5. There are dots on the hash marks for -5, -2, 2, and 5. Above the left side of the number line is the expression -2 > -5, and on the left side is 5 > 2.

If we let a size 12{a} {} and b size 12{b} {} represent two numbers, then a size 12{a} {} and b size 12{b} {} are related in exactly one of three ways: Either

Equality symbol

a = b a  and  b  are equal ( 8 = 8 )

Inequality symbols

a > b a   is greater than  b ( 8 > 5 ) a < b a   is less than  b ( 5 < 8 ) Some variations of these symbols are a b a   is not equal to  b ( 8 5 ) a b a   is greater than or equal to  b ( a 8 ) a b a   is less than or equal to  b ( a 8 )

Sample set b

What integers can replace x so that the following statement is true?

- 3 x < 2 size 12{"-3"<= " x"<" 2"} {}

A number line containing hash marks for numbers -5 through 5. There are dots on the hash marks for -3, -2, -1, 0, 1.

The integers are -3, -2, -1, 0, 1.

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Draw a number line that extends from -3 to 5. Place points at all whole numbers between and including -1 and 3.

A number line containing hash marks for numbers -3 through 5. There are dots on the hash marks for 0, 1, 2, and 3.

-1 is not a whole number

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Practice set b

What integers can replace x so that the following statement is true? - 5 x < 2 size 12{"-5"<= " x"<" 2"} {}

-5, -4, -3, -2, -1, 0

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Draw a number line that extends from -4 to 3. Place points at all natural numbers between, but not including, -2 to 2.

A horizontal line with arrows on the end.

A number line with hash marks for the numbers -4 to 3, and a dot on the hash mark for 1.

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Exercises

For the following 8problems, next to each real number, note all collections to which it belongs by writing N size 12{N} {} for natu­ral number, W size 12{W} {} for whole number, or Z size 12{Z} {} for integer. Some numbers may belong to more than one collec­tion.

Is the number 0 a positive number, a negative number, neither, or both?

Neither

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An integer is an even integer if it is evenly divisi­ble by 2. Draw a number line that extends from -5 to 5 and place points at all negative even integers and all positive odd integers.

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Draw a number line that extends from -5 to 5. Place points at all integers that satisfy - 3 x < 4 size 12{"–3"<= " x "<" 4"} {} .

A number line with hash marks for the numbers -5 to 5. There is a solid dot on the hash mark for -3, an open dot on the hash mark for 4. There is a thick line drawn in between the dots on the line.

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Is there a largest two digit number? If so, what is it?

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Is there a smallest two digit number? If so, what is it?

Yes, 10

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For the pairs of real numbers in the following 5 problems, write the appropriate symbol (<,>, =) in place of the □.

For the following 5 problems, what numbers can replace m so that the following statements are true?

- 1 m - 5 size 12{"–1 "<= " m "<= "–5"} {} , m size 12{m} {} an integer.

{-1, 0, 1, 2, 3, 4, 5}

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- 7 < m < - 1 size 12{"–7 "<" m "<"–1"} {} , m size 12{m} {} an integer.

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3 m < 2 size 12{ - 3<= m<2} {} , m size 12{m} {} a natural number.

{1}

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- 15 < m - 1 size 12{"–15 "<" m"<= " –1"} {} , m size 12{m} {} a natural number.

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5 m < 5 size 12{ - 5<= m<5} {} , m size 12{m} {} a whole number.

{0, 1, 2, 3, 4}

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For the following 10 problems, on the number line, how many units are there between the given pair of numbers?

Are all positive numbers greater than zero?

yes

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Are all positive numbers greater than all nega­tive numbers?

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Is 0 greater than all negative number?

yes

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Is there a largest natural number?

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Is there a largest negative integer?

yes, -1

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Exercises for review

( [link] ) Convert 6 5 8 size 12{6 { {5} over {8} } } {} to an improper fraction.

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( [link] ) Find the value: 3 11 size 12{ { {3} over {"11"} } } {} of 33 5 size 12{ { {"33"} over {5} } } {} .

9 5 or 1 4 5 or 1 . 8 size 12{ { {9} over {5} } " or 1" { {4} over {5} } " or 1" "." 8} {}

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( [link] ) Find the sum of 4 5 + 3 8 size 12{ { {4} over {5} } + { {3} over {8} } } {} .

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( [link] ) Convert 30.06 cm to m.

0.3006 m

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( [link] ) Find the area of the triangle.

A triangle with base 16mm and height 3mm

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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