<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

Is there an integer that is not a natural number?

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

Got questions? Get instant answers now!

Is there an integer that is a whole number?

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

Got questions? Get instant answers now!

Practice set a

Is every natural number a whole number?

yes

Got questions? Get instant answers now!

Is every whole number an integer?

yes

Got questions? Get instant answers now!

Is every integer a real number?

yes

Got questions? Get instant answers now!

Is there an integer that is a whole number?

yes

Got questions? Get instant answers now!

Is there an integer that is not a natural number?

yes

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Yes, 10

Got questions? Get instant answers now!

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}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

{1}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

{0, 1, 2, 3, 4}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Are all positive numbers greater than all nega­tive numbers?

Got questions? Get instant answers now!

Is 0 greater than all negative number?

yes

Got questions? Get instant answers now!

Is there a largest natural number?

Got questions? Get instant answers now!

Is there a largest negative integer?

yes, -1

Got questions? Get instant answers now!

Exercises for review

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

Got questions? Get instant answers now!

( [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} {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( [link] ) Convert 30.06 cm to m.

0.3006 m

Got questions? Get instant answers now!

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

A triangle with base 16mm and height 3mm

Got questions? Get instant answers now!

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

Ask