Real Numbers (subsets of real numbers and ordering real numbers)

Real Numbers (subsets of real numbers and ordering real numbers)

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This tutorial is designed to teach you about Real numbers ( subsets of real numbers and ordering real numbers). The presentation starts with notes and examples followed by practice problems.

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Definitions to know

Natural (or counting) numbers: N= {1,2,3…} Whole numbers: W={0,1,2,3…} Integers: {…-3,-2,-1,0,1,2,3…} Rational numbers: Q={all numbers that can be expressed

in the form m/n, where m and n are integers and n is not zero}. The decimal form of a rational number is either a terminating or repeating decimal.

Irrational number: I={all nonterminating, nonrepeating decimals}. Any number that is not a perfect square has an irrational square root.

Real numbers: R={all rationals and irrationals}

Shortcut

Real Numbers- (R) Irrational Numbers- (I) Rational Numbers- (Q) Integer- (Z) Whole Numbers- (W) Natural Numbers- (N)

Real Numbers

Real numbers can be divided into two basic groups: Irrational numbers and Rational numbers.

Rational Numbers can further be divided into 3 groups: integers, whole numbers, and natural numbers.

N atu ra l N u m b ers(N )

W h o le N u m b ers(W )

In teg ers(Z )

R ation a l N u m b ers(Q )

Irra tion a l N u m b ers(I)

R eal N um bers(R )

Rational Numbers

Rational numbers are all numbers that can be expressed in the form m/n, where m and n are integers and n is not zero}. The decimal form of a rational number is either a terminating or repeating decimal.

A natural number is also always : a whole number, integer, rational number, and real number, etc.

Although a natural number is always an integer; an integer is not always a natural number, etc.

Example: 2, 8/2, 15, 83= real/rational/

integer /whole /natural -2,-8/2,-15,-83= real/rational/

integer

Natural Numbers (N)

Integers (Z)

Whole numbers (W)

Rational numbers (Q)

Irrational Numbers

Irrational numbers are all nonterminating, nonrepeating decimals. Any number that is not a perfect square has an irrational square root.

Examples: 0.078651685…. 1760566….. 17

Examples of Real Number’s Subsets

3/4= real/ rational 0= real/rational/integer/whole .34345646….= real/irrational 1= real/rational/integer/whole/natural = real/irrational 1/3= real/ rational 91,215,225,201,544=

real/rational/integer/whole/natural

Practice 1(subsets of real numbers)

Which of the following numbers is irrational?

A

B

C

D

-1/2

3.63

121

6/55

Practice 2 (subsets of real numbers)

36 is what kind of number?

A

B

C

D

Real/ Irrational

Real/rational/integer/ whole/natural

Real/rational/integer/whole

Real/rational

Practice 3 (subsets of real numbers)

Which of the following is a rational number? A

B

C

D

1/5

3

25/115

Practice 4 (subsets of real numbers)

-6 is what kind of number? A

B

C

D

Real/rational/integer

Real/irrational

Real/rational

Real/rational/integer/ whole

Ordering Real Numbers

Real numbers are ordered from least to greatest.

Example: 1, -4, 49, -16/2, 19/5

ordered from least to greatest would be:

-16/2 (-8), -4, 1, 49 (7), 19/5(3.8)

Practice 5 (ordering real numbers)

Order the following numbers from smallest to largest:

-8, 14/3, 10, 5

A

C

D

B -8, 5, 14/3, 10

-8, 10, 5, 14/3

-8, 14/3, 5, 10

-8, 14/3, 10, 5

Practice 6 (ordering real numbers)

Order the following numbers from least to greatest:

144, -47, 19/6, -13

A

B

C

D

-47, -13, 19/6, 144

-47, -13, 144, 19/6

-13, -47, 144, 19/6

-47, 144, 19/6, -13

References

Information retrieved 11/30/05 from: http://www.regentsprep.org/Regents/math/math-topic.cfm?TopicCode=rational

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