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THE REAL NUMBER SYSTEM

“Numbers rule the universe.”

- Pythagoras

3 Mathematics Division, IMSP, UPLB

Learning Objectives

At the end of the lesson, you should be able to

identify subsets of the set of real numbers

recognize the various forms of rational numbers

distinguish rational numbers from irrational numbers

locate numbers on the real number line


NUMBER!!!???

evolved over time by expanding the notion of what we mean by the word “number.”

Question: What is the difference between “numbers” and “numerals”?

at first, “number” meant something you could count, like

– how many girlfriends I have

– how many legs an insect has

These are called …


SET OF NATURAL NUMBERS

All natural numbers are truly natural.

We find them in nature.


The set of

NATURAL NUMBERS,

(also called

COUNTING NUMBERS)

is denoted by

N = {1, 2, 3, 4, 5, 6, 7, …}


P = the set of prime numbers (divisible only by 1 and itself) What is the smallest prime number? Can an even number be prime? Name some more prime numbers.

Special Subsets of N


C = the set of composite numbers What is the smallest composite number? Are all odd numbers composite? Name some more composite numbers. Are P and C disjoint?


The Number Zero

Is zero a number?

How can the number of nothing be a

number?

0 is a special number because it does

not quite obey the same laws as other

numbers (e.g. We can’t divide by zero)

Zero is also used as a place-holder


SET OF WHOLE NUMBERS

W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9…}

What is the only difference between this

set and the set of natural (or counting)

numbers?


What is N W ? N W ?

Are N and W disjoint?

What is W – N? N – W?


NEGATIVE NUMBERS

Negative numbers – used to represent

losses, debt, depth, etc.

These are the natural numbers with

the negative (minus) sign


Opposites of the positive numbers

When a positive number and its

negative are added the result is 0.

These pairs of numbers are called

additive inverses of one another.

NEGATIVE NUMBERS


Time to think: What is the additive inverse of 5?

What is the additive inverse of –23?

NEGATIVE NUMBERS

5

23


SET OF INTEGERS

Z = {…,-4,-3,-2,-1,0,1,2,3,4,…}

Special Subsets of the Set of Integers

1. Set of Negative integers:

–N = Z– = {…-4, -3, -2, -1}

2. The set consisting of Zero alone: {0} 3. Set of Positive integers:

Z+ = N = {1, 2, 3, 4, …}

What is the notation for the set of nonpositive integers? nonnegative integers?


Other Special Subsets of Z

E = the set of even integers

= {x | x = 2k where k є Z}

Which of the following numbers is even? Why?

a) 146

b) 2313

c) –1887640


O = the set of odd integers = {x | x = 2k + 1 where k є Z} = {x | x = 2k – 1 where k є Z} Which of the following is odd? Why?

a) 12345

b) 24670

c) –32146987


Time to think:

Is E = O? Why?

Is E O? Why?

Are E and O disjoint?


SET OF RATIONAL NUMBERS

A rational number is a number that can be expressed as the ratio or quotient of two integers p and q where q ≠ 0. The set of rational numbers is denoted as

0,, qZqpq

pQ


Examples of Rational Numbers

1a) 0.25

4

1b) 0.5

2

11c) 5.5

2

20d) 4

5

2e) 0.666...

3


About Rational Numbers

Integers are rational numbers

Fractions are rational numbers

(similar to the word “fracture” suggesting breaking something up)

• proper fraction

• improper fraction

• mixed numbers


About Rational Numbers

Furthermore, rational numbers are numbers with decimals that are

– terminating

– non-terminating but repeating

Example: (Can you convert the following to fractions?)

0.25 0.1111… 0.125125…

0.645 0.00222 0.547123


POSITIVE, NEGATIVE RATIONAL NUMBERS

What are the notations?

Positive rational numbers

Negative rational numbers

Nonpositive rational numbers

Nonnegative rational numbers

(Zero is neither positive nor

negative)


Positive rational numbers can be written

in any of three forms, any of three

notations

as fractions

as decimals

as percents

Each form has its own special

characteristics

The forms are interchangeable.

But the ways of doing standard operations

with the three notations are very different.


FRACTIONS are used to name

part of a whole object or part of a whole collection of objects or to compare two quantities.

Note that (for mixed numbers):

2

3

2

13

2

13

2

13

2

13

2

13


can be thought of as

2 divided by 3

so that when you divide

a numerator by a denominator

you can express that fraction

as a DECIMAL.

The fraction


A PERCENT is a fraction with 100 in the

denominator.

The word percent comes from the Latin word

“per centum” . “per” meaning FOR and

“centum” meaning ONE HUNDRED.

60% means 60 out of 100. A percent

represents a portion of something and that

something is the whole thing, or 100 %.


Multiple Representations

Fractions and mixed numbers – appear in recipes

Decimals – occur in scientific measurements

Percentages – used in commerce


SET OF IRRATIONAL NUMBERS

are those real numbers that can not be expressed as the ratio of two integers

denote the set of irrational numbers as Qc (the complement of Q) if U=R

can also be described as numbers with decimals that are nonterminating and nonrepeating


Examples of Irrational Numbers

Non-terminating, non-repeating decimals

a) 1.01001000100001…

c) 3.141592653589… p

d) 2.7182818284590… e

b) 1.414213562… 2


WARNING!!!

7

22p



The square roots of all

positive numbers which are

not perfect squares are

irrationals. (Actually, if an

integer is not an exact kth power

of another integer then

its kth root is irrational.)

Determine if rational or irrational:

3

5

3

16

8

7



log23 is also irrational

The golden ratio (also called

as golden mean, golden

section, divine proportion,

golden number) and its

reciprocal are irrational.


The Golden Ratio

...61803.12

51


The Reciprocal of the Golden Ratio

...61803.011





40

Some rational and irrational numbers are of the

form wheren p

: principal th root of . sometimes called radicals n p n p

radicand

index

p

n

: principal square root of , 0.p p p

FYI about RADICALS

41

2We know that 2 4 so 4 2.

2

Also, 2 4 so is 4 2?

QUESTION:

42

Principal Root

Definition.

If is even and is non-negative,

we define as the positive th

root of .

n

n p

p n

p

Therefore, 4 2 and 4 2

43

NOTE:

If is negative, is

undefined.

np p

Suppose is positive. If is odd,

n n

p n

p p

(Since we assume U=R)

44

Example

Determine the value of the following

radicals.

1. 9 3

32. 8 2

33. 27 3 27 3

44. 625 5

5. 4 undeis fined

45

xn=p

If is odd then is solution tothe .nnn p x p

33 8 is the solution to 8x

3 32 8 so 8 2

2 4 and 4 are the solutions to 4x

2 22 4 so 4 2 ; ( 2) 4 so 4 2

If is even then and

ar solutions toe the . ( 0)

n n

n

n p p

x p p

46

Example

Find the solution(s) to the following.

21. 16x 4, 4

32. 125x 5

53. 4x 5 4

24. 9x no solutions

25. 3x 3, 3


Set of Real Numbers, R

R is the union of the

set of rational numbers and

set of irrational numbers



Real Number Line

One-dimensional coordinate

system

1–1 correspondence between the

set of points on a line and the set

of real numbers



Locating numbers in the number

line:

Find the following numbers in the

number line shown below:

a) 1 c)

b) -2 d)

2

3

0



SUMMARY A real number is either rational or irrational.

If it is a rational number, it is either an integer or a non-integer fraction.

If it is an integer, it is is either a whole number or a negative integer.

If it is a whole number, it is either a counting number or zero.

There is a 1-1 correspondence between the set of real numbers and the set of points on the line.


FYI:

The cardinality of N, W, Z–, Z, E, O, Q is 0ּא (countable)

The cardinality of Qc and R is (uncountable) 1ּא


FYI:

Example of a set outside R is the set of imaginary numbers (I).

The union of R and I is the set of complex numbers.

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Ako, si , ay nangangakong magsisipag mag-aral hindi lang ...jfrabajante.weebly.com/uploads/1/1/5/5/11551779/4_math_17.pdf · Ang aking kontrata: Ako, si _____, ay nangangakong magsisipag