unit 1 real numbers

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Unit 1: Real Numbers Mathematics 4º ESO – Option B

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Unit 1: Real Numbers

The thousandth decimal of

sleeps there though

no one may ever try to compute it

William James, 1909

Human beings share the desire to organize and classify. Ancient astronomers

classified stars into groups called constellations. Modern astronomers continue

to classify stars by such characteristics as color, mass, size, temperature, and

distance from Earth. In mathematics it is useful to place numbers with similar

characteristics into sets.

1. Set of Numbers

1.1 Natural Numbers

The numbers we use to count are called the counting numbers or

natural numbers. Because we begin counting with the number 1, the

set of natural numbers begins with 1. The set of natural numbers isfrequently denoted by :

}

(The set is in roster notation. The numbers are separated by commas

and are inside set brackets. The number 5 is followed by an ellipsis, a

series of three dots. An ellipsis tells us that this set of numbers is

infinite. The list of numbers continues on forever, without end).

Every natural number greater than 1 can be classified as either aprime number or a composite number.

A prime number is a natural number greater than 1 that has exactly

two factors (or divisors), itself and 1.

A composite number is a natural number that is divisible by a

number other than itself and 1.





1.2 Whole Numbers

Another important set of numbers, the whole numbers, help to

answer the question, How many?

} Note that the set of whole numbers contains the number 0 but that the

set of counting numbers does not. If a student were asked how many

books have read this term, the answer would be a whole number. If

the student has read no books, he would answer zero.

Although we use the number 0 daily and take it for granted, the

number zero as we know it was not used and accepted until the

sixteenth century .

1.3 Integer Numbers

If the temperature is 12ºF and drops 20 º, the resulting temperature is

– 8ºF. This type of problem shows the need for negative numbers. The

set of integers consists of the negative integers, 0, and the positive

integers.

}

The term positive integers is yet another name for the natural

numbers.

We can obtain an image of integer numbers by representing them on

a number line.

To construct the number line, arbitrarily select a point for zero to serve

as the starting point. Place the positive integers to the right of 0,

equally spaced from one another. Place the negative integers to the

left of 0, using the same spacing.

The arrows at the ends of the number line indicate that the line

continues indefinitely in both directions.





Note that for any natural number, n , on the number line, the opposite

of that number, - n , is also on the number line. For example:

The number line can be used to determine the greater (or lesser) oftwo integers. Two inequality symbols that we will use in this chapter

are and . The symbol is read “is greater than”, and the symbol is read “is less than” On the number line, the numbers increase

from left to right.

The absolute value of a number is its distance from 0 on a number

line. The symbol or notation for absolute value is | |. A distance is a

number that is greater than or equal to 0. Since an absolute value is a

distance, the absolute value od an integer is also greater than or equal

to 0.

||

1.4 Rational Numbers

The numbers that fall between the integers on the number line are

either rational or irrational. In this section, we discuss the rational

numbers.

Rational numbers are used to express a part of a whole, a part of a

quantity.

The number below the fraction line is called denominator, and

expresses the number of parts into which the whole is divided. The





number above the fraction line is called the numerator, and expresses

the number of parts taken.

A more formal definition of rational numbers could be:

So any number that can be expressed as a quotient of two integers

(denominator not zero) is a rational number:

When the numerator and denominator have a common divisor, we can

reduce the fraction to its lowest terms (or simplest form):

A fraction is said to be in its lowest terms (or reduced, or simplified)

when the numerator and the denominator are relatively prime.

Now we can introduce the definition of equivalent fractions:

The set of rational numbers, denoted by , is the set of all numbers

of the form, where p and q are integers, and .

Two fractions are said to be equivalent when simplifying

both of them produces the same fraction, which cannot be

further reduced.

Equivalent fractions look different but represent the same

portion of the whole.

Equivalent fractions have the same numerical value. They

are represented by the same rational number.

Equivalent fractions are represented by the same point on

the number line.

We can test if two fractions are equivalent by cross-

multiplying (or cross-product) their numerators and

denominators.





Proper and Improper Fractions. Mixed Numbers.

Rational numbers less than 1 or greater than -1 are represented by

proper fractions. A proper fraction is a fraction whose numerator is

less than its denominator:

Consider the number . It is an example of a mixed number. It is

called a mixed number because it consists of an integer, 2, and a

fraction, and it is equal to . The mixed number means

( ).

Rational numbers greater than 1 or less than -1 that are not integers

may be represented as mixed numbers, or as improper fractions. An

improper fraction is a fraction whose numerator is greater than its

denominator.

Fractions and Decimal Numbers

a) Converting Fractions to Decimal Numbers

To obtain the decimal number which is related to a fraction we only

have to divide the numerator by the denominator:

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Note the following important property of the rational numbers.

Every rational number, every fraction, when expressed as a

decimal number will be either a terminating or a recurring (or

repeating) decimal number.

DecimalNumbers

Decimal Part The fraction in its lowestterms

Terminating

or Exact

It Does not go on

forever.

You can writedown all its digits

If the only factors of the

denominator are 2 or 5 orcombinations of 2 and 5 thenthe fraction will be aterminating decimal.

Recurring

orRepeating

It Does go on

forever.

It Repeats a blockof digits.

If the denominator hasn’t any

2 or 5 factors then the fractionwill be a recurring decimal.

If the denominator has anyfactors other than 2 and/or 5then the fraction will be arecurring decimal.





b) Converting Decimal Numbers to Fractions

To convert a terminating decimal number to a quotient of

integers, we observe the last digit to the right of the decimal point.

The position of this digit will indicate the denominator of the

quotient of integers. The numerator will be the decimal numberwithout the decimal point.

Converting a repeating decimal number to a quotient of integers

is more difficult than converting a terminating decimal.

When the repeating digits are directly to the right of the decimal

point, as the number , we use the following algorithm.

As the numerator, we use

the decimal number without

the decimal point,

whole part + decimal part

without dot.

As the denominator, we use

10, 100, 1000…, according to

the number of the decimal

places.

3 hundredths

means 100 as a

denominator

Whole part + decimal part

without decimal point.

Whole part

As the denominator: we use

9, 99, 999…, according to the

number of repeating digits.

As the numerator





Sometimes the repeating digits are not directly to the right of the

decimal point. For example ; then we have to follow the

algorithm below.

1.5 Irrational Numbers

A Piece of the

In December 2002, Yasumasa Kanada and others at the University of

Tokyo announced that they had calculated to decimal places, beating their previous record

set in 1999. Their computation of consumed more than 600 hours oftime on a Hitachi SR8000 supercomputer.

This record, like the record for the largest prime number, will most

likely be broken in the near future (it might already be broken as you

read this). Mathematicians and computer scientists continue to

improve both their computers and their methods used to find numbers

like the most accurate approximation for or the largest prime

number.

Whole part + decimal part

without decimal point.

Whole part and

non-repeating

decimal art

As the denominator: we use as

many nines as repeating digits after

the decimal point, and as many

zeroes as non-repeating digits.

As the numerator

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The formula is known as the Pythagorean theorem.

The School of Pythagoras found that the solution of the formula,

where and , is not a rational number.

There is no rational number that when squared will equal 2. This fact

prompted (caused) a need for a new set of numbers, the irrational

numbers.

The points on the number line that are not rational numbers are

referred to as irrational numbers. Recall that every rational number is

either a terminating or a repeating decinal number. Therefore,

Irrational numbers, when represented as decimal numbers, will be

non-terminating , non-repeating decimal numbers. They have an

unlimited amount of decimal digits.

Types of representing irrational numbers

A non-repeating decimal number such as can

be used to indicate an irrational number. Notice that no number

or set of numbers repeat on a continuous basis, and the three

dots at the end of the number indicate that the number

continues indefinitely.

Non-repeating number patterns can be used to indicate

irrational numbers. For example:

The square roots of some numbers are irrational:

√ √ √ √





Another important irrational numbers:

... Euler’s Number

Golden Ratio

2. Real Numbers

Now that we have discussed both the rational and the irrational numbers,

we can discuss the real numbers and the properties of the real number

system. The union of the rational numbers and the irrational numbers is the

set of real numbers, symbolized by .

The relationship between the various sets of numbers in the real number

system can be illustrated with a tree diagram.

{

{

√

2.1 The order on the Real Numbers

The set of real numbers is simply ordered. It obeys the following lawsof order:

For two given real numbers, a and b , exactly one of the following

relations is true:





2.2 Properties of the Real Number System

We are prepared to consider the properties of the real number system.

Properties Addition Multiplication

Closure:If an operation isperformed on any twoelements of a set andthe result is an elementof the set, we say that

the set is close underthat given operation.

Commutative

Associative Identity There exists a unique

real number 0 such that

There exists aunique realnumber 1 suchthat

Inverse For each real number ,there is a unique real

number – such that

For each nonzero

real number

,

there is a uniquereal number ⁄ such that

Distributive





2.3 The Real Number Line

The real numbers can be represented geometrically by a coordinate

axis called a real number line.

The number associated with a point on a real number line is called the

coordinate of the point. The point corresponding to zero is called the

origin. Every real number corresponds to a point on the number line,

and every point on the number line corresponds to a real number.

Irrational numbers can only be represented on the number line

approximately, but there are some exceptions:

We can represent certain square roots using the Pythagorean

Theorem graphically (using a compass and a ruler).

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2.4 Absolute Value

The absolute value of a real number a, denoted ||, is the distance

between a and zero on the number line.

A more formal definition of Absolute Value:

||

2.5 Interval Notation

When we want to refer to a particular subset of real numbers we use

interval notation. For example, if we want to indicate that the

solutions of an inequality are all the numbers between 1 and 7, we

need to use special notation: | } The table below explores the different possibilities:





3. Rounding and Error

3.1 Rounding Numbers

Rounding a number is another way of writing a number

approximately.

Quite often, an approximate answer is acceptable. Rounding gives

approximate answers. Rounding is very common for numbers in

everyday life, for example:

Populations are often expressed to the nearest million.

The number of people attending a pop concert may be

expressed to the nearest thousand.

Inflation may be expressed to the nearest whole

number, or the nearest tenth of a percentage.

There are several different methods for rounding, but here we will

only look at the common method, the one used by most people.

How to Round Numbers

Decide which is the last digit to keep

Leave it the same if the next digit is less than 5 (this is

called rounding down )

But increase it by 1 if the next digit is 5 or more (this is

called rounding up )

Bear in mind the following place value diagram





For example:

Any number can be rounded to a given number of decimal places

(written d.p.)

(Round up to 2 decimal places or to the nearest hundredths)

(Round down to 4 d.p.)

Any number can be rounded to a given number of significant figures

(written s.f.)

(Round up to the nearest thousand or 2 s.f.)

(Round down to 4 s.f. or to the nearest unit)

3.2 Estimation and Approximation

Estimation and approximation are important elements of the non-

calculator examination paper. You will be required to give an

estimation by rounding numbers to convenient approximations,

usually one significant figure.

For example:

If we have to estimate the value of

We would do the following approximation .

(Using a calculator, the actual answer is 1.9939407… so the estimated

answer is a good approximation)





3.3 Determining Rounding Error

Absolute and relative error are two types of error measurement. The

differences are important.

Absolute error is the difference between the Exact Value and theapproximate value.

| | Sometimes is impossible to know the exact value of a number, then

the Absolute Value depends on the approximation.

Relative error is the absolute error divided by the magnitude of the

exact value. The percent error is the relative error expressed in termsof per 100.

We need to know the value or the percentage of the relative error to

determine the accuracy of different measurements or approximations.

So it is a comparative tool.

As an example:

If the exact value is 50 and the approximation is 49.9, then the

absolute error is || and the relative error is

.

The relative error is often used to compare approximations of

numbers of widely differing size; for example, approximating the

number 1,000 with an absolute error of 3 is, in most applications,

much worse than approximating the number 1,000,000 with an

absolute error of 3; in the first case the relative error is 0.003 and in

the second it is only 0.000003.





Another example:

If you measure a beaker and read, 5mL.

If you know that the correct reading should have been 6mL.

Then, this means that your % error (Approximate error) would have

been

|| or 16.66666..% error.

unit 1 real numbers

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