Unit 1 Real Numbers (4º ESO)

8/3/2019 Unit 1 Real Numbers (4 ESO)

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Unit 1: Real Numbers Mathematics 4 ESO Option B

IES Albayzn (Granada) Pgina 2

1.2 Whole Numbers

Another important set of numbers, the whole numbers , help toanswer the question, How many?

=0,1,2,3,4, Note that the set of whole numbers contains the number 0 but that theset of counting numbers does not. If a student were asked how manybooks have read this term, the answer would be a whole number. Ifthe 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 12F and drops 20 , the resulting temperature is 8F. This type of problem shows the need for negative numbers. Theset of integers consists of the negative integers, 0, and the positiveintegers.

=4,3,2,1,0

The term positive integers is yet another name for the naturalnumbers.

We can obtain an image of integer numbers by representing them ona number line .

To construct the number line, arbitrarily select a point for zero to serveas 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 linecontinues indefinitely in both directions.


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Note that for any natural number, n , on the number line, the oppositeof 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 chapterare >and is read is greater than , and the symbol


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number above the fraction line is called the numerator, and expressesthe 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:

56=0.833...105=2 254=6.2537=0.42857 When the numerator and denominator have a common divisor, we canreduce the fraction to its lowest terms (or simplest form):

618=39==0.3333 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 0.

Two fractions are said to be equivalent when simplifyingboth of them produces the same fraction, which cannot befurther 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 onthe number line.

We can test if two fractions are equivalent by cross-multiplying (or cross-product ) their numerators anddenominators.


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Proper and Improper Fractions. Mixed Numbers.

Rational numbers less than 1 or greater than -1 are represented byproper fractions . A proper fraction is a fraction whose numerator isless than its denominator:

47 511 Consider the number 2. It is an example of a mixed number . It iscalled a mixed number because it consists of an integer, 2, and a

fraction , and it is equal to 2+. The mixed number 4 means 4+.

Rational numbers greater than 1 or less than -1 that are not integersmay be represented as mixed numbers, or as improper fractions . Animproper fraction is a fraction whose numerator is greater than itsdenominator.

=2+34 Fractions and Decimal Numbers

a) Converting Fractions to Decimal Numbers

To obtain the decimal number which is related to a fraction we onlyhave to divide the numerator by the denominator:

4033=1.2121


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Note the following important property of the rational numbers.

Every rational number, every fraction, when expressed as adecimal number will be either a terminating or a recurring (orrepeating) decimal number.

DecimalNumbers

Decimal Part The fraction in its lowestterms

Terminating or Exact

3.75 It Does not go onforever.

You can writedown all its digits

If the only factors of thedenominator are 2 or 5 orcombinations of 2 and 5 thenthe fraction will be aterminating decimal.

4512=154=152=3.75 Recurring

orRepeating

0.555 2.0666

It Does go on

forever.

It Repeats a blockof digits.

If the denominator hasnt any

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

2545=59=53=0.555 If the denominator has anyfactors other than 2 and/or 5then the fraction will be a

recurring decimal.6230=3115=3135=2.066


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b) Converting Decimal Numbers to Fractions

To convert a terminating decimal number to a quotient ofintegers, we observe the last digit to the right of the decimal point.The position of this digit will indicate the denominator of thequotient of integers. The numerator will be the decimal numberwithout the decimal point.

Converting a repeating decimal number to a quotient of integersis more difficult than converting a terminating decimal.

When the repeating digits are directly to the right of the decimalpoint, as the number 6.21, we use the following algorithm.

4.8=483100

As the numerator , we usethe decimal number withoutthe decimal point,

whole part + decimal partwithout dot.

As the denominator , we use10, 100, 1000, according tothe number of the decimalplaces.

3 hundredths means 100 as adenominator

6.21=621699

Whole part + decimal partwithout decimal point.

Whole part

As the denominator : we use9, 99, 999, according to thenumber of repeating digits.

As the numerator


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Sometimes the repeating digits are not directly to the right of thedecimal point. For example 12.142; then we have to follow thealgorithm below.

1.5 Irrational Numbers

A Piece of the

In December 2002, Yasumasa Kanada and others at the University ofTokyo 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 mostlikely be broken in the near future (it might already be broken as youread this). Mathematicians and computer scientists continue toimprove both their computers and their methods used to find numberslike the most accurate approximation for or the largest primenumber.

12.142=121421214900

Whole part + decimal partwithout decimal point.

Whole part and

non-repeatingdecimal art

As the denominator : we use asmany nines as repeating digits afterthe decimal point, and as manyzeroes 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 =1and =1, is not a rational number.

+= 1+1= 2= There is no rational number that when squared will equal 2. This factprompted (caused) a need for a new set of numbers, the irrationalnumbers .

The points on the number line that are not rational numbers arereferred to as irrational numbers. Recall that every rational number iseither a terminating or a repeating decinal number. Therefore,

Irrational numbers , when represented as decimal numbers, will benon-terminating , non-repeating decimal numbers. They have anunlimited amount of decimal digits.

Types of representing irrational numbers

A non-repeating decimal number such as 5.1263953 canbe used to indicate an irrational number. Notice that no numberor set of numbers repeat on a continuous basis, and the threedots at the end of the number indicate that the numbercontinues indefinitely.

Non-repeating number patterns can be used to indicate

irrational numbers. For example:6.101101110111 0.525225222 7.2468101214 The square roots of some numbers are irrational :

2, 3, 5, 24


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Another important irrational numbers :

=3.1415926535897 =2.7182818284590

... Eulers Number

=1.6180339887498Golden 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 numbersystem. 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 numbersystem can be illustrated with a tree diagram.

() (

)

5

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 followingrelations is true:

< , = , >


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2.2 Properties of the Real Number System

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

Properties

, , Addition MultiplicationClosure: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

+0=0+ = There exists aunique realnumber 1 suchthat

1=1 Inverse For each real number ,

there is a unique realnumber such that+()=()+ =0

For each nonzero real number ,there is a uniquereal number 1 such that

= =1 Distributive ( +)= +


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2.3 The Real Number Line

The real numbers can be represented geometrically by a coordinateaxis 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 theorigin . 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 lineapproximately, but there are some exceptions:

We can represent certain square roots using the PythagoreanTheorem 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 distancebetween a and zero on the number line.

A more formal definition of Absolute Value:

| |=


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The table below explores the different possibilities:

3. Rounding and Error

3.1 Rounding Numbers

Rounding a number is another way of writing a numberapproximately.

Quite often, an approximate answer is acceptable. Rounding givesapproximate answers. Rounding is very common for numbers ineveryday life, for example:

Populations are often expressed to the nearest million.

The number of people attending a pop concert may beexpressed to the nearest thousand.

Inflation may be expressed to the nearest whole

number, or the nearest tenth of a percentage.


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There are several different methods for rounding , but here we willonly 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 iscalled rounding down )

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

. . (Round down to 4 d.p.)


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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 anestimation 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

=2.

(Using a calculator, the actual answer is 1.9939407 so the estimatedanswer is a good approximation)

3.3 Determining Rounding Error

Absolute and relative error are two types of error measurement. Thedifferences 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, thenthe Absolute Value depends on the approximation.


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Relative error is the absolute error divided by the magnitude of theexact 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 todetermine 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 |5049.9|=0.1and the relative error is .=0.002. The relative error is often used to compare approximations ofnumbers of widely differing size; for example, approximating thenumber 1,000 with an absolute error of 3 is, in most applications,

much worse than approximating the number 1,000,000 with anabsolute error of 3; in the first case the relative error is 0.003 and inthe 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 havebeen

| |== 0.16666 or 16.66666..% error.

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Unit 1 Real Numbers (4º ESO)