Decimals

Index The set of reals Decimals Reading real numbers Powers of ten Approximating a quantity Exercises

Decimals

Matematicas 2o E.S.O.Alberto Pardo Milanes

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1 The set of reals

2 Decimals

3 Reading real numbers

4 Multiplying and dividing by 10, 100, 1000, etc

5 Approximating a quantity

6 Exercises

Alberto Pardo Milanes Decimals


The set of reals



The set of reals

Sets of numbers

IN is the set of natural numbers.The set Z of natural numbers, negative numbers, and zero are allcalled integers.A rational number is a number that can be expressed as a fractionp/q where p and q are integers and q 6= 0. The set of rationalnumbers is named Q.There are numbers that can’t be expressed as a fraction. Anirrational number is a num-ber that can’t be expressed as a fraction p/q for any integers p and q.

Example: π, φ,√2, . . . can’t be expressed as a fraction as they

are irrational numbers.The set R of rational and irrational numbers is named the set ofreal numbers.



Decimals



Decimals

Decimal expansion

The decimal expansion of a number is its representation in thedecimal system.

Example: the decimal expansion of 252 is 625, of π is 3.14159 . . . ,and of 1/9 is 0.1111 . . .

The decimal expansion of a number may terminate, becomeperiodic or continue infinitely without repeating.A finite decimal (orterminating decimal) is a number that has a finite decimal expansion.

Example: 1/8 = 0.125.



Decimals

Decimal expansion

A decimal number is a repeating decimal if at some point itbecomes periodic: there is some finite sequence of digits that isrepeated indefinitely. The repeating portion of a decimal expansionis conventionally denotedwith a vinculum (a horizontal line placed above multiple quantities).

Example: 5/3 = 1,66666666 · · · = 1.6, read it as one point sixrecurring.

Note the possibility of repeating decimals that begin with anon-repeating part.

Example: 61/30 = 2,03333333 · · · = 2.03, read it as two pointzero, three recurring.



Decimals

Decimal expansion

Irrational numbershave decimal expansions that neither terminate nor become periodic.

Example: π = 3. 14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535948128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962 . . .

A fraction in lowest terms with a prime denominator other than 2or 5 always produces a repeating decimal.



Reading real numbers




Remember that the value of a digit depends on its place orposition in the number and the decimal point shows where thefractional part of a number begins. Different places of a figuregives different names:

Examples:

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In 42.5 → five are the tenths and four are the tens.In 3,267.2558 → three are the thousands and eight are the tenthousandths.In 2,656,711.3 → two are the millions.Look at the following examples to learn how to read decimalnumbers:

Examples:321.7 → Three hundred twenty-one and seven tenths.5,062.57 → Five thousand sixty-two and fifty-seven hundredths.43.27 → Forty-three point two seven.$4.76 → Four dollars and sixty-seven cents.3.42 → Three point forty-two recurring.12.37 → Twelve point three, seven recurring.



Multiplying and dividing by 10,100, 1000, etc



Multiplying and dividing by 10, 100, 1000, etc

When you multiply a number by 10 the digits move one place tothe left, making the number bigger.

Example: 1.414213 · 10 = 14.14213

When you multiply a number by 100 the digits move two places tothe left, when you multiply a number by 1000 the digits movethree places to the left,. . . When you divide a number by 10 thedigits move one place to the right, making the number smaller,when you divide a number by 100 the digits move two places tothe right, when youdivide a number by 1000 the digits move three places to the right,. . .

Examples:63.256 · 100 = 6325.668.63 : 10 = 6.8631234.5 : 100 = 12.345



Approximating a quantity




Rounding off and truncating a decimal are techniques used toestimate or approximate a quantity.Instead of having a long string of figures, we can approximate thevalue of the decimal to a specified decimal place.To truncate a decimal, we leave our last decimal place as it isgiven and discard all digits to its right.

Examples:Truncate 123,237.23 to the tens place: 123,230.Truncate 35.77 euros to euros: 35 euros.Truncate 1.123 to the tenths: 1.1




After rounding off, the digit in the place we are rounding will eitherstay the same (referred to as rounding down) or increase by 1(referred to as rounding up). To round off a decimal first find therounding place, then look at the digit to the right of the placebeing rounded and:

• If the digit is 4 or less, the figure in the place we are roundingremains the same (rounding down).

• If the digit is 5 or greater, add 1 to the figure in the place weare rounding (rounding up).

After rounding, discard all digits to the right of the place we arerounding.

Examples:Round 123,237.23 to the tens place:123,240 we are rounding up.Round 123,234.23 to the tens place:123,230 we are roundingdown.Round 45.79 euros to the nearest euro: 46 we are rounding up.



Exercises



Exercises

Exercise 1

A mint costs .05 euros. How much would a roll of ten mints cost?



Exercises

Exercise 2

Lucy won a long-jump contest with a distance of 6.45 m.Samantha jumped 2.02 m less than Lucy, while Jenny jumped .73m farther than Samantha, but 1.2 m less than Mary. How manymeters did Mary jump?



Exercises

Exercise 3

A peach costs .62 euros. How much would two dozen peaches cost?



Exercises

Exercise 4

Round 7.601 to the nearest whole number:Round 68.94 to the nearest tenth:Round 1.25396 · 100 to the nearest hundredth:Truncate 1.787 to a whole number:Truncate 2.24 to a tenth:Truncate 2585.2 : 100 to a hundredth:



Exercises

Exercise 5

Mike swims 50 m every 32.54 seconds. Based on rounding,estimate the time he needs to swim 1000 m.



Exercises

Exercise 6

Anna bought 12 CDs with the same price each. The total cost was255.00 euros. What was the price of each CD?



Exercises

Exercise 7

Kate earns 8.60 euros per hour working part-time. She worked atotal of 15.7 hours one week. How much money did She earn?



Exercises

Exercise 8

Salmon cost 4.31 euros per kg at the supermarket. What is theprice for a 800 g piece of salmon?



Exercises

Exercise 9

A can of mushrooms costs .0361 euros per ounce. To the nearestcent, how much does an ounce of mushrooms cost?



Exercises

Exercise 10

During the month of February, Alfred spent 14.78 euros on gas thefirst week, 15.35 euros during the second week, 15.94 euros duringthe third week, and 14.07 euros during the fourth week. Which isclosest to the total amount of money Alfred spent on gasolineduring February?35 euros 50 euros 60 euros 100 euros



Exercises

Exercise 11

The length of a new swimming pool being built at the communityrecreation center is listed as 26 feet. What is the length of the newpool in yards? What is the length of the new pool in meters? (Note1 yard=3 feet=0.9144 meters).



Exercises

Exercise 12

The speed of light is about 186,282 miles per second. Earth isabout 92,976,000 miles from the sun. How long does it takes thesun´s light to reach the Earth to the nearest hundredth of aminute?



Exercises

Exercise 13

Renting a hall for a meeting will cost euros 5,000. Each personattending the meeting will be charged with euros 20.75. How manypeople will have to attend if you want all the expenses to becovered?



Exercises

Exercise 14

A circle is a set of points that are a given distance from a givenpoint, the centre. Circumference is the distance around a circle.The diameter of a circle is the distance across a circle, through itscenter. If you divide the circumference of a circle by its diameterthe result is always the number π. Find a circle at home (awindow, a lamp, a round table,. . . ), measure its circumference andits diameter and write it down. Divide the circumference of yourcircle by its diameter and round the result to the nearesthundredth. π will appear!


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Decimals