Unit4

Indice Decimal expansion Reading decimal numbers Operations with decimals Approximating a quantity Exercises

Decimals

Matematicas 1o E.S.O.Alberto Pardo Milanes

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1 Decimal expansion

2 Reading decimal numbers

3 Operations with decimals

4 Approximating a quantity

5 Exercises

Alberto Pardo Milanes Decimals


Decimal expansion



Decimal expansion

What is the decimal expansion of a number?

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 of1

9is 0.1111 . . .Numbers can be placed to the left or right of a decimal point, toindicate values greater than one or less than one. The number tothe left of the decimal point is a whole number.



Decimal expansion

Rational numbers and irrationals

The decimal expansion of a number may terminate, becomeperiodic, or continue infinitely without repeating.

Rational numbers are numbers that are fractions. There are somenumbers that can be written as fraction, called Irrational numbers.

All rational numbers have either finite decimal expansions (finitedecimals) or repeating decimals.

However,irrational numbers, neither terminate nor become periodic(continue infinitely without repeating).



Decimal expansion

Finite decimalA finite decimal is a positive number that has a finite decimalexpansion.Example: 1/2 = 0.5 is a finite decimal.

Recurring decimalA decimal number is a repeating/recurring decimal if at somepoint it becomes periodic: there is some finite sequence of digitsthat is repeated indefinitely. The repeating portion of a decimalexpansion is conventionally denoted with a vinculum (a horizontalline placed above multiple quantities).Example: 1/3 = 0.33333333 · · · = 0. 3 is a recurring decimal.Note that there are repeating decimals that begin with anon-repeating part.Example: 1/30 = 0.03333333 · · · = 0.03 is a recurring decimal thatbegin with a non-repeating part.



Decimal expansion

IrrationalsThe decimal expansion of an irrational number never repeats orterminates.Example: π =3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129 . . .is an irrational.



Decimal expansion




Decimal expansion




Decimal expansion




Decimal expansion




Decimal expansion




Decimal expansion




Decimal expansion




Decimal expansion




Decimal expansion




Decimal expansion




Decimal expansion




Reading decimal numbers




When reading and writing decimals take note of the correct placeof the last digit in the number. A decimal point means “and”.Remember that the value of a digit depends on its place orposition in the number. Look at the names of the different placesof a figure (Place underlined - name of position)7,654,321.234567 Millions7,654,321.234567 Hundred thousands7,654,321.234567 Ten thousands7,654,321.234567 Thousands7,654,321.234567 Hundreds7,654,321.234567 Tens7,654,321.234567 Ones (units) position7,654,321.234567 Tenths7,654,321.234567 Hundredths7,654,321.234567 Thousandths7,654,321.234567 Ten thousandths7,654,321.234567 Hundred Thousandths

7,654,321.234567 Millionths




Examples:Look at the following examples to learn how to read decimalnumbers:321.7 → Three hundred twenty-one and seven tenthsor three hundred twenty-one point seven5,062.57 → Five thousand sixty-two and fifty-seven hundredthsor five thousand sixty-two point five seven43.27 → Forty-three point two seven0 → Zero5.07 → Five point oh seven0.0305 → Nought point oh three oh fiveor point oh three oh fivee4.67 → Four euros and sixty-seven centsor Four euros sixty-seven5o → Five Celsius degrees3.4 → Three point four recurring



Operations with decimals




Adding and subtracting

Addition and subtraction of decimals is like adding and subtractingwhole numbers. The only thing we must remember is to line up theplace values correctly.Examples:

To add 12.35 + 5.287:

1 2 .3 5+ 5 .2 8 7

1 7 .6 3 7

To subtract 12.993− 2.28 :

1 2 .9 9 3- 2 .2 8

1 0 .7 1 3




Multiplying and dividing

When multiplying numbers with decimals, we first multiply themas if they were whole numbers. Then, the placement of the numberof decimal places in the result is equal to the sum of the number ofdecimal places of the numbers being multiplied.

Example: To multiply 2.81 by 3.1:

2 .8 1× 3 .1

2 8 18 4 3

8 .7 1 1Division with decimals is easier to understand if the divisor is awhole number. In this case, when the decimal point appears in thedividend, we put it on the divisor.

Example: To divide 3.42 by 5:3 4. 2 /50 4 2 6. 8

2Alberto Pardo Milanes Decimals



Multiplying and dividing

If the divisor has a decimal in it, we can make it a whole numberby moving the decimal point the appropriate number of places tothe right. If you move the decimal point to the right in the divisor,you must also do this for the dividend.Example: To divide 13.34 by 3.2 we divide 133.4 by 32.



Approximating a quantity





Rounding off and truncating a decimal are techniques used toestimate or approximate a quantity. Instead of having a long stringof figures, we can approximate the value of the decimal to aspecified decimal place.

TruncatingTo truncate a decimal, we leave our last decimal place as it isgiven and discard all digits to its right.Example:Truncate 123,235.23 to the tens place:123,230.Truncate 123,235.23 to the tenth:123,235.2





Rounding offAfter 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), then we discard all digits to its right.To round off a decimal look at the digit to the right of the placebeing rounded:• 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 we arerounding (rounding up).• After rounding, discard all digits to the right of the place we arerounding.Examples:Round 123,235.23 to the tens place:123,240 we are rounding up.Round 123,234.23 to the tens place:123,230 we are rounding down.



Exercises



Exercises

Exercise 1

We know that 234 · 567 = 132,678. Find 2.34 · 5.67:



Exercises

Exercise 1

We know that 234 · 567 = 132,678. Find 2.34 · 5.67:

2.34 · 5.67 = 13.2678



Exercises

Exercise 2

Carmen earns e4.60 an hour working part-time as a private tutor.Last week she worked 6 hours. How much money did Carmenearn?



Exercises

Exercise 2


Data: She earns:e4.60 an hourLast week she worked 6 hours.

4.60 · 6 = 27.60

Answer: Carmen earns e27.60working part-time as a privatetutor.



Exercises

Exercise 2



4.60 · 6 = 27.60




Exercises

Exercise 2



4.60 · 6 = 27.60




Exercises

Exercise 3

What is the cost of 3 pounds of jellybeans if each pound costse2.30?



Exercises

Exercise 3


Data: Each pound costs e2.30.

3 · 2.30 = 6.90

Answer: 3 pounds of jellybeans coste6.90.



Exercises

Exercise 3



3 · 2.30 = 6.90




Exercises

Exercise 3



3 · 2.30 = 6.90




Exercises

Exercise 3



3 · 2.30 = 6.90




Exercises

Exercise 4

The length of a swimming pool is 16 feet. What is the length ofthe pool in yards? What is the length of the pool in meters?(Note 1 yard=3 feet=0.9144 meters).



Exercises

Exercise 4


Data: The length is 16 feet.1 yard=3 feet=0.9144 meters

16 : 3 = 5.333333(16 : 3) · 0.9144 = 5.333333 · 0.9144 = 4.8768

Answer: The length of thepool is 5.333333 yards.The length of the pool is4.8768 meters.



Exercises

Exercise 4



16 : 3 = 5.333333(16 : 3) · 0.9144 = 5.333333 · 0.9144 = 4.8768




Exercises

Exercise 4



16 : 3 = 5.333333(16 : 3) · 0.9144 = 5.333333 · 0.9144 = 4.8768




Exercises

Exercise 4



16 : 3 = 5.333333(16 : 3) · 0.9144 = 5.333333 · 0.9144 = 4.8768




Exercises

Exercise 5

The highest point in Alabama is Cheaha Mountain. It stands just abit higher than 730 meters. What is this elevation in miles?(Note 1 km=5/8 miles)



Exercises

Exercise 5


Data: Cheaha Mountain is 730 meters high.1 km=5/8 miles

730 m = 0.73 km 0.73 · 5 = 3.65

3.65 : 8 = 0.45625

Answer: Cheaha Mountainis 0.45625 miles high.



Exercises

Exercise 5



730 m = 0.73 km 0.73 · 5 = 3.65

3.65 : 8 = 0.45625




Exercises

Exercise 5



730 m = 0.73 km 0.73 · 5 = 3.65

3.65 : 8 = 0.45625




Exercises

Exercise 5



730 m = 0.73 km 0.73 · 5 = 3.65

3.65 : 8 = 0.45625




Exercises

Exercise 6

Round 7.601 to the nearest whole number:Truncate 68.94 to the tenth:Round 68.94 to the nearest tenth:Truncate 125.396 to the hundredth:Round 125.396 to the nearest hundredth:



Exercises

Exercise 6

Round 7.601 to the nearest whole number:7.601 ' 8Truncate 68.94 to the tenth:68.94 ' 68.9Round 68.94 to the nearest tenth:68.94 ' 68.9Truncate 125.396 to the hundredth:125.396 ' 125.39Round 125.396 to the nearest hundredth:125.396 ' 125.40



Exercises

Exercise 6




Exercises

Exercise 6




Exercises

Exercise 6




Exercises

Exercise 6




Exercises

Exercise 7

A can of beans costs e0.0726 per ounce. To the nearest cent, howmuch does an ounce of beans cost? How much does ten ounces ofbeans cost?



Exercises

Exercise 7


Data: A can of beans costs e0.0726 per ounce.

0.0726 ' 0.070.0726 · 10 = 0.726 ' 0.72

Answer: To the nearestcent, an ounce of beanscost e0.07, ten ounces ofbeans e0.72.



Exercises

Exercise 7



0.0726 ' 0.070.0726 · 10 = 0.726 ' 0.72




Exercises

Exercise 7



0.0726 ' 0.070.0726 · 10 = 0.726 ' 0.72



Unit4

Education

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repeatingrecurring decimal

decimal numbers3 operations

recurring decimal thatbegin

irrational numbers

repeating decimals

positive number

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