Decimals, rationals, irrationals, rounding off, truncating

1. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Decimals
Matematicas 2o E.S.O.
Alberto Pardo Milanes
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2. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
1 The set of reals
2 Decimals
3 Reading real numbers
4 Multiplying and dividing by 10, 100, 1000, etc
5 Approximating
6 Exercises
Alberto Pardo Milanes Decimals

3. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
The set of reals
Alberto Pardo Milanes Decimals

4. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
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 all
called integers.
A rational number is a number that can be expressed as a fraction
p=q where p and q are integers and q6= 0. The set of rational
numbers is named Q.
There are numbers that can't be expressed as a fraction. An
irrational number is a number that can't be expressed as a fraction
p=q for any integers p and q.
p
Example: , ,
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 of
real numbers.
Alberto Pardo Milanes Decimals

5. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Decimals
Alberto Pardo Milanes Decimals

6. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Decimals
The decimal expansion of a number is its representation in the
decimal system.
Example: the decimal expansion of 252 is 625, of is 3.14159 : : : ,
and of 1=9 is 0.1111 : : :
The decimal expansion of a number may terminate, become
periodic or continue in

7. nitely without repeating.
A

8. nite decimal (or terminating decimal) is a number that has a

9. nite decimal expansion.
Example: 1=8 = 0.125.
Alberto Pardo Milanes Decimals

10. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Decimals
A decimal number is a repeating decimal if at some point it
becomes periodic: there is some

11. nite sequence of digits that is
repeated inde

12. nitely. The repeating portion of a decimal expansion
is conventionally denoted with a vinculum (a horizontal line placed
above multiple quantities).
Example: 5=3 = 1;66666666 = 1.6, read it as one point six
recurring.
Note the possibility of repeating decimals that begin with a
non-repeating part.
Example: 61=30 = 2;03333333 = 2.03, read it as two point
zero, three recurring.
Alberto Pardo Milanes Decimals

13. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Decimals
Irrational numbers have decimal expansions that neither terminate
nor become periodic.
Example: = 3. 14159265358979323846264338327950288419716
9399375105820974944592307816406286208998628034825342117
0679821480865132823066470938446095505822317253594081284
8111745028410270193852110555964462294895493038196442881
0975665933446128475648233786783165271201909145648566923
4603486104543266482133936072602491412737245870066063155
8817488152092096282925409171536436789259036001133053054
8820466521384146951941511609433057270365759591953092186
117381932611793105118548074462379962 : : :
A fraction in lowest terms with a prime denominator other than 2
or 5 always produces a repeating decimal.
Alberto Pardo Milanes Decimals

14. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Reading real numbers
Alberto Pardo Milanes Decimals

15. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Reading real numbers
Remember that the value of a digit depends on its place or
position in the number and the decimal point shows where the
fractional part of a number begins. Dierent places of a

16. gure
gives dierent names:
Examples:
Billions
hundred-millions
ten-millions
millions
hundred-thousands
ten-thousands
thousands
hundreds
tens
units
tenths
hundredths
thousandths
ten-thousandths
hundred-thousandths
millionths
ten-millionths
hundred-millionths
1 6 1 8 0 3 3 9 8 8 . 7 4 9 8 9 4 8 4
In 42.5 !

17. ve are the tenths and four are the tens.
In 3,267.2558 ! three are the thousands and eight are the ten
thousandths.
In 2,656,711.3 ! two are the millions.
Alberto Pardo Milanes Decimals

18. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Reading real numbers
Look at the following examples to learn how to read decimal
numbers:
Examples:
321.7 ! Three hundred twenty-one and seven tenths.
5,062.57 ! Five thousand sixty-two and

19. fty-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.
Alberto Pardo Milanes Decimals

20. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Multiplying and dividing by 10,
100, 1000, etc
Alberto Pardo Milanes Decimals

21. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Multiplying and dividing by 10, 100, 1000, etc
When you multiply a number by 10 the digits move one place to
the left, making the number bigger.
Example: 1.414213 10 = 14.14213
When you multiply a number by 100 the digits move two places to
the left, when you multiply a number by 1000 the digits move
three places to the left,. . . When you divide a number by 10 the
digits move one place to the right, making the number smaller,
when you divide a number by 100 the digits move two places to
the right, when you divide a number by 1000 the digits move three
places to the right,. . .
Examples: 63.256 100 = 6325.6 68.63 : 10 = 6.863
1234.5 : 100 = 12.345
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22. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Approximating
Alberto Pardo Milanes Decimals

23. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Approximating
Rounding o and truncating a decimal are techniques used to
estimate or approximate a quantity. Instead of having a long string
of

24. gures, we can approximate the value of the decimal to a
speci

25. ed decimal place.
(Truncating)
To truncate a decimal, we leave our last decimal place as it is
given and discard all digits to its right.
Examples:
Truncate 123,237.23 to the tens place: 123,230.
Truncate 35.77 to euros: 35 euros.
Truncate 1.123 to the tenths: 1.1
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26. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Approximating
(Rounding o)
After rounding o, the digit in the place we are rounding will either
stay the same (referred to as rounding down) or increase by 1
(referred to as rounding up). To round o a decimal

27. rst

28. nd the
rounding place, then look at the digit to the right of the place
being rounded and:
If the digit is 4 or less, the

29. gure in the place we are rounding
remains the same (rounding down).
If the digit is 5 or greater, add 1 to the

30. gure in the place we
are rounding (rounding up).
After rounding, discard all digits to the right of the place we are
rounding.
Alberto Pardo Milanes Decimals

31. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Approximating
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 rounding
down.
Round 45.79 to the nearest euro: 46 we are rounding up.
Alberto Pardo Milanes Decimals

32. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Alberto Pardo Milanes Decimals

33. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 1
A mint costs 5 euro cents. How much would a roll of ten mints
cost?
Alberto Pardo Milanes Decimals

34. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
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 .73
m farther than Samantha, but 1.2 m less than Mary. How many
meters did Mary jump?
Alberto Pardo Milanes Decimals

35. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 3
A peach costs .62 euros. How much would two dozen peaches cost?
Alberto Pardo Milanes Decimals

36. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
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:
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37. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 5
Mike swims 50 m every 32.54 seconds. Based on rounding,
estimate the time he needs to swim 1000 m.
Alberto Pardo Milanes Decimals

38. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 6
Anna bought 12 CDs with the same price each. The total cost was
255.00 euros. What was the price of each CD?
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39. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 7
Kate earns 8.60 euros per hour working part-time. She worked a
total of 15.7 hours one week. How much money did She earn?
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40. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 8
Salmon cost 4.31 euros per kg at the supermarket. What is the
price for a 800 g piece of salmon?
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41. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 9
A can of mushrooms costs .0361 euros per ounce. To the nearest
cent, how much does an ounce of mushrooms cost?
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42. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 10
During the month of February, Alfred spent 14.78 euros on gas
the

43. rst week, 15.35 euros during the second week, 15.94 euros
during the third week, and 14.07 euros during the fourth week.
Which is closest to the total amount of money Alfred spent on
gasoline during February?
35 euros 50 euros 60
euros 100 euros
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44. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 11
The length of a new swimming pool being built at the community
recreation center is listed as 26 feet. What is the length of the new
pool in yards? What is the length of the new pool in meters? (Note
1 yard=3 feet=0.9144 meters).
Alberto Pardo Milanes Decimals

45. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 12
The speed of light is about 186,282 miles per second. Earth is
about 92,976,000 miles from the sun. How long does it takes the
suns light to reach the Earth to the nearest hundredth of a
minute?
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46. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 13
Renting a hall for a meeting will cost 5,000 euros. Each person
attending the meeting will be charged with 20.75 euros. How
many people will have to attend if you want all the expenses to be
covered?
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47. Index The set of reals Decimals Reading real numbers : by 10, 100, 1000, etc Approximating a quantity Exercises
Exercises
Exercise 14
A circle is a set of points that are a given distance from a given
point, the centre. Circumference is the distance around a circle.
The diameter of a circle is the distance across a circle, through its
center. If you divide the circumference of a circle by its diameter
the result is always the number . Find a circle at home (a
window, a lamp, a round table,. . . ), measure its circumference and
its diameter and write it down. Divide the circumference of your
circle by its diameter and round the result to the nearest
hundredth. will appear!
Alberto Pardo Milanes Decimals