Decimals

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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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

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 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.

Decimals

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

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

nite decimal expansion.
Example: 1=8 = 0.125.

Decimals
A decimal number is a repeating decimal if at some point it
becomes periodic: there is some

nite sequence of digits that is
repeated inde

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.

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.

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

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 !

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.

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

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.

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 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

Approximating

Approximating
Rounding o and truncating a decimal are techniques used to
estimate or approximate a quantity. Instead of having a long string
of

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

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

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

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

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

gure in the place we
are rounding (rounding up).
After rounding, discard all digits to the right of the place we are
rounding.

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.

Exercises

Decimals

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