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| Decimal
Numbers |
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Decimal representation of rational
numbers or fractions
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Expanded form of
decimal number, decimal fractions
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Whole number part or integer part (integral part) and decimal
places |
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Terminating decimals
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Recurring decimals (Infinite
decimals, period) |
| Purely recurring decimals
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|
Mixed recurring decimals
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Converting decimal number to a
fraction |
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Converting terminating decimal to a fraction |
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Converting the purely recurring decimal
to a fraction |
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Converting the mixed recurring decimal
to a fraction |
|
Adding and subtracting decimals |
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Multiplying and dividing decimals
by power of ten
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Place
values and place holders
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Multiplying and dividing decimals
by decimal units
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Decimal numbers scientific notation
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Multiplication of decimal numbers
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Division of decimal numbers |
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| Decimal representation of rational
numbers or fractions
|
| In order to convert a rational number represented as a fraction into
decimal form, one may use long division. |
| By dividing the numerator
by the denominator we get a terminating or a recurring decimal. |
| If the final remainder is 0 the quotient is a whole number or a finite
or terminating decimal, i.e. a decimal with |
| a limited number of digits after the decimal point. |
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|
| Sometimes when dividing, the division will never stop as there is
always a remainder. |
| These fractions convert to a recurring,
(periodic or infinite) decimals, and they have an unlimited number
of digits after the decimal point. |
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| A rational number is either a terminating decimal or recurring
decimal. |
| We can determine which fraction will convert to terminating
decimal and which to recurring decimal only if the |
| given fraction is
expressed in its lowest terms, that is, when its numerator and the
denominator have no |
| common factor other than 1. |
|
| Expanded form of decimal number,
decimal fractions
|
| The integer and fractional parts of a decimal number are separated
by a decimal point. |
| Converting decimals to fractions involves
counting the number of places to the right of the decimal point. |
| This
will give the corresponding place value, which then determines the
number of zeros that will be used in |
| forming the denominator. |
| The
numerator of the fraction equivalent is the number without the decimal point, and the denominator is 1 |
| followed by the number of
zeros corresponding to the number of decimal places. |
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| Terminating decimals
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| The
irreducible fractions, i.e., the vulgar fractions in lowest terms
whose the only prime factors in a
|
| denominator are 2 and/or 5
can be converted to terminating decimals. |
| That is, the terminating
decimals represent rational numbers whose fractions in the lowest
terms are of the |
| form a/(2n
· 5m). |
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| Examples: |
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 |
| |
|
 |
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| Recurring decimals (Infinite decimals, period) |
| Purely recurring decimals |
| The irreducible fractions, i.e., the vulgar fractions in lowest terms
whose prime factors in the denominator are |
| other than 2 or 5, that is,
the prime numbers from the sequence (3, 7, 11, 13, 17, 19, ...) convert to the |
| purely recurring decimals, i.e., the decimals which
start their recurring cycle immediately after the decimal |
| point. |
|
| Examples: |
|
 |
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|
 |
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| Mixed recurring decimals
|
| The irreducible fractions, i.e., the vulgar fractions in lowest terms
whose denominator is a product of 2's |
| and/or 5's besides the prime
numbers from the sequence (3, 7, 11, 13, 17, 19, ...) convert to the
mixed |
| recurring decimals, i.e., the decimals that have some extra
digits before the repeating sequence of digits. |
| The repeating
sequence may consist of just one digit or of any finite number of
digits. The number of digits in |
| the repeating pattern is called the
period. All recurring decimals are infinite decimals. |
|
| Examples: |
|
 |
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|
 |
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| All fractions can be written either as terminating decimals or as
recurring/repeating decimals. |
|
| Converting decimal number to a
fraction |
| Converting
terminating decimal to a fraction |
| Every terminating decimal and recurring decimal can be
converted to a fraction a/b,
a Î
Z,
b
Î
N. |
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| Examples: |
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 |
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 |
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| Converting the purely recurring decimal
to a fraction |
| When converting the purely recurring decimal less than one to
fraction, write the group of repeating digits to |
| the numerator, and to
the denominator of the equivalent fraction write as much 9’s as is
the number of digits |
| in the repeating
pattern. |
|
| Examples: |
|
 |
|
|
| Converting the mixed recurring decimal
to a fraction |
| When converting the mixed recurring decimal less than one to
fraction, write the difference between the |
| number formed by the
entire sequence of digits, including the digits of the recurring part,
and the number |
| formed only by the digits of the non-recurring pattern to its numerator. |
| To the denominator of the equivalent
fraction write as much 9’s as is the number of digits in the repeating |
|
pattern and add as much 0’s as is the number of digits in the non-recurring pattern. |
|
| Examples: |
|
 |
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|
 |
|
|
| Adding and subtracting decimals |
| To add or subtract decimals, insert zeros in empty decimal place
values so that all of the numbers have the |
| same number of decimal
places. |
| Write the corresponding decimal places one under another,
i.e., the decimal points should be vertically |
| aligned, and then follow
the rules for adding or subtracting whole numbers. |
| Line up the
decimal point in the answer. |
|
| Examples: |
|
a)
7.9 + 12 + 0.147 =
20.047,
b)
215.04 + 3.756 + 17.3 = 236.096 |
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|
7.900
215.040 |
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|
12.000
3.756 |
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|
+
0.147
+ 17.300 |
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|
20.047
236.096 |
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c)
41.56
− 9.178 =
32.382,
d)
5.08
− 0.9937 = 4.0863 |
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|
41.560
5.0800 |
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|
−
9.178
−
0.9937 |
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32.382
4.0863 |
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|
| Multiplying and dividing decimals
by power of ten
|
| Place
values and place holders
|
| To the left of the decimal point, the digits represent 1's, 10's, 100's,
1000's, and so on. |
| Digits to the right of the decimal point represent
1/10's, 1/100's, 1/1000's, and so forth. |
| To find the value of a decimal
place, we divide the value of the decimal place to the left of it by 10. |
| The place value of each column is one tenth of the place value of
the column to its left. Or, we can say that |
| each decimal unit contains
10 units of the lower place value thus, |
| 0.1 = 10 · 0.01,
0.01 = 10 · 0.001, and so on. |
| |
| Multiplying and dividing decimals
by decimal units
|
| To multiply decimals by powers of
10, such as 10, 100, 1000 etc. move the decimal point right as many
|
| places as there are
zeros in the power. |
| To divide decimals by powers of
10, move the decimal point left as many places as there are zeros in the |
| power. |
|
| Examples: |
|
a)
0.58 ·10 =
5.8,
b)
79.4 ·100 =
7940,
c)
0.001 ·10000 = 10, |
|
|
d)
30.07 ¸
10 = 3.007,
e)
0.08 ¸
100 = 0.0008,
f)
12.0 ¸
1000 = 0.012. |
|
|
| Multiplying and dividing decimals
by decimal units
|
| To multiply decimal by the decimal
units; 0.1, 0.01, 0.001, ... Move the decimal point of the decimal left |
| as many places as there are
zeros in the decimal unit. |
| Divide decimal by the decimal units by
moving the decimal point of the decimal right as many places as |
| there are zeros in the decimal unit. |
|
| Examples: |
|
a)
83.05 · 0.1 = 8.305, b)
547.2 · 0.001 = 0.5472, c)
0.9 · 0.0001 = 0.00009, |
|
|
d)
0.75 ¸
0.001 = 750,
e)
4.035 ¸
0.01 = 403.5,
f)
29.0 ¸
0.1 = 290. |
|
|
| Decimal numbers scientific notation
|
| Scientific
notation is used to express very large or very small numbers. |
| A
number in scientific notation is written as the product of a
number (the coefficient) and a power of 10 (the |
| exponent), i.e., |
| coefficient ´
10exponent |
| The
coefficient should have exactly one non-zero digit to the left of the decimal point. The
exponent indicates, |
| how many places the decimal point was moved to
the left or to the right. If the decimal point was moved to |
| the
left, the exponent is positive, if moved to the right the
exponent is negative. |
|
| Converting
from a number to scientific notation: |
| Examples: |
|
a)
302,567,908 = 3.02567908 · 108 b)
0.000040635 = 4.0635 · 10−5. |
|
| Converting
from scientific notation to a decimal number: |
| Examples: |
|
a)
2.09085 · 107 = 20908500
b)
7.81 · 10−5
= 0.0000781 |
|
|
| Multiplication of decimal numbers
|
| To multiply decimal numbers, ignore the decimal points and multiply
the digits. Count the total number of |
| decimal places in both decimal
numbers being multiplied. |
| Place a decimal point in the answer so
that it has as many digits to the right of the decimal point as the total |
|
number of decimal places in the two decimals being multiplied. |
|
| Examples: |
|
a)
0.589 · 47.8,
b)
79.6 · 0.00503,
c)
629.7 · 1.03 |
|
|
2356 3980
6297 |
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|
4123
000
0000 |
|
|
4712
2388
18891 |
|
|
28.1542
0.400388
648.591 |
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|
| Division of decimal numbers |
| To divide decimal numbers, first make the divisor into a whole
number by multiplying both the dividend and |
| the
divisor by the same
power of 10 (such as 10, 100, 1000, . . .) or move a decimal point in
the divisor to the |
| right end, and in the dividend the same number of
places. |
| To divide a decimal by a whole number, divide as you would
for whole numbers. Place a decimal point in the |
| answer so that it
lines up with the decimal point in the dividend. |
| If after dividing you
have a remainder, add a zero to the dividend and continue to divide
until there is no |
| remainder or the decimals repeat. |
|
| Examples: |
|
a)
467.5 ¸ 17 = 27.5,
b)
2.773 ¸ 0.47 = (2.773 ·
100) ¸
(0.47 · 100) = |
|
|
−
34
= 277.3 ¸
47 = 5.9 |
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|
127
− 235 |
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− 119
423 |
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85
− 423 |
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|
− 85
0 |
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|
0 |
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c)
0.15059 ¸ 3.7 = 1.5059 ¸
37 = 0.0407, d)
10.8 ¸ 0.004 =
10800 ¸
4
=
2700 |
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|
150
− 8 |
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− 148
28 |
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259
− 28 |
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|
− 259
0 |
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|
0 |
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| Beginning
Algebra Contents |
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|
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
