(a + b+ c)^{2} = a^{2} + b^{2} + c^{2} + 2 (ab + bc + ca).
(a^{3} + b^{3}) = (a + b) (a^{2} - ab + b^{2})
(a^{3} - b^{3}) = (a - b) (a^{2} + ab + b^{2}).
(a^{3} + b^{3} + c^{3} - 3abc) = (a + b+ c) (a^{2} + b^{2} + c^{2} - ab - bc - ac).
When a + b + c = 0, then a^{3} + b^{3} + c^{3} = 3abc.
Decimal Fractions :
Fractions in which denominators are powers of 10 are known
as decimal fractions.
Thus, ^{1}/_{10} = 1 tenth = .1; ^{1}/_{100} = 1 hundredth = .01; ^{99}/_{100} = 99 hundredths = .99; ^{7}/_{1000} = 7 thousandths = .007, etc.
Conversion of a Decimal Into Vulgar Fraction :
Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after
the decimal point. Now, remove the decimal point and reduce the fi action to its
lowest terms.
Thus, 0.25 = ^{25}/_{100} = ^{1}/_{4} ; 2.008 = ^{2008}/_{1000} = ^{251}/_{125}
Annexing zeros to the extreme right of a decimal fraction does not change its value.
Thus, 0.8 = 0.80 = 0.800, etc.
If numerator and denominator of a fraction contain the\same number of decimal
places, then we remove the decimal sign.
Thus, ^{1.84}/_{2.99} = ^{184}/_{299} = ^{8}/_{13} ; ^{.365}/_{.584} = ^{365}/_{584} = ^{5}/_{8}
Operations on Decimal Fractions :
Addition and Subtraction of Decimal Fractions : The given numbers are so
placed under each other that the decimal points lie in one column. The numbers
so arranged can now be added or subtracted in the usual way.
Multiplication of a Decimal Fraction By a Power of 10 : Shift the decimal
point to the right by as many places as is the power of 10.
Thus, 5.9632 x 100 = 596.32; 0.073 x 10000 = 0.0730 x 10000 = 730.
Multiplication of Decimal Fractions : Multiply the given numbers considering
them without the decimal poirit. Now, in the product, the decimal point is marked
off to obtain as many places of decimal as is the sum of the number of decimal
places in the given numbers.
Suppose we have to find the product (.2 x .02 x .002).
Now, 2 x 2 x 2 = 8 Sum of decimal places = (1 + 2 + 3) = 6
∴ .2 x .02 x .002 = .000008.
Dividing a Decimal Fraction By a Counting Number : Divide the given
number without considering the decimal point, by the giv,en counting number.
Now, in the quotient, put the decimal point to give as many places of decimal as
there are in the dividend.
Suppose we have to find the quotient (0.0204 = 17). Novv, 204 = 17 = 12
Dividend contains 4 places of decimal. So, 0.0204 - 17 = 0.0012
Dividing a Decimal Fraction By a Decimal Fraction : Multiply both the
dividend and the divisor by a suitable power of 10 to make divisor a whole number.
Now, proceed as above.
Thus, ^{0.00066}/_{0.11} = ^{0.00066 x 100}/_{0.11 x 100} = ^{0.066}/_{11} = .006
Annexing zeros to the extreme right of a decimal fraction does not change its value.
Thus, 0.8 = 0.80 = 0.800, etc.
If numerator and denominator of a fraction contain the\same number of decimal
places, then we remove the decimal sign.
Thus, ^{1.84}/_{2.99} = ^{184}/_{299} = ^{8}/_{13} ; ^{.365}/_{.584} = ^{365}/_{584} = ^{5}/_{8}
Comparison of Fractions :
Suppose some fractions are to be arranged in ascending
or descending order of magnitude. Then, convert each one of the given fractions in
the decimal form, and arrange them accordingly.
Suppose, we have to arrange the fractions ^{3}/_{5} , ^{6}/_{7} and ^{7}/_{9} in descending order.
Now, ^{3}/_{5} = 0.6 ^{6}/_{7} = 0.857, ^{7}/_{9} = 0.777.....
Since 0.857> 0.777..... > 0.6, so ^{6}/_{7} > ^{7}/_{9} > ^{3}/_{5}
Recurring Decimal :
in a decimal fraction, a figure or a set of figures is repeated
continuously, then such a number is called a recurring decimal.
In a recurring decimal, if a single figure is repeated, then it is expressed by putting
a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set.
Thus, ^{1}/_{3} = 0.333..... = 0.3 ; ^{22}/_{7} = 3.142857142857..... = 3.142857.
Pure Recurring Decimal :
A decimal fraction in which all the figures after the
decimal point are repeated, is called a pure recurring decimal.
Converting a Pure Recurring Decimal Into Vulgar Fraction :
Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.
Thus, 0.5 = ^{5}/_{9} ; 0.53 = ^{53 }/_{99} ; 0.067 = ^{67}/_{999} ; etc.
Mixed Recurring Decimal :
decimal fraction in which soMe figures do not repeat
and some of them are repeated, is called a mixed recurring decimal.
e.g., 0,17333..... = 0.173.
Converting a Mixed Recurring Decimal Into Vulgar Fraction :
In the numerator, take the difference between the number formed by all the digits after decimal point
(taking repeated digits only once) and that formed by the digits which are not repeated.
In the denominator, take the number formed by as many nines as there are repeating
digits followed by as many zeros as is the number of non-repeating digits.
Thus, 0.16 = ^{16 - 1}/_{90} = ^{15}/_{90} = ^{1}/_{6} ; 2273 = ^{2273 - 22}/_{9900} = ^{2251}/_{9900}