Download Lecture 1 Numbers, fractions

Mark A. Magumba
Basic Mathematics
Basic Principles
• Commutative Property of addition and multiplication
– A + B = B + A and A * B = B * A
• Associative property of addition and multiplication
– (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
• Distributive property
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a × (b + c) = a × b + a × c
a × (b + c) = a × b + a × c
a × (b - c) = a × b - a × c
(a + d) × (b + c) = (a × b + a × c) + (d × b + d × c)
(a - d) × (b + c) = (a × b + a × c) - (d × b + d × c)
Numbers
• Integers
– Are numbers without fractional parts or the set of whole numbers
• Natural Numbers
– include the series 0, 1, 2, 3….e.t.c.
– Some disagreement over whether zero should be included
– Is the set of non negative integers
• Even numbers
– Numbers divisible by 2 e.g. 0, 2, 4, 6…..
• Odd numbers
– Numbers indivisible by 2 e.g. 1, 3, 5, 7
• Prime numbers
– Numbers with only two factors i.e. 1 and themselves e.g. 3, 5, 7, 11, 13
– In other words it is only divisible by 1 and itself
• Real numbers
– Represents continuous values including integers and fractional values
Factorization
• Factors of a number are those numbers that
you multiply to get it for instance the factors
of 6 are 2, 3, 1 and 6 itself
• Multiples of a number are the numbers you
get after multiplying that number with
another number for instance the first multiple
of 6 is itself as a result of multiplying it by 1
• The set of factors and multiples excludes 0
Fractions and Ratios
• Adding/subracting fractions
a/b – x/y = ((a*y)-(x*b))/(b*y)
a/b + x/y = ((a*y) + (x*b))/(b*y)
For same denominator
e.g.
a/y – b/y = (a-b)/ y
a/y + b/y = (a+b)/ y
When
denominators
different
• Multiplying fractions
a/b * x/y = ax/ by
• Dividing fractions
a/b ÷x/y = a/b * y/x = ay/=bx
Types of fractions
• Proper fraction
– where the numerator is less than the denominator (numerator
is the figure on top, denominator is the figure at the bottom)
e.g. ½, ¾, 5/7 e.t.c
• Improper fraction
– Where the denominator is larger than the numerator e.g. 4/3,
7/5, 10/9 e.t.c
– It is bad practice to give improper fractions as final answers,
improper fractions ought to be converted to mixed fractions
• Mixed fractions
– Contains a whole number and a fraction e.g. 1½ , 6¾
– They are the result of an improper fraction e.g. 3/2 = 1½
Mixed fractions
• To convert an improper fraction e.g. a/b (a>b)
to a mixed fraction use the formula
a/b = Qr /b where Q is the quotient or the
result you get from dividing a by b, r is the
remainder, the denominator is maintained
e.g. for 3/2 you divide 3 by 2 which gives 1 and a
remainder of 1, applying our formula Q = 1, r =
1 and b = 2 so 3/2 = 1½
Simplifying fractions
• A simplified fraction is a form of a fraction that
maintains its value but with smaller numbers e.g.
½ = 2/4 = 4/8, in other words the simplified form
of 4/8 is ½ and the simplified form of any
improper fraction is its mixed fraction
• For any fraction a/b, if u multiply the numerator
and the denominator by the same value the
resulting fraction is equivalent to the original
fraction in other words a/b = (a*y)/(b*y)
Converting whole numbers to
improper fractions
• To convert any whole number A to an
improper fraction with a denominator of b
multiply A by the denominator to get the
numerator and maintain the denominator that
is a whole number A expressed in halves will
result in
(A*2)
multiply A by 2
2
maintain denominator
Converting whole numbers to
improper fractions
• For instance 8 expressed in terms of halves is
(8*2)/2 or 16/2, expressed as quarters it would
be (8*4)/4 or 32/4 as thirds it would be (8*3)/3
or 24/3
• To convert a mixed fraction to an improper
fraction you have to first covert the whole
number then add the fraction for instance to
convert 2½ to an improper fraction first get the
equivalent of 2 in halves which is 4/2 then add
the ½ and 4/2 + ½ = 5/2
Operations on Mixed fractions
• It is not often easy to do operations on mixed
fractions directly and it is advisable to convert
them to improper fractions first
Percentage
• A percentage is a fraction of 100
• For instance 30% means 30/100
• To increase something by 30% is to increase it
by 30/100 (3/10 or 0.3) of its current value
• For instance 130 is 30% greater than 100
• And 100 is 30% less than 130