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 – 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 Multiplying fractions a/b * x/y = ax/ by Dividing fractions a/b ÷x/y = a/b * y/x = ay/=bx When denominators different

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 = Q r / 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