books are the result of the mathematicians’efforts to solve mathematical problems with ease.Methods of arriving at solutions may differ, butthe end results or answers are always the same.These different approaches to mathematicalproblems make the study of mathematics moreinteresting, either by individual study or as agroup.You can supplement your study of mathe-matics with the following training manuals:1. Mathematics, Vol. 1, N A V E D T R A10069-D12. Mathematics, Vol. 2-A, N A V E D T RA100623. Mathematics,Vol. 2-B, N A V E D T RA100634. Mathematics, Vol. 3, N A V E D T R A10073-A1TYPES OF NUMBERSPositive and negative numbers belong to theclass called REAL NUMBERS. Real numbers andimaginary numbers make up the number systemin algebra. However, in this training manual, wewill deal only with real numbers unless otherwiseindicated.A real number may be rational or irrational.The word rational comes from the word ratio. Anumber is rational if it can be expressed as thequotient, or ratio, of two whole numbers.Rational numbers include fractions like 2/7,whole numbers (integers), and radicals if theradical is removable. Any whole number isrational because it could be expressed as aquotient with 1 as its denominator. For instance,8 equals 8/1, which is the quotient of two integers.A number like v’TZ is rational since it can beexpressed as the quotient of the two integers inthe form 4/1. An irrational number is a realnumber that cannot be expressed as the ratio oftwo integers. The numbersand 3.1416 (n) are examples of irrational numbers.An integer may be prime or composite. Anumber that has factors other than itself and1 is a composite number. For example, thenumber 15 is composite. It has the factors 5and 3. A number that has no factors except itselfand 1 is a prime number. Since it is advantageousto separate a composite number into primefactors, it is helpful to be able to recognize a fewprime numbers. The following are examples ofprime numbers: 1, 2, 3, 5, 7, 11, 13, 17, 19, and23.A composite number may be a multiple of twoor more numbers other than itself and 1, and itmay contain two or more factors other than itselfand 1. Multiples and factors of numbers are asfollows: Any number that is exactly divisible bya given number is a multiple of the given number.For example, 24 is a multiple of 2, 3, 4, 6, 8, and12 since it is divisible by each of these numbers.Saying that 24 is a multiple of 3, for instance,is equivalent to saying that 3 multiplied bysome whole number will give 24. Any number isa multiple of itself and also of 1.FRACTIONS, DECIMALS,AND PERCENTAGESThe most general definition of a fraction statesthat “a fraction is an indicated division. ” Anydivision may be indicated by placing the dividendover the divisor with a line between them. By theabove definition, any number, even a so-called“whole” number, may be written as a commonfraction. The number 20, for example, may bewritten as 20/1. This or any other fractionthat amounts to more than 1 is an IMPROPERfraction. For example, 8/3 is an improperfraction, The accepted practice is to reduce animproper fraction to a mixed fraction (a wholenumber plus a proper fraction). Perform theindicated division and write the fractional part ofthe quotient in its lowest term. In this case,8/3 would be 2 2/3. A fraction that amounts toless than 1 is a PROPER fraction, such as thefraction 1/4.To refresh your memory, we are including thefollowing rules in the solution of fractions:1. If you multiply or divide both thenumerator and denominator of a fraction by thesame number, the value does not change. Theresulting fraction is called an EQUIVALENTfraction.2. You can add or subtract fractions only ifthe denominators are alike.3. To multiply fractions, simply find the prod-ucts of the numerators and the products of thedenominators. The resulting fractional productmust be reduced to the lowest term possible.1-2
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