TYPES OF NUMBERS

books are the result of the mathematicians’ efforts to solve mathematical problems with ease. Methods of arriving at solutions may differ, but the end results or answers are always the same. These different approaches to mathematical problems make the study of mathematics more interesting, either by individual study or as a group. You can supplement your study of mathe- matics with the following training manuals: 1. Mathematics, Vol. 1, N A V E D T R A 10069-D1 2. Mathematics, Vol. 2-A, N A V E D T RA 10062 3. Mathematics, Vol. 2-B, N A V E D T RA 10063 4. Mathematics, Vol. 3, N A V E D T R A 10073-A1 TYPES OF NUMBERS Positive and negative numbers belong to the class called REAL NUMBERS. Real numbers and imaginary numbers make up the number system in algebra. However, in this training manual, we will deal only with real numbers unless otherwise indicated. A real number may be rational or irrational. The word rational comes from the word ratio. A number is rational if it can be expressed as the quotient, or ratio, of two whole numbers. Rational numbers include fractions like 2/7, whole numbers (integers), and radicals if the radical is removable. Any whole number is rational because it could be expressed as a quotient 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 be expressed as the quotient of the two integers in the form 4/1. An irrational number is a real number that cannot be expressed as the ratio of two integers. The numbers and 3.1416 (n) are examples of irrational numbers. An integer may be prime or composite. A number that has factors other than itself and 1 is a composite number. For example, the number 15 is composite. It has the factors 5 and 3. A number that has no factors except itself and 1 is a prime number. Since it is advantageous to separate a composite number into prime factors, it is helpful to be able to recognize a few prime numbers. The following are examples of prime numbers: 1, 2, 3, 5, 7, 11, 13, 17, 19, and 23. A composite number may be a multiple of two or more numbers other than itself and 1, and it may contain two or more factors other than itself and 1. Multiples and factors of numbers are as follows: Any number that is exactly divisible by a given number is a multiple of the given number. For example, 24 is a multiple of 2, 3, 4, 6, 8, and 12 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 by some whole number will give 24. Any number is a multiple of itself and also of 1. FRACTIONS, DECIMALS, AND PERCENTAGES The most general definition of a fraction states that “a fraction is an indicated division. ” Any division may be indicated by placing the dividend over the divisor with a line between them. By the above definition, any number, even a so-called “whole” number, may be written as a common fraction. The number 20, for example, may be written as 20/1. This or any other fraction that amounts to more than 1 is an IMPROPER fraction. For example, 8/3 is an improper fraction, The accepted practice is to reduce an improper fraction to a mixed fraction (a whole number plus a proper fraction). Perform the indicated division and write the fractional part of the quotient in its lowest term. In this case, 8/3 would be 2 2/3. A fraction that amounts to less than 1 is a PROPER fraction, such as the fraction 1/4. To refresh your memory, we are including the following rules in the solution of fractions: 1. If you multiply or divide both the numerator and denominator of a fraction by the same number, the value does not change. The resulting fraction is called an EQUIVALENT fraction. 2. You can add or subtract fractions only if the denominators are alike. 3. To multiply fractions, simply find the prod- ucts of the numerators and the products of the denominators. The resulting fractional product must be reduced to the lowest term possible. 1-2