Lesson BASICS - Types of Numbers
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<b>Introduction</b> This Lesson will discuss some of the different types of numbers. The following are some of the sets of numbers: <b> 1. Integers 2. Rationals 3. Irrationals 4. Real </b> There are a few more subsets of numbers, but they are not particularly interesting or are not applicable to a "beginners guide": such as transcendentals or complex numbers. <b>Integers</b> These are whole numbers, from minus infinity through zero to plus infinity: all the whole numbers. <b>Rationals</b> Rational numbers are those that can be written as a fraction of 2 integers, a/b <b>EXAMPLES</b> 23 is rational as it can be written as 23/1 or 230/10 or (-115)/(-5) etc -3 is rational --> 30/(-10) etc 0.123 is rational --> 123/1000 etc 11.43 is rational --> 1143/100 etc How about 0.3737? Again rational... 3737/10000 But, what about 0.37373737.. where the number repeats for ever, never stopping? How can we possibly write that as a fraction of 2 integers? Well, we can: 37/99. Check it on your calculator. <b>How to find the fraction of a repeating decimal</b> <b>Q</b> Find the fraction that is 0.258258258... <b>A</b> we need the repeating "unit" moved in front of the decimal point. Here the repeating unit is 258, so if we multiplied the decimal, call it d, by 1000, we would end up with 258.258258258... so we have 1000d = 258.258258258... and d = __0.258258258... now, subtracting, gives 999d = 258 --> d = 258/999 --> d = 86/333 <b>Irrationals</b> Logically, any number that cannot be written as the fraction a/b is termed irrational. Irrational numbers are those that continue, never stopping and do not repeat. This way we can never write them down as such into a fraction... as soon as we stop writing the number, it is no longer the real number, just an approximation to it. By this i mean, let there be a number 0.1232375912075... that never stops nor repeats. If i could write it as 12323/100000, then this is not equal to the original number: this is 0.12323. The same goes for all versions of the number...you basically never get to the end of the number, so you cannot write it as the fraction of two integers. So, how do we write irrationals then? Well we give them names instead. <b>EXAMPLES</b> e --> 2.71828182... {{{pi}}} --> 3.14159265... {{{sqrt(2)}}} --> 1.41421356.. {{{sqrt(3)}}} --> 1.73205080.. {{{sqrt(5)}}} --> 2.23606797.. etc but what about {{{sqrt(4)}}}? Well, this is +2 or -2 which are rationals. <b>Reals</b> Real numbers is the combination of rational and irrational numbers: Reals are the whole set of numbers that you will come across normally. There is a whole family of numbers that exist beyond real numbers..the imaginary numbers, but this is beyond the scope of this Introduction.
