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M7_notes - 5.pdf - M7 class 5 notes Number Theory 1. The...

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M7 class 5 notes1Number Theory1. The Real Number SystemThe real number system evolved over time by expanding the notion of what we mean by the word“number.” At first, “number” meant something you could count, like how many sheep a farmer owns.These are called thenatural numbers, or sometimes thecounting numbers.1)Natural Numbersor “Counting Numbers”1, 2, 3, 4, 5, . . .The use of three dots at the end of the list is a common mathematical notation to indicate that the listkeeps going forever.At some point, the idea of “zero” came to be considered as a number. If the farmer does not have anysheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus thenumber zero thewhole numbers.2)Whole NumbersNatural Numbers together with “zero”0, 1, 2, 3, 4, 5, . . .3)IntegersWhole numbers plus negatives. . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .About Negative NumbersHow can you have less than zero? Well, do you have a checking account? Having less than zeromeans that you have to add some to it just to get it up to zero. And if you take more out of it, itwill be even further less than zero, meaning that you will have to add even more just to get it upto zero.The strict mathematical definition goes something like this:For every real numbern, there exists itsopposite, denoted –n, such that the sum ofnandnis zero, orn+ (–n) = 0Note that the negative sign in front of a number is part of the symbol for that number: Thesymbol “–3” is one object — it stands for “negative three,” the name of the number that is threeunits less than zero.The number zero is its own opposite, and zero is considered to be neither negative nor positive.
M7 class 5 notes2The next generalization that we can make is to include the idea of fractions. While it is unlikely that afarmer owns a fractional number of sheep, many other things in real life are measured in fractions, like ahalf-cup of sugar. If we add fractions to the set of integers, we get the set ofrational numbers.4)Rational NumbersAll numbers of the form, whereaandbare integers (butbcannot be zero)Rational numbers include what we usually callfractionsNotice that the word “rational” contains the word “ratio,” which should remind you of fractions.The bottom of the fraction is called thedenominator. Think of it as thedenomination—it tells you whatsize fraction we are talking about: fourths, fifths, etc.The top of the fraction is called thenumerator. It tells youhow manyfourths, fifths, or whatever.RESTRICTION: The denominator cannot be zero! (But the numerator can)If the numerator is zero, then the whole fraction is just equal to zero. If I have zero thirds or zerofourths, than I don’t have anything. However, it makes no sense at all to talk about a fractionmeasured in “zeroths.”Fractions can be numbers smaller than 1, like 1/2 or 3/4 (calledproper fractions), or they can benumbers bigger than 1 (calledimproper fractions), like two-and-a-half, which we could alsowrite as 5/2

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