By Keith Hirst
Quantity and geometry are the rules upon which arithmetic has been outfitted over a few 3000 years. This booklet is worried with the logical foundations of quantity structures from integers to complicated numbers. the writer has selected to advance the tips by way of illustrating the ideas used all through arithmetic instead of utilizing a self-contained logical treatise. the belief of facts has been emphasized, as has the representation of recommendations from a graphical, numerical and algebraic viewpoint. Having laid the rules of the quantity process, the writer has then grew to become to the research of endless procedures regarding sequences and sequence of numbers, together with energy sequence. The e-book additionally has labored examples all through and contains a few feedback for self-study initiatives. additionally there are instructional difficulties aimed toward stimulating staff paintings and discussion.
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Additional info for Numbers, Sequences and Series
Example text
Our contemporary notion of number has therefore developed over a period of at least 3000 years. Our present conceptions of the real number system were formulated only about 100 years ago, and we shall discuss that in Chapter 5. There are two points of view we can take in moving from a discussion of the integers to a consideration of the rational number system. 3. The second is to construct the rationals from the integers, through an abstract mathematical approach. Both of these will, of course, reflect our long-standing acquaintance with rational numbers and fractions from everyday mathematics and from school.
2 Proof by Mathematical Induction The axiom of induction is the basis of an important method of proof known as 'Proof by Mathematical Induction'. This involves proving statements involving a variable n, typical examples being (i) for all n E N, 13 + 23 + 33 + 1 + 2n, (ii) for all n ~ 3, 2n (iii) for all n ~ 4, n! > 2n • ~ ... + n3 = n2(n + 1)2/4, Each of these statements is of the form P(n), and the variable n ranges over all natural numbers from some initial number onwards. To explain the procedure we shall take this initial number to be 1, without any loss of generality.
5 X 82 + 2 x 8 + 1. This gives Example 10 The number 2E5A is expressed in the hexadecimal system. Convert it to base ten. In hexadecimal A and E represent 10 and 14 respectively in decimal. So in base ten 2E5A represents 2 x 163 + 14 X 162 + 5 x 16 + 10 = 11866. Example II The number 654 is expressed in base ten. Convert it to base seven. We apply continued division by 7 as follows, using the quotient/remainder notation. 654 = 93 x 7 + 3, (1) (2) (3) 93 = 13 x 7 + 2, 13 = 1 x 7 + 6, I=Ox7+1. We stop when the quotient first becomes zero.