Page 26 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 1] NUMBERS 17

(d)Itis possible for L to have no largest number and for R to have no smallest number. What
type of number does the cut define in this case?
(a)Let a be the largest rational number in L, and b the smallest rational number in R. Then either a ¼ b or
a < b.
We cannot have a ¼ b since by definition of the cut every number in L is less than every number
in R.
1
We cannot have a < b since by Problem 1.9, ða þ bÞ is a rational number which would be greater
2
than a (and so would have to be in R) but less than b (and so would have to be in L), and by definition a
rational number cannot belong to both L and R.
2
2
(b)Asanindication of the possibility, let L contain the number and all rational numbers less than , while
3
3
2
2
R contains all rational numbers greater than .Inthis case the cut defines the rational number .A
3
3
similar argument replacing 2 3 by any other rational number shows that in such case the cut defines a
rational number.
2
(c) Asanindication of the possibility, let L contain all rational numbers less than , while R contains all
3
2
2
rational numbers greaters than . This cut also defines the rational number . A similar argument
3 3
shows that this cut always defines a rational number.
(d)Asanindication of the possibility let L consist of all negative rational numbers and all positive rational
numbers whose squares are less than 2, while R consists of all positive numbers whose squares are
greater than 2. We can show that if a is any number of the L class, there is always a larger number of
the L class, while if b is any number of the R class, there is always a smaller number of the R class (see
Problem 1.106). A cut of this type defines an irrational number.
From (b), (c), (d)itfollows that every cut in the rational number system, called a Dedekind cut,
defines either a rational or an irrational number. By use of Dedekind cuts we can define operations
(such as addition, multiplication, etc.) with irrational numbers.
Supplementary Problems
OPERATIONS WITH NUMBERS
1
3
1.35. Given x ¼ 3, y ¼ 2, z ¼ 5, a ¼ , and b ¼ ,evaluate:
2 4
2
xy 2z 2 3a b þ ab 2 2 2
ð2x yÞð3y þ zÞð5x 2zÞ; ; ; ðax þ byÞ þðay bxÞ :
2ab 1 2a 2b þ 1 ðay þ bxÞ þðax byÞ
ðaÞ ðbÞ ðcÞ 2 2 ðdÞ 2 2
Ans. (a) 2200, (b) 32, (c) 51=41, (d)1
1.36. Find the set of values of x for which the following equations are true. Justify all steps in each case.
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
4fðx 2Þþ 3ð2x 1Þg þ 2ð2x þ 1Þ¼ 12ðx þ 2Þ 2 2 2x þ 2 ¼ x þ 1
ðaÞ ðcÞ x þ 8x þ 7
1 1 1 1 x 3
ðbÞ ¼ ðdÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼
8 x x 2 4 x 2x þ 5 5
2
Ans. (a)2, (b)6; 4(c) 1; 1(d) 1 2
x y z
1.37. Prove that þ þ ¼ 0giving restrictions if any.
ðz xÞðx yÞ ðx yÞðy zÞ ðy zÞðz xÞ
RATIONAL AND IRRATIONAL NUMBERS
3
1.38. Find decimal expansions for (a) , (b) p ffiffiffi 5.
7
_ _ _ _ _ _
Ans. (a)0:4285711, (b)2.2360679 ...
5
7
8
4
2