Real Numbers (5)
Mind Warm Ups
Problem 41 : true or false? For any elements a, b and c in an ordered ring R:
(i) Either a < b; or a = b; or a > b
(ii) a < b and b < c imply a < c
(iii) a < b implies a + c < b + c
(iv) a < b and 0 < c imply ac < bc
(v) a < b and c < 0 imply ac > bc
(vi) a ≠ 0; implies a2 > 0.
Ans: all six statements of problem 42 are true. If a, b and c are elements of an ordered field, the "less than" relation has in addition to the statements in problem 42, the following properties:
(vii) 0 < a < b implies 0 < b-1 < a-1
(viii) a < b < 0 implies b-1 < a-1 < 0.
Problem 42: suppose a and b are elements of an ordered ring R, prove that |a + b| ≤ |a| + |b|.
Ans: the absolute value of a is defined as |a| = a if a≥0; and |a| = -a if a≤0. the following basic properties of the absolute value of elements of an ordered ring are derived from this definition:
(1) |a| ≥0; |a| = 0 if and only if a = 0
(2) |ab| = |a||b|; here is the proof:
|ab| = ab by definition. |a| = a and |b| = b, for a and b >0 also by definition. Hence (2).
A similar reasoning is applied to the situation where a or b is <0.
(3) |a + b| ≤ |a| + |b|; here is the proof:
---> -|a| ≤ a ≥|a|; -|b| ≤ b ≥|b|;
sum a and b --> -(|a| +|b|)≤(a + b)≤(|a| + |b|)
Therefore; |a + b| ≤ |a| + |b|. This inequality is known as the Triangle Inequality. We can also arrive at this inequality from (|a + b|)2 = (a + b)2. Try it.
(4) |a| - |b| ≤ |a - b|. Prove it. Hint: |a| = |a - b + b|.
Problem 43: given a sequence (an) in an ordered field F, a in F, positive elements ε and n0 in F. What does the following math statement mean?
|an - a| < ε ; whenever n ≥ n0
Ans: a sequence is basically a list of elements from a given population. Usually (not compulsory) there is a rule or pattern associated with the order of the list; for example, the list (1,2,3,4,5,6,7) is a sequence of +ve integers from 1 to 7. The rule is that each successive number is 1 unit more than the number that precedes it. This sequence is finite, the last element is 7. We can express the sequence of +ve integers as infinite by including three dots after 7, that is, (1,2,3,4,5,6,7...).
The math statement in problem 43 means that the sequence (an) converges to a, in other words, the limit of the sequuence is a, that is, lim an = a.
In other words, the interval between an and a will always be less than the number ε no matter how small we make ε. Which means that no element in the sequence = a. So (an - a) is called a convergent sequence.
Problem 44 : how is a Cauchy sequence different from a convergent sequence?
Ans 44: convergent sequence --> |an - a| < ε whenever n ≥ n0
Cauchy sequence --> |an - am| < ε whenever m, n ≥ n0
The elements of a Cauchy sequence get close to each other whereas the elements of a convergent sequence get close to a fixed point. A convergent sequence is always a Cauchy sequence but a Cauchy sequence may or may not converge in some ordered fields.
Problem 45 : is the real number system R an ordered and complete field? suppose (an) is a sequence of real numbers, express (an) as :
(i) a nondrecreasing sequence
(ii) a nonincreasing sequence
Ans: yes. the real number system R is an ordered and complete field. An ordered field F is complete if every Cauchy sequence of elements in F converges to an element in F.
(i) an ≤ an+1
(ii) an ≥ an+1
Problem 46 : if n is a +ve integer (n ≠ 0), what is the value of n that produces:
(i) the upper bound of a nonincreasing sequence
(ii) the lower bound of a nondecreasing sequence
Ans 46: (i) 1; (ii) 1.
Problem 47 : if n is a +ve integer (n ≠ 0),
(i) what is the lower bound of the sequence (an) = 1/n?
(ii) what is the sum of the lower bound of (an) and its limit?
Ans: (i) 0; (ii) try it.
Problem 48 : what does the following math statement mean?
a ≅ b (mod m)
Ans: the math statement of problem 48 was introduced by the great mathematician Carl Friedrich Guass (04/30/1777 - 02/23/1855). He used it to represent the following definition: two integers a and b shall be said to be congruent for the modulus m when their difference a - b is divisible by the integer m.
When a and b are not congruent, they are called incongruent for the modulus m, that is,
a ≇ b (mod m).
Problem 49 : determine whether or not a and b are congruent modulus m, if a is an even number, b an odd number and m a +ve integer.
Ans: try it. In general,
a ≅ b (mod m) can be expressed as b = a + km; where k is some multiple of m.
Problem 50 : explain the following basic properties of congruences:
(i)Determination; (ii)Reflexivity; (iii) Symmetry; (iv) Transitivity
Ans:
(i)Determination implies that either a and b are congruent modulus m or they are not. That is, a ≅ b (mod m) or a ≇ b (mod m)
(ii)Reflexivity implies that a is congruent to itself. That is, a ≅ a (mod m)
(iii)Symmetry implies that when a ≅ b (mod m) then b ≅ a (mod m)
(iv)Transitivity implies that when a ≅ b (mod m); b ≅ c (mod m); then a ≅ c (mod m).
Peter Oye Sagay