Real Numbers (4)
Mind Warm Ups
Problem 31 : Suppose x is a perfect square and its root is y. What is the result of dividing x by y?
Ans: in general a number b is a perfect square if b = a2, where a is a non zero integer. For exmple, the numbers 9, 16 and 25 are perfect squares. So if x is a perfect square, it can be expressed as the square of an integer. Since its root is given as y, x = y2. So x/y = y = √x.
Problem 32: x and y are perfect squares, each < 100. x + y = z is also a perfect square. Determine x and y if 9<√z<11.
Ans: √z = 10. So z = 100. Therefore, x is either 36 or 64. x = 36, implies y = 64; x = 64 implies y = 36.
Problem 33: the numbers 6 and 28 are called perfect numbers in numerology. Why?
Ans: In numerological terminology, a perfect number is a number that is the sum of its divisors (the number itself is excluded as a divisor). 6 and 28 are called perfect numbers because they are equal to the sum of their divisors: 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14.
Problem 34 : What other perfect number do you know?
Ans: In general, P is a perfect number if P = 2p-1(2p - 1), where p is a prime number that gives rise to a Mersenne Prime. The perfect number 28 is the case where p is 3. When p is 5 we have the perfect number 496. As an excercise express 496 as a sum of its divisors.
Problem 35: what is the difference between a commutative ring and a commutative ring with unit?
Ans: a ring is an algebraic structure with the following specified characteristics:
(1) the structure is formed from a population or set of numbers, say R
(2) the operation of addition and multiplication are defined in R such that adding or multiplying any two numbers a and b in R results in a number in R
(3) (a + b) + c = a + (b + c) for all a, b, c in R
(4) a + b = b + a for all a, b in R.
(5) 0 is in R such that a + 0 = a for all a in R
(6) each a in R has its additive inverse -a such that a + (-a) = 0
(7) (ab)c = a(bc) for all a, b, c in R
(8) a(b + c) = ab + ac and (b + C)a = ba + ca for all a, b, c in R.
Addition and multiplication are called binary operations on R when (2) holds. Notice that multiplicative commutativity is not established. When multiplication is commutative, that is, when ab = ba for all a, b in R, the ring is called a commutative ring. A ring is called a ring with unit if there is an element 1 ≠ 0 in R such that the product of a and 1, a(1) = a = 1(a).
So the difference between a commutative ring and a commutative ring with unit is that the latter has the unit element 1 such that the product of a and 1, a(1) = a = 1(a) holds for all a in R.
Problem 36: what additional characteristic does R, a commutative ring with unit, need to become a field F?
Ans: a field F is a commutative ring with unit that has a multiplicative inverse. In other words, for each element a ≠ 0 in F, there is an element a-1 in F called the multiplicative inverse of a such that a(a-1) = 1. In a field F, the division a/b = ab-1.
Problem 37 : show that for a in field F, (a-1)-1 = a.
Ans: (a-1)-1 = 1/(a-1) = a(a-1)/(a-1) = a .
Problem 38 : x and y are prime numbers. x is also an even number. y wants to be like x. It can be x, if its value is reduced by the value of the neutral element of the Abelian group under the multiplicative operation. What is the value of y?
Ans: x = 2 since 2 is the only even number that is also odd. The value of the neutral element in an Abelian Group under the operation of multiplication is 1. Therefore y = 3.
In general, a group G is a non empty set with the following characteristics:
(1) A binary operation o is possible on G
(2) There is an identity element or neutral element in G such that aoI = a, for all a in G, where I is the Identity element.
(3) The element a in G has an inverse for all a in G such that ao(inverse of a) = I, the neutral element.
G is an Abelian group (named after mathematician Neil Hendrik Abel,8/1802-4/1829), if it is multiplicatively commutative. In other words, ab = ba for all a, b in G.
Problem 39 : P is the positive cone of an ordered ring R and b-a is in P for a, b in P. Is a > b?
Ans: a < b. The elements of a positive cone of an ordered ring are positive. So b - a > 0 which implies that b > a. In general, a ring R is called an ordered ring if it has the following characteristics:
(1) It has a subset P that does not include 0.
(2) If a ≠ 0, then only a or -a is in P
(3) a, b in P implies a + b is in P and ab is in P.
P is called the positive cone of R and the elements of P ar said to be positive.
Problem 40 : what is the absolute value |a|, for a > 0 and for a < 0.
Ans: for a > 0, |a| = a; for a < 0, |a| = -a.
Peter Oye Sagay