Website owner: James Miller
ORIGIN OF THE CONCEPT OF IRRATIONAL NUMBERS
Origin of the concept. The origin of the concept of an irrational number lies in the following fact discovered by the ancient Greeks.
Fact. In a square with sides of unit length there is no rational number equal to the length of its diagonal.
Proof. Consider a square with sides of length 1 and a diagonal of length a. See Fig. 1. According to the Pythagorean Theorem
a2 = 12 + 12 = 2 .
But there is no rational number whose square is equal to 2. We prove this as follows:
A rational number is a number that can be written as the quotient of two integers. Thus we wish to prove that there is no quotient p/q whose square is equal to 2 where p and q are positive integers.
Prove. There does not exist two integers p and q, one of which is odd, for which (p/q)2 = 2.
Here p and q cannot both be even numbers, for if they were both could be divided by 2 and the fraction could be reduced to lower terms. Hence either p or q is an odd number.
Suppose there are positive integers p and q such that (p/q)2 = 2 . If (p/q)2 = 2, then
1) p2 = 2q2
Thus p2 is an even number since twice any number is an even number i.e. the factor 2 in 2q2 reveals it as even. But if p2 is an even number then p must also be an even number. (Why? The square of any odd number is also odd and if p were odd its square would be odd). Since p is even let b = p/2. Then
2) p2 = 4b2
From 1) and 2) we then have
3) 2q2 = 4b2
or
4) q2 = 2b2
Thus q2 is even and q must also be even. Thus, based on our original suppositions, we have deduced that both p and q are even. But we already know that either p or q must be odd.. This contradiction proves that there are no positive integers p and q such that (p/q)2 = 2 .
Prove. The square of any odd number is odd.
Proof. The odd numbers are given by the algorithm
2n + 1 where n = 0, 1, 2, ....
The square of an odd number is given by
(2n + 1)2 = 4 n2 + 4n + 1
which must be odd since 4 n2 + 4n is even.
Q. What is the significance of the fact that there is no rational number that gives the length of the diagonal of a square with unit sides?
A. A rational number, defined as a quotient of two integers, is simply a devise invented by man for designating continuous quantities. Man came up with this idea as a technique for naming any continuous quantity. The significance of the above fact is simply that this devise of using the quotient of two integers for naming continuous quantities is incapable of naming some quantities. Those quantities that it is unable to name are the irrational numbers. Th rational numbers plus the irrational numbers constitute the real numbers.
The irrational numbers
Fact. The square roots of all natural numbers that are not “perfect squares” (i.e. the square of some positive integer) are irrational numbers. Thus, just as there is no rational number whose square is 2, there is no rational number whose square is 3, 5, 6, 7, 8 or any integer that is not a perfect square. Consequently, while the square roots of 1, 4, 9, 16, 25, 36, etc. are rational numbers, the square roots of 2, 3, 5, 6, 7, 8, 10, etc are all irrational numbers. In the same way, the cube roots of natural numbers that are not perfect cubes are all irrational numbers.
We shall prove the above assertion for the natural number 5 i.e. we shall prove that there is no quotient of two integers p/q whose square is 5. Before presenting the proof, we note the following:
1) The unique factorization theorem of arithmetic states:
Unique factorization theorem. Any positive integer greater than 1 is a prime or can be expressed as a product of primes. Except for the order of the factors, this expression is unique.
Syn. Fundamental theorem of arithmetic
Examples. 12 = 2∙2∙3, 84 = 2 ∙2 ∙3 ∙7
2) A consequence of the unique factorization theorem is that if a natural number p is expressed as a product of its primes, then a factorial representation of p2 will contain the same primes with each prime occurring twice as many times. This means that every prime factor of p2 occurs an even number of times. For example,
84 = 2 ∙2 ∙3 ∙7 and 842 = 22 ∙22 ∙32 ∙72
The proof is as follows: Assume that there is a rational number p/q whose square is 5; i.e. assume that
where p and q are natural numbers. Under this assumption
p2 = 5q2 .
Now, from 2) above, we know q2 must contain 5 as a factor an even number of times, if it contains it at all. Consequently, 5q2 must contain 5 as a factor an odd number of times. Also, from 2) above we know that p2 must contain 5 as a factor an even number of times, if it contains it at all. However, if p2 were really equal to 5q2 that would imply that it contained the factor 5 an odd number of times, a contradiction of known fact. Thus the supposition that there is a quotient of two integers p/q whose square is 5 is proved false.
Sums and products of irrational and rational numbers. It can be shown that the sum of a rational number and an irrational number is an irrational number. It can also be shown that the product of a rational and an irrational number is an irrational number. Thus and are irrational numbers.
Decimal representation of rational and irrational numbers.
Theorem 1. The rational numbers are precisely those decimal numbers that either terminate or are periodic.
One can easily see that a terminating decimal number such as 3.23 is a rational number since it is just another way of writing 323/100, a quotient of two natural numbers. Other rational numbers repeat. Example: 2/3 = 0.66666 ... where the 6's repeat indefinitely and 24/55 = 0.4363636 ... where the block 36 repeats indefinitely.
Theorem 2. The irrational numbers are precisely those infinite decimals that are not repeating.
Theorem 3. An irrational number can be approximated to any desired degree of accuracy by a rational number.
References
Ross R. Middlemiss. Algebra for College Students.
Richard Courant, Herbert Robbins. What is Mathematics?
Hawks, Luby, Touton. Second-Year Algebra
Jesus Christ and His Teachings
Way of enlightenment, wisdom, and understanding
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
On Self-sufficient Country Living, Homesteading
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
Theory on the Formation of Character
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life