SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

The positive rational numbers (numbers given by n+a/b where n, a and b are positive integers) were the numbers used for centuries, up to the time of the invention of 0. What are the algebraic rules that are followed by the positive rationals? They obey the following laws: Let S be the set of all positive rational numbers. For every a,b,c in set S: 1. Closed under Addition. a,b in S implies a+b in S 2. Associative Law under Addition. a+(b+c) = (a+b)+c 5. Commutative Law under Addition. a+b = b+a 6. Cancellation Law under Addition. a+x = b+x implies a=b 7. Closed under Multiplication. a,b in S implies ab in S 8. Associative Law under Multiplication. a(bc) = (ab)c 9. Identity Element under Multiplication. 1a =a1 = a 10. Inverse exists for every element under Multiplication. aa' = a'a = 1 where a' is the inverse of a 11. Commutative Law under Multiplication. ab = ba 12. Cancellation Law under Multiplication. ax = bx implies a=b 13. Left Distributive Law -- Multiplication over Addition a(b+c) = ab + ac 14. Right Distributive Law -- Multiplication over Addition (a+b)c = ac + bc Note that there is no identity element under addition. And no inverse elements exist under addition. To get an identity element, zero had to be appended to the positive rationals. To get inverse elements, negative numbers had to be appended. The reason the most of these laws hold is intuitively obvious when viewed in the right way. In the case of addition we can think in terms of adding sets of objects. In the case of multiplication we can view the product "ab" as an area and the product "abc" as a volume to confirm that the associative, commutative, and distributive laws hold.

More from SolitaryRoad.com:

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

[ Home ] [ Up ] [ Info ] [ Mail ]