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SN 020 Geometry 13 Inequalities:
By William E. Steinman:
An inequality is a statement that quantities are not equal.
If two quantities are not equal, the first is either greater than or less than the other.
Profound!
We us three symbols in dealing with inequalities.
> is the symbol meaning is greater than.
< is the symbol meaning is less than.
Examples:
42 > 24
15 < 51.
Inequalities of the same order can be combined.
For example if a > b and b > c, we can write a > b > c.
This simply means b is less than a and greater than c.
Obviously then, a is greater than c. Thus a > c.
Way back in SN 011we leaned that axioms are statement that are accepted as true without proof. Sometimes they are called postulates. We think of these things as being self evident. Here are some axioms.
A quantity may be substituted for its equal in any inequality.
If x > y and y = 10 then x > 10.
If the first of three quantities is greater than the second and the second is greater than the third, then the first is greater than the third.
See above in example a > b > c therefore a > c.
The whole is greater than any of its parts.
If equals are added to unequals, the sums are unequal in the same order.
If unequals are added to unequals of the same order, the sums are unequal in the same order.
If equals are subtracted form unequals, the differences are unequal in the same order.
If unequals are subtracted from equals, the differences are unequal in he same order.
If unequals are multiplied by the same positive number, the products are unequal in the same order.
If unequals are multiplied by the same negative number, the results are unequal in the opposite order.
If unequals are divided by the same positive number, the results are unequal in the same order.
If unequals are divided by the same negative number, the results are unequal in the opposite order.
Again we belabor the obvious, but this time it is not quite as obvious. You may have to think about these.
We have some rules about definitions.
All terms in a definition must have been previously defined or left undefined by agreement.
The term being defined should be placed in the next larger class or set.
For example Homo sapiens should be placed in the inclusive set animal.
The term being defined should be distinguished from all other members of the class.
The distinguishing characteristics of a defined term should be as few as possible.
Point, line, and surface are the terms in geometry that are not defined. We agree on that so we can build the definitions of all other geometric terms.
So, we can say a square is a kind of polygon.
It is a polygon with four equal sides.
We can defined a polygon in terms of a geometric figure.
A polygon is a closed plane figure bounded by straight line segments.
This takes us as far as we can go, for a line and a plane are, by agreement, undefined.
Recall that we learned in SN 011 that assumptions, which are called postulate or axioms, are agreed upon or taken as self evident. They have no proofs. These assumptions are then used to prove the theorems that form the basis of geometry. We can then use proven theorems to prove new theorems and so on. What we cannot do is violate the logical sequence. We cannot use a new theorem to prove a previous theorem. Okay!
That is quite enough for today!