This is Math Newsletter number 5; Wednesday, August 25, 2010.
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Meaning of Equality

What does it mean for two different number expressions to be
equal?

It means that the two different number expressions are each
reducible to the same number.

What does it mean that 2 + 5 = 3 + 4?

2 + 5 reduces to 2 + (4 + 1) = (2 + 4) + 1 = 6 + 1 = 7
3 + 4 reduces to 3 + (3 + 1) = (3 + 3) + 1 = 6 + 1 = 7

There are many degrees of equality.

If we say that two things are identical, we mean that they
are the same in all intrinsic characteristics. They would
still be different by being in different places or different
times.

In a weaker use of "identical", we refer to identical twins
who have the same genetic makeup, but different environmental
experiences.

Another way in which two things could be equal is if they are
different names for the same thing. Some writers use this
point of view when they say that "2 + 5" and "3 + 4" are
different names for the number 7.

Another example of the "different names for the same thing"
is if I say "Let p be a variable name for a number. At the
beginning of an algorithm let p take on the value of 7. Let
A be a constant equal to 7.

Then at the beginning of the algorithm, p = A.

When we introduce remainder arithmetic, we see yet another
level of equality.

9 and 2 have the same remainder when divided by 7.

9 = 2 modulo 7.

All concepts of equality, (worthy of the name),
have certain characteristics in common.

Consider the set of objects to which this particular concept
of equality is being applied.

It might be the set of integers. It might be a larger set of
numbers. It might be other objects which are not usually
considered to be numbers.

The thing that all concepts of equality have in common is
that the equality relation partitions the set to which the
equality relation is applied.

All elements within the same partition are considered to be
the same with respect to everything that matters.

For example, with the "have the same remainder, when divided
by 7" relation, the integers, {0,7,-7,14,-14,21,-21,...} are
the same with respect to "remainder when divided by 7".

All the integers {2,9,16,23,...} are the same with respect to
"remainder when divided by 7".

In a social setting, when one states that men and women have
equal rights, we really mean the following. Your gender is
not to be considered important with respect to determining
what your rights are to be.

We also may extend the concept of equality to games.

Here are two games that I will call equal.

game 1: Tic Tac Toe.

A pair of vertical lines is crossed perpendicularly by a pair
of horizontal lines, producing 9 squares.
Players take turns placing either an "O" or an "X" into one
of the squares. Player who makes "three in a row", a
straight line, of three of their symbols, wins the game.

game 2: fifteen.

Blocks or pieces of paper, numbered 1 to 9, are placed
between the two players. Players take turns selecting a
number from the pieces. Once a piece is taken, it is no
longer available for the other player to take. The first
player to be able to have three of eir pieces add to 15, wins
the game.

These two games are equal.

Here is the proof of why I can say that these two games are
equal.

8 1 6
3 5 7
4 9 2

Here are the 9 numbers placed on the tic tac toe board.

As can easily be seen, every straight line of numbers that
would have won a tic tac toe game has the numbers add up to
15.

And, it is also true the other way around.

Every set of three numbers on this tic tac toe board,
that add to 15, lie on a straight line.

So now you know that every winning tactic for "tic tac toe"
can be translated into a winning tactic for "fifteen".

In later issues we may discuss equality of different number
systems. When is it valid to say that one number system is
equal to another number system? Mathematicians use the word
"isomorphic" in such cases. Historically we say that the two
equal number systems are "isomorphic", meaning that they have
the same body of relationships.

Kermit Rose