Reality Check: You Already have

All the Money You'll ever Need?

Let's substitute some real numbers into the variables M and N.
Let M=3 and N=5. So, **(N-M)** is 2, the average
(A) is 4 and 2A is 8.

**(N + M) (NM) = 2A (NM)**
(1)

(5 + 3) x 2 = 8 x 2

16 = 16

**N ^{2} - M^{2} = 2AN-2AM**
(2)

5

25 - 9 = 40 - 24

16 = 16

**N ^{2} - 2AN = M^{2}
- 2AM** (3)

25 - 40 = 9 - 24

-(16) = -(16)

**N ^{2} - 2AN + A^{2}
= M^{2} - 2AM + A^{2}** (4)

25 - 40 + 16 = 9 - 24 + 16

0 = 0

**(N - A) ^{2}
= (M - A)^{2}** (5)

(5 - 4)

(1)

1 = 1

And in equation (6), taking the square root of both sides leads us to:

** (N - A) ^{2} =
(M - A)^{2}**

or:

(1)^{2} = (-1)^{2} **--->**
( 1 ) = ( -1 )

**Hold everything! You can't do that !** This
is the error in the proof, and here's the reason why:

Since *the square of a real number n*
(or

**Never take the square
root of a negative number.**

**(**
Note: This is similar to another Mathematical Rule that you must **never divide
by zero**. **)**

Apart from this constant rule which we finally ran into, I'd also encourage
anyone working with *real-life applications* of Mathematics to examine
and state the conditions under which any particular problem must agree with
before trying to solve it or provide a proof.

For example, although we finally ran into a mathematical rule showing this
*proof* to be incorrect, there was clearly a questionable operation
performed prior to equation (**6**): Although you might not think so,
there's nothing wrong with using a negative number to represent the money here.
A negative value would symbolize a *debt* or *payment you might owe*,
thus it is possible to assign a negative value, for example, to cash you have
borrowed or must have in the future to complete some project. However, most
of the equations in this puzzle, such as equation (**5**),
have a real problem: It would be similar to saying that a person can eliminate
their debt to one creditor by simply multiplying it by yet another debt; and
not only that, but gain the same amount of cash in the process! Sorry, but life
doesn't work that way!! Our *proof* was still mathematically correct *at
that point,* **but** it violated how you could apply that math to increasing
personal wealth in real life. (Many people have been fooled into parting with
their own money because they believed a *scammer's lies* that "seemed
too good to be true!")

So, it's always a good idea when a problem deals with money (or more specifically,
the lack of it), to state that we should never multiply two negative numbers
together! [Accountants have often used red ink
rather than *a minus sign* to symbolize debts versus credits for better
understanding; we hate credit card bills, because they often use *minus signs*
in very confusing ways!]

Whenever you're using mathematics to handle problems with physical objects,
you must make sure that all the ** mathematical operations** agree
with reality for the present

**Back to The Starman's Math Index**