*Nothing comes from nothing; nothing ever will.* – Oscar Hammerstein

Boy, is that lyric ever wrong. (Bonus points, if you can name the song without Googling it.)

Recently, I read Charles Seife intriguingly titled book, *Zero: The Biography of a Dangerous Idea* and while I know it was recommended to me, I

cannot recall who provided me with the recommendation. I thought it was Ron Baker, but no, he has not read it. So, in a sense, even this review comes from nothing.

In this short, but fun read, Seife traces the history of the zero from its humble beginnings as an idea, through its use as a placeholder, to its current status as the only number capable of destroying everything (including temporarily disabling the USS Yorktown on the high seas. The humble zero shook the foundations of philosophy, caused arguments, even wars, and now is beginning to play its part in understanding the very origins and substance of the universe.

While I highly recommend reading the book, I did extract a few nuggets for reading here.

First, there are at least three ways that we can prove that something comes from nothing or to express it mathematically that 0 = 1.

Proof A:

Let a = 1 and b = 1; therefore b = a.

b^{2} = ab, Premise 1

a^{2} =a^{2}, Premise 2

a^{2} – b^{2} =a^{2} – ab, subtract premise 1 from premise 2

(a + b) (a – b) =a(a – b), Factor

a + b = a, Divide both sides by (a – b)

b = 0 Subtract a from both sides

Since a = 1 and b = a, therefore 0 = 1.

Proof B:

Let a = (1 – 1) + (1 – 1) + (1 – 1)… = 0

Let b = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1)… = 1

The ellipses in the previous two equations indicate the infinite repetition of the parenthetical express. Therefore the two are arithmetically the same, they are just grouped differently, therefore a = b, therefore 0 = 1.

Proof C: (hat tip to Joe Santoro)

0^{0} = 1, therefore 0 = 1.

Why is this so important? Well, because if you can prove that 0 = 1, you can then prove that 0 = anything, certainly any number. For an example of this simply replace the 1 that leads the Proof B, let b example with any whole number, fraction, or decimal number. (By the way, there are an infinite number of decimal numbers between any two whole numbers.) In short, proving that 0 = 1 is the ultimate in understanding the idea of subjective value – value is what the customer says it is.

What is fascinating about the book was learning that some of the foundational mathematical concepts that we accept today are based on this idea that 0 = 1. Calculus is based on the idea than we can compute the area of an irregular object as long as we allow our estimations to set 0 = 1. Had this been explained in this manner to me back in high school or college perhaps I would not have dreaded the course so much. I equated calculating the area under a curve to torture. Theory is important.

One last idea I had while reading this book was to create a new formula (tongue in cheek of course) for computing value. Seife clearly points out that calculating speed in the old “A train leaves New York” type word problems is not always as simple as distance = rate times time. What train moves at an absolute constant speed? Any train would begin slowly, speed up, slow down around curves and crossings, etc. It does no good to calculate speed at a constant rate, i.e., 140 mph for 3.5 hours. What was required for mathematicians was a higher degree of understanding. They achieved this through Newton’s calculus. (Remember 0 = 1.)

An example would be calculating the velocity of a falling object, which falls at 32 feet per second per second. After one second, the object is falling at 32 feet per second after two seconds it is falling 96 feet per second and so on. This is expressed as v = gt^{2}/2 where v is velocity, g the force of gravity and t is time. As Seife puts it, “Rate times time equals distance is not a universal law; it doesn’t apply under all conditions.”

Look at the last sentence again and replace the word **distance **with **value**. See where I am going. Imagine a formula for a professional firm using the “falling object” model (which, by the way, while still wrong, is less wrong than value = rate x hours). It would look something like this — value = rate x hours^{2}/2. Picture explaining this to a customer, “Well we bill you $150 for the first hour, than $300 for the second, $675 for the third, etc., since we spent a total of ten hours on your matter, our bill comes to $7,500, any questions.”

Again, while still wrong, the above is less wrong. What is needed is for professionals to make the leap that 0 = 1 (that value is subjective). VeraSage has been sounding the alarm on this for years. Dare I say Ron Baker = Issac Newton.

We now have the math! Worshipers of the ABH give up! We have proven you wrong again, this time with your own beloved mathematical formulae.

Man I hate to do this to you……

Proof A:

Let a = 1 and b = 1; therefore b = a.

b2 = ab, Premise 1

a2 =a2, Premise 2

a2 – b2 =a2 – ab, subtract premise 1 from premise 2

(a + b) (a – b) =a(a – b), Factor

a + b = a, Divide both sides by (a – b)

Let’s stop right there…..

a=b from line one

Therefore a – b = 0

YOU CAN’T DIVIDE BY 0!!!!!!!!!!!!!!!!!!!!!!

Basic 4th grade elementary students know that. Dividing by 0 produces the result of infinity. Infinity is NOT a number. It is a theoretical concept.

Proof B is nonsense. The second line is a lie. b does not equal 1 in that equation. b = 0. Doesn’t matter how you group the numbers.

Proof C is very misleading. Any number raised to the exponent of 0 is 1 by definition. That does not make the root number equal to 1. 1,000,000,000,000,000,000 ^0 = 1.

45,265,987 ^ 0 = 1.

-12,365.235874 ^ 0 = 1

So sorry, Proofs debunked.

Liked the article though.

Eric,

Thanks for the response, and I hate to do this to you, but…

Proof A is the foundational construct of calculus. You are right division by 0 is not allowed, at least in a Fourth grade class, but that is because the teach can’t explain it to the students. In higher math it is not only allowed by assumed.

Also, infinity is a number, it has been proven, you must read the book for the details. It is like e (the square root of -1), it can’t be calculated, but must exist or other things cannot be true.

Your interpretation of Proof B is wrong. If you group the numbers the second way an go on infinitely, the value is whatever the first number is. Similarly, I could substitute any number for 1 in the entire sequence.

You say Proof C is misleading, why, because it proves the point? The point is that any number raised to the 0 power is 1, even 0, even infinity. The point is that value is subjective, even for numbers is subjective.

Ed, Ed, Ed……

In NO math is division by zero allowed. Not in algebra, not in calculus, not in geometry, not even in chaos theory.

Dividing by zero makes no sense in our mathematical construct. Putting it in laymen terms, dividing by zero says take this set of objects (the numerator) and equally distribute the pieces into zero buckets (the denominator). How many pieces to I allocate per bucket?

Doesn’t make sense. There are no buckets to allocate the pieces. The answer is not any number. Once you divide by zero you leave the realm our our mathematical system.

Proof B I must admit I was very unclear. Basically, you cannot start at the left of the formula and get to the right of the formula, through the equal sign, without coming to the end of the numbers to calculate.

If, indeed the numbers are the same, the rightmost number of formula b will be -1, bringing the answer to 0.

If the rightmost number is not -1, the formulas are different. Therefore the answers should be different.

If you never get to the rightmost number, you cannot move through the equal sign.

For Proof C, writing an exponent is a short hand method for writing a formula.

2^3 is actually 2 X 2 X 2 = 8

10^2 is actually 10 X 10 = 100

5^-3 is actually 1/5 X 1/5 X 1/5 = 1/125

So

4^0 is actually X =

(ie write down the number 4, 0 times and multiply). Also doesn’t mak a lot of sense. Mathematicians have defined #^0 to be equal to 1 because of the exponent graph.

Because it is a definition, like 3 is defined to be a whole number between 2 and 4 and 1 is NOT a prime number, it really stands beyond proof. You either accept the definition or not. And if you change the definition, you change the mathematical foundations, you get very different results tha if you stayed inside the mathematical rules.

Now, isn’t that really your point? Changing the rules (billable time vs value based fees) changes the results. And isn’t it proven that using value based fees produce far superior relationships and results?

One last go round on this. I am actually more interested in feedback about the new “value” equation of value = rate x hours^2/2, but here goes.

First, Eric read the book. When Newton worked out his formula for derivatives (calculus) he divided by zero. As Seife puts it, “Newton’s method of fluxions was very dubious. It relied upon an illegal mathematical operation, but it had one huge advantage. It worked.”

In a sense proof A & B are mathematical tricks, A relies on zero and B relies on infinity. Both prove the subjectivity of even math.

Proof C is a definition, you are correct, but a definition that proves that something (1) can come from nothing.

BTW, the most perplexing indeterminate is infinity raised to the zero power. 😉