Euler’s Equation

by Neil James Hudson

e

     When I first met Euler’s equation, I thought it was proof of the existence of God.  e, the base of natural logarithms, underpinning the whole of mathematics.  i, the square root of minus one, the unit of complex numbers.  π, the relationship of a circumference to a diameter, from which geometry is made.  1, unity, the foundation of all numbers.  And we all know about 0.

     e i π+1=0, said God, and there was light.

     So when Euler’s equation fell apart, I knew we were in trouble.  We held an emergency meeting at the maths department at the university, which was fast approaching a 0 of its own.  But as long as we were still there and were still being paid, there was work to be done.

     Professor Hazlitt chaired the meeting, the only one of us who had actually done any work of real note.  “It seems to have happened at about 10:30 this morning,” he said.  “Before then, the equation seemed to hold.  But now, no matter how hard we try, we just can’t get the terms to fit together.”

     “Could we just have missed a flaw in the proof all these years?” I asked.

     Hazlitt shook his head.  “The equation definitely worked yesterday: it was holding everything else together.  But now something’s changed.  Mathematics has changed.”

     “Euler’s equation was the proof,” I said.  “It was God’s covenant with mankind, like the rainbow after the flood.  It said, the universe isn’t just chance, it’s designed.  All the basic elements of reality fit together like a jigsaw.  The equation is God’s signature, just to let us know he’s still here. But now the equation doesn’t hold; God has left the building.”

     “Frankly, Dr Carlton, I had hoped for something more helpful,” said Hazlitt.  “If we can identify the change, we may be able to rescue mathematics.”

     “It’s not given to us to mend the universe,” I said, and indeed the meeting agreed to do no more than to monitor the situation.

#

i

     But by the next morning, -1 had a real square root.  It was its own square root, like 1.  Multiply -1 by -1 and you got -1.  This hadn’t happened before.  Complex mathematics was wiped out at a stroke.

     “This can’t be happening,” said Hazlitt, and I felt pity for him.  Complex numbers had been his speciality, and now there weren’t any.

     “Mathematics is our best description of the universe,” I said.  “The universe is getting simpler.  We’re winding down.”

     “But isn’t there something we can be doing?  Shouldn’t we be praying, or trying to, I don’t know, get our souls in order?”

     “The game’s over,” I said.  “It’s as if the exam’s just finished, and we’ve handed in our papers.  You can carry on working through the problem if you want, but it won’t affect your grade any more.  It’s too late to be good..”

     “I don’t suppose you’re upset. This is what you’ve been waiting for.”

     I allowed myself a small smile.  “I didn’t expect it to be like this.  Frankly, I don’t know what’s going on.  I expected God to finish it, not wind things up.  I think there may be another entity at large in the universe. If God can make an equation, I can only think of one being who could unmake it.”

     Professor Hazlitt left angrily, leaving me to my thoughts. At least I now understood something that had been puzzling me. For centuries, people had been obsessed with the number of the Beast. Everyone tried to understand the number itself, but no one understood the real significance.

     Before long, the Beast would be the only thing left with a number.

#

π

     There seemed little left for me to do but to go home.  I was depressed:  I didn’t think I’d scored well enough in my own personal exam, and in any case I wasn’t sure who was in charge here.  God, I felt, may have broken his own covenant, and if you couldn’t trust God, who could you trust?

     Numbers were falling apart everywhere.  Things that were supposed to be equal were greater than each other.  The basic relationships that underpinned the universe had become exes.

     Wearily I got in my car and started the engine.  As I tried to reverse out of the car park though, the car juddered as if it were moving over a pile of rocks rather than the flat tarmac surface.  I got out, knowing already what I would find.

     The wheels were out of shape.  It took me a while to see it, but the circumferences were completely out of proportion to the diameters.

     Well, that’s geometry buggered, I thought.  I could see no choice but to return to the maths department.  Every shape I looked at seemed wrong, and I wondered how long it would take before the Moon fell down.

     We had failed humanity. We were mathematicians; people should have looked to us for answers. Instead we just described everything, and expected it to work as it should.  We had never looked at how to keep the system going.

     I was interrupted by Professor Hazlitt, bursting into the room with panic on his face. “Dr Carlton, how many of us are there in this room?”

     “Well, there’s me, that’s one,” I said.  “And there’s you.  That’s another one.  So….that’s more than one.”

     “But how many more?”

     “Let me think,” I said.  “There’s at least one more than one, but….how many ones are there?”

     And then there was only one anyway.

#

1

     Because everything was one.  I’d been right, there were no other numbers left.  There were no distinctions to be made between me and anything else:  it was all one.

     Before now, I’d often thought of the differences between myself and Jasmine.  But now there no such differences.  We were no longer separate, discrete, countable.  We weren’t even we:  we were I.  Every atom that had ever joined with another was now the same atom.  The molecules were just one atom.  I myself was that same atom, and far from being a small part in a large universe, I was the universe.

     Was I God then?  All I knew was, there was no God other than me.  That would imply a separate entity, another number.  Another universe, in fact, to house another atom.

     God could not exist to create the universe:  God was the universe.  The distinction could not apply.  No distinction could apply.  Could I see?  I didn’t know.  I couldn’t draw the line between what I was looking at, and the person looking.

     I was total and complete existence.  I stretched across the universe, engulfing all.

     I had a bad thought.

     Euler’s equation was collapsing, coming apart at the seams.  Term by term, the universe had been unpicked until there was only 1 left.

     But there wasn’t only 1 left.  Even when 1 was all there was, there was still something other, something not 1.  Even God had a Devil.  And 1 had its

#

0

     e i π +1=0, someone had written on the blackboard.  “You’d better believe it,” I wrote underneath.

     The numbers healed.  From our non-existence we were returned to unity, then discreteness.  Geometry returned to its standards, real numbers realised that they weren’t the only option, and finally logarithms returned to their natural ways, and Euler’s equation, the key to the universe, fitted together once again.

     And I realised what had happened.

     I was quite wrong to view the equation as a covenant.  It was a warning.  Everything was there:  logarithms, complex numbers, geometry, real numbers.

     And there, right on the other side of the equation for all to see, was a big zero.

     When we had been non-existent, when everything had been zero, nothing had actually changed.  We were still equal to all the terms on the other side of the equals sign.  Those terms contained the universe.  Which means:

     The universe isn’t real.

     In the beginning, God created the Heavens and the Earth.  And this isn’t it.  So to warn us, He gave us Euler’s equation.  He fitted the terms of the universe together to show us that it all added up to nothing.

     It’s easy to see our mistake now.  The equation is three-dimensional.  We can see the terms lying flat, but there are other equations at right angles to the equals sign.  We can’t see them because they’re edge-on.  What we have to do is tilt the equation so we can see the other three-dimensional terms.

     These terms define the real universe.  If we can unravel them, we can find out what the real universe is like.  And if we can describe it, we may work out to get there.

     Professor Hazlitt retired, dreading the challenge of the new mathematics.  It doesn’t matter; ultimately, neither of us exists anyway. I have a new team, and we’re working to find the equations that describe the real universe. Somewhere, there is a world with a God, where real people can work and love.  Where Euler’s equation doesn’t make 0.

And this time, we won’t just describe it. People look to us for answers now, and we will find them.  With our constants and our mathematical relationships, we will find God.

~

Bio

Neil James Hudson is the author of around fifty short stories and the novel “On Wings of Pity”. His story collection “The End of the World: A User’s Guide” can be ordered from his website at neiljameshudson.net. He is currently working on a long series of vignettes under the title “One Hundred Pieces of Millia Maslowa”, some of which should be published in the coming months. He lives in the middle of nowhere on the North York moors, and works as a charity shop manager in York.

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