# To infinity and beyond

What’s a bigger number, the total number of atoms in Dwayne “The Rock” Johnson’s body or the total number of microprocessor clock cycles since the dawn of time?

Think of a big number. A really big number. Grains of sand on a beach. Bigger. On all the beaches. Bigger still. Stars in the universe. The number of atoms in Dwayne Johnson’s body. Microprocessor clock cycles since the dawn of time????

A few things I inherited from my father – other than a prematurely receding hairline and a predilection for dad jokes — were a love of math and an insatiable curiosity for bizarre and random statistics. As a young kid, I became fascinated with the concept of “countable infinities,” an esoteric and weird mathematical concept proposed by Georg Cantor (an esoteric and weird mathematician from the late 1800s). OK, I was a weird young kid —but perhaps not as weird as Georg — who believed his theory of transfinite numbers had been communicated to him directly by God.

I won’t dig into the theory of countable infinities — mainly because this allows me to elegantly sidestep the fact that I don’t really understand it (mentioning it as if I did makes me sound much smarter than I am), but it did get me thinking about __really__ big numbers. In ancient times, the philosophers would ponder huge numbers in terms of physical quantities – how many grains of sand on the beach? How many stars in the sky? Today, we spend too much time glued to our smartphones to bother going to the beach or looking at the sky. Accordingly adapted to our modern time, one more contemporaneously huge number that occurred to me is microprocessor clock cycles.

Just a bit of background on the CPU clock. As a mechanical engineer, my grasp on this is tenuous, and most certainly flawed, but as near as I understand and can explain, the true magic of a computer lies not in its brilliance but rather the ability to subdivide and break the amazing things it does down into gazillions of ridiculously simple tasks. It’s true, the fundamental capabilities of a computer are all distilled down into a set of remarkably mundane logical tasks – all basically comparing a pair of zeros and ones and passing on the results. That’s it. That a computer can do these tasks doesn’t seem all that amazing.

What’s amazing to me is that zillions of these simple tasks can be combined almost instantaneously in an artful way that allows me to type this blog or watch a silly cat gif. But they can. What makes cat gifs possible - is that computers are really, really, really, really fast at doing those simple comparisons. Incomprehensibly fast. Unfathomably fast. Mind-blowing fast. In every computer, there is a “clock” – basically the metronome conducting each one of these basic operations. And that clock **FLIES!** Tempus fugit. On steroids.

Example – that smartphone in your pocket most likely has a clock speed around two GHz. We sometimes hear those numbers without taking the time to think about them. So, let’s break that down. A Hertz – just means “times per second.” A Giga is a one followed by nine zeros. So, two Giga Hertz means your smartphone is executing one of those basic comparison operations **2,000,000,000 TIMES PER SECOND**. Not just this second but *every *second.

Think about that for a second. No wait — don’t. It probably makes your head hurt. That means in the time it’s taken to read this; your smartphone has quietly gone off and executed about 12,000,000,000 clock cycles. Oh, and it’s not like YOUR phone is so special (although that * is* a lovely Hello Kitty case). Oops… there goes another 6,000,000,000 ticks of the clock. Where was I? Oh yeah, how fast the clock in your phone is. It is. So is the phone in your neighbor’s pocket. The computer on your desk is likely a bit faster. By the way, the main processor in your phone isn’t the only thing with a ticking clock. The Wi-Fi chip, PMIC, cellular module, Bluetooth module, are all ticking away too. Not to mention multi-core processors… Whew. And it turns out that phones and computers are just the tip of the iceberg when it comes to CPUs and their attendant clocks.

There are __literally__ billions of microprocessors manufactured every year and only a tiny fraction of these end up running cat videos on smartphones and computers. The remainder serve as the brains of far more pedestrian applications – the clock on your microwave, your thermostat, heart lung pumps, traffic signals, airplane navigation, your fitness monitor. Just about every conceivable modern product that runs on electricity has some sort of microprocessor humming away in its guts. Depending on its vintage, your car might have 50 or more — most likely running a bit slower than your smartphone but still buzzing away with their clocks ticking away at fairly scalding speeds.

Every one of these microprocessors is adding up clock cycles at a furious and unfathomable rate. So, still being a somewhat weird adult, I got to wondering – if you were to add up all of the clock cycles on all of the microprocessors built since the dawn of time, how incomprehensibly big would that number be? How does it compare to the other really big numbers of antiquity? Like the stars in the sky or atoms in the Rock’s body?

### Are you math-curious?

Doing this requires a bit of math – and making a handful of completely indefensible and certainly incorrect assumptions. If you’re math-curious, you can follow along and feel free to spot mistakes and poke holes in my very questionable reasoning. ** If you’re not so curious, you can skip to the end*** and see what I came up with.

Let’s start with how many microprocessors have been made since the dawn of time. One of the fascinating things about this particular “countable infinity” is that unlike grains of sand in the world or stars in the universe, it has its genesis in my lifetime (so does the number of Atoms in Dwayne “The Rock” Johnson, but more on that later).

It’s generally agreed that the first commercial microprocessor was the Intel 4004 —invented in 1971 — less than 50 years ago. So, one useful data point is the assumption that the total number of microprocessor clock cycles in 1970 was essentially zero. So that’s a start. From there on it gets a bit fuzzier. I found an article referenced by Wikipedia that claims the total number of microprocessors manufactured in 2008 was around 10 billion(!).

There’s no doubt that the number of microprocessors manufactured has been increasing at a staggering rate. But how staggering? To estimate the total number of microprocessors made since the dawn of time I made the heroic assumption that the number of microprocessors manufactured is following a second-order polynomial equation. Please note that this is a mathematical trick. By taking an assumption that I pulled completely out of thin air and giving it a fancy name, I’m tricking you into thinking I know more about what I’m talking about than I actually do. But I don’t. It’s sort of a total guess. Got it? Onward.

So with a bit of guesswork, let’s assume that the total number of microprocessors manufactured per year can be represented by the following equation: **CPUs_per_year = 7.305*10^5*(year-1971)^2 **and looks like this.

This has the advantage of predicting zero CPUs manufactured in 1971 and 10 billion in 2008, which we know to be roughly true. And intuitively, given the increase in electrical stuff, it seems and looks plausible.

Next, I had to guess at the clock speed for all those CPUs. This is even fuzzier yet. Because the early processors were slooow — and later processors have gotten faster. That 1971 Intel 4004 clocked in at a breathtaking 74,000 hertz. But I wasn’t able to find much literature on historical embedded micro clock speeds.

There’s plenty of information out on personal computer clock speeds, which passed a megahertz in 1995, a GHz in 2000, and interestingly, have plateaued at around 4-5 GHz recently. So, the increases in PC clock speed have been dramatic, nonuniform and lumpy. But embedded micros, which form the bulk of the microprocessor population, run slower and presumably evolve less chaotically. So, I assumed that the __average__ microprocessor was running about 1/10^{th} as fast as its PC contemporary – and took the huge leap that the increase in average embedded clock speed since 1971 has been roughly linear and thus can be described by the equation: **Average_clock_speed = 1*10^7*(year-1971) +7400**.

As long as I’d made two highly simplified and undoubtedly flawed assumptions, I kept going and assumed that the average life of a CPU was 2.5 years before it was shut down forever and stopped counting. Probably more for your microwave clock and a lot less for your Fitbit. But as good a guess as any. Thus the equation summarizing the total clock cycles since the dawn of time involved integrating *number of micros built X clock speed X average life X seconds in a year over the last 48 years* and looked like this.

For the math-skippers ...

** *Pick back up here.** I did it all logically and perfectly. Trust me.

What did I get for the total number of all CPU clock cycles since the dawn of time?

**764,000,000,000,000,000,000,000,000**

That’s a seven followed by 26 zeros. And counting. A number so big and inconvenient to write that us engineers truncate it to 7.64 X 10^26 using scientific notation. That’s a big, big, big number. And getting bigger.

So how does that compare to the other big numbers I mentioned?

Grains of sand in the world? A paltry 7.5 X 10^18. By my estimate, there have been 100 million more __times__ total CPU clock cycles than grains of sand.

How about stars in the universe? Well that’s a very big number indeed. By one estimate, 1 X 10^24. But still dwarfed by our clock cycle count, which is now up to 764,000,000,124,675,000,000,000,000 and growing.

But atoms in the Rock’s body? Now that’s a REALLY, REALLY, REALLY big number. Using this as a guide and assuming that the Rock weighs about 260 pounds, I’d estimate the Rock is packing in around 1.2 X 10^28 atoms. Since we’re trying to show really big numbers, I’ll eschew the more convenient scientific notation and share that’s 12,000,000,000,000,000,000,000,000,000 atoms.

So, Dwayne —the Rock — wins. What else is new? He always wins. He has roughly 15 times more atoms in his body than CPU clock cycles since the dawn of time.

But take heart —if my equation is correct (and it’s not) and the pace of clock speed and microprocessor innovation remains the same (which it won’t), the number of CPU clock cycles since the dawn of time will surpass the number of atoms in Dwayne “The Rock” Johnson’s body around the year 2070.

Thanks, Dad … you just cost me a wasted Sunday morning.

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