
Understanding Extreme Numbers - dnetesn
http://nautil.us/blog/-how-to-understand-extreme-numbers
======
jerf
This is not just an academic idea for programmers, it is something that you
will need to understand to get to the highest levels of engineering. In my
career, I have cared about how many nanoseconds an operation takes, and I have
optimized processes that required several hours to run. That's 9 orders of
magnitude down from one second, and if for convenience we go with my hours-
long process taking 10000 seconds (2.78 hours), that's four orders of
magnitude up.

Obviously, there are people with far more experience than I on the upside; if
you work with a process that will run on a 10,000 (CPU) node cluster that will
run for 1,000,000 seconds (11.6 days) that's 10 orders of magnitude up from
1CPU for 1sec... and they're actually still quite likely to care about
nanosecond level performance, too! Which means that someone working in this
situation is spanning _19 orders of magnitude_ in their day-to-day work.

19 orders of magnitude down from 1 meter is about five orders of magnitude
smaller than a hydrogen atom. 19 orders of magnitude up from one meter is
about 1000 light-years. It's a pretty big span.

There are not a lot of engineering disciplines, or really, disciplines in
general, where spanning this many orders of magnitude in a single job is
possible. When it comes time to start optimizing things, you need to know
where you stand in the world to make correct decisions:
[https://stackoverflow.com/questions/4087280/approximate-
cost...](https://stackoverflow.com/questions/4087280/approximate-cost-to-
access-various-caches-and-main-memory/4087315#4087315)

~~~
wiredfool
Two that have stuck with me:

A nanosecond is about a foot. A pico second is the size of a flake of pepper.

Adm. Grace Hopper gave me a nanosecond back in high school, and had a pack of
picoseconds at the talk. (But didn't give them away. too small)

~~~
dahart
That's extra helpful because these units are approximately the speed of light.
Light travels a foot in one nanosecond, and it also traverses the width of a
(small, ~.01 inch) pepper flake in a picosecond.

When I think about light travelling one foot in a nanosecond, it suddenly puts
CPU speeds into proportion for me -- CPUs are now routinely clocking 3-4
cycles per light-foot! A foot is a long way in CPU die lengths, but it's
surprising to think about how close CPU's are actually getting to the speed of
light already.

~~~
AnimalMuppet
I remember reading (clear back in the late 1970s, in Datamation magazine) how
much fun IBM was having trying to make 2 mainframe CPUs operate coherently
with sub-nanosecond clock speeds, when the two CPUs were several feet apart.
Just from a speed-of-light perspective, the problem is obvious.

------
dahart
> That means they correctly distinguish between numbers within the millions or
> billions, but assume that “million,” “billion,” and “trillion” are equally
> spaced on a number line.

Maybe normal people are good at log scales! Our number system makes log_10
pretty natural, and our naming system makes log_1000 somewhat automatic.

I think it's only fair to point out that log scales are how scientists and
engineers understand extreme numbers; nobody really has an "intuitive"
understanding of 10^36 or Graham's number.

I can feel this myself; it's easy to think of 1 million as a thousand squared
and feel like it's tractable, it's easy to think of a billion as a thousand
cubed and feel like it's relatively small. But when I actually plot or
visualize that many things, especially if the individuals have variation, I
can see that thinking of it in exponents of a thousand is misleading, and that
it's hard to get any sense of how many individuals are there and of the
complexity of variations and sub-groups that are possible.

------
d--b
> But about 35 percent of people in the study used what the authors call a
> “segmented linear heuristic.” That means they correctly distinguish between
> numbers within the millions or billions, but assume that “million,”
> “billion,” and “trillion” are equally spaced on a number line.

That kind of logarithmic thinking is a reason why I believe people don't react
more strongly when they hear that Jeff Bezos is worth 100 billion dollars.

The wealth of the super rich is so astronomical that people can't make sense
of it. It doesn't help to say he's 100,000 times a millionaire, because even
if we more or less know how wealthy a millionaire is, we can't imagine what
_multiplied_ by 100,000 means.

------
BrandoElFollito
Background: I used to be an physicist (PhD) and graduated as an engineer
(applied physics). I also actively promote science with children. I am
interested in the interpretation of numbers and quantities for many years.

I do not like this article because it fails to

\- show that we do not "feel" numbers larger than a few

\- and that we have a tendency to think in orders of magnitude (5, a few,
lots, plenty of many).

To start with: "many treated 980 million and 2 billion as nearly identical".
They were right, 980 million (0.98 billion) is nearly identical to 2 billion
when you look at orders of magnitude.

Missing this fact is one of the big problems we have at school, namely the
incapacity to estimate. We fetish exact numbers (3.14) instead of telling that
pi is 3, pi squared is 10 and g is 10. This way, children can understand
orders of magnitude and we do not end with (real case when I was teaching) a
skydiver who lands at 573.426625432423 km/h. I cannot deny that the answer is
precise to the upth-st digit.

Quantities such as "a million" do not make sense outside of the context. In
context they just become a number, not a quantity. For instance what is the
size of a hair multiplied by a million? Or a glass? I have no idea. Sure, I
can estimate that a glass is 5 cm = 5 x 10^-2 m, which makes it 5 x 10^-2 x
10^6 = 5 x 10^4 m. 50 km. But I had no idea before because I don't know what
"a million times" means outside of the numbers.

We are good at estimating small orders of magnitude (from experience, not more
than 3) and that's all. The rest is calculus and this is easy. But also
detached from reality, a scientist does not ponder on 10^9. This is jst 10^9,
a number.

------
zentiggr
Not claiming to be exceptional, just noting that a perspective I developed
seems less common than I expected...

I think I've just had a cognitive bias of my own revealed, and found a source
of confusion in conversation... I guess I've been sucking up any and all
science that interests me since I could read, I have no trouble perceiving
scale differences... Planck's constant, universe-scale light-year
measurements, Ackermann function values, they are all on a very long and
flexible scale of powers of ten, and that whole scale is mostly intuitive. I
sometimes even convert everyday numbers into E notation so I know I'm scaling
things right.

------
qwerty456127
Thanks to existence of the short and long scales I have no idea what people
mean when they say "a billion" or something like that today in the globalized
Internet-connected world. Please avoid using such ambiguous words at all
costs, use scientific notation instead.

~~~
sjcsjc
Aside from my mother, who died 18 years ago, I know no-one else who uses
billion meaning 10^12 (long scale, as you point out). I grew up, and still
live, in the UK.

I'd be very interested to know if anyone here uses long scale, or knows anyone
who does. Is it still in common usage anywhere?

~~~
danbruc
We here in Germany - and according to Wikipedia most countries in continental
Europe - use the long scale. But when I switch to English I also switch to the
short scale, I can not remember to have ever encountered the long scale in
English in the wild.

~~~
sjcsjc
Thanks. Obviously I didn't read the Wikipedia entry properly. I must say I'm
surprised. I had no idea that these terms were still equivocal.

------
mncharity
I suggest the article is missing a key point.

> Of course, our ancient human ancestors didn’t live among billions of people,
> or incur trillions of dollars of debt. The orders of magnitude in their
> immediate surroundings were limited to what they could experience firsthand.

A square kilometer might have a billion trees. With how many leaves? One
person's year might have a billion heartbeats. A pre-industrial farmer might
tend a million plants per year. A sack of rice might have a million grains.

> It’s not surprising that we can intuitively visualize a 6-inch or 6-mile
> distance,

Asking people how tall a skyscraper is, I get answers like "a mile?". And even
for horizontal miles, who is this "we" who can "intuitively visualize" miles -
people who drive?

> Things that are so far removed from our daily experience [...] are
> inherently hard to understand,

People are regularly innumerate and clueless about things that _are_ part of
their daily experience. Far remove is not a necessary condition. Nor is it a
sufficient one - experience need not be direct. How often do kids see real
horses? Or elephants? Or dinosaurs?

So I suggest the article is missing a key point.

Large quantities, large differences in scale, are inherent in the universe.
It's what makes estimation and approximation possible. People simplify with
units, landmarks and anchors, and rules of thumb. With experience and stories.
But education, and press, leave people largely innumerate, making almost no
effort to teach the needed landmarks. In science, and history, and civics, and
finance. The main exceptions being a few units of time, and of currency, which
are taught down to K.

Think of those little lists in introductory physics. For some measure (eg,
Volts), here are a few values. Usually poorly chosen. And _never_ built upon,
into a feel for reasonable values.

And then we get science folks saying nanometers are _unimaginably_ small. And
journalists that a trillion dollars is unimaginably large.

Yes, people are largely innumerate. But we collectively make almost no effort
to change that.

[http://bionumbers.hms.harvard.edu/](http://bionumbers.hms.harvard.edu/)
started not from the movement to teach introductory biology more
quantitatively, but out a graduate student's realization that they had no
quantitative feel for their own field. A characterization repeatedly made by
physicists about their own doctoral candidates. How many people here recognize
a Newton-meter of torque, as being the reopening a 2L soda bottle?

So a takeaway shouldn't be researchers doing something novel, to address a new
modern problem. But rather their getting around to doing something long known
and established, to address a long standing problem. A problem long,
pervasively, unnecessarily, and abjectly neglected.

~~~
bumbledraven
> A square kilometer might have a billion trees.

There are 1 million square meters in a square kilometer, so having a billion
would require 1000 trees per square meter.

> One person's year might have a billion heartbeats.

A person's heart beats around once per second, and there are only around 3 *
10^7 seconds per year. That's far from a billion (10^9).

~~~
mncharity
Excellent. Thank you for the belay. Yes, that should have been "million trees"
(order 1 tree/m^2 for pine forest; though orchard is order 0.1), and "a
person's life" respectively.

Meta: failure mode on both was switching/editing between several candidate
examples in place, and not checking that the final edit was consistent.
Including a sketch of the calculation is one way to make this less likely.

