Very Small Numbers

Very Small Numbers or Extremely Small Numbers refer to those small numbers that people fail to fully grasp or understand their scale and significance. For instance, Many people will interpret 99.9% and 99.99% as the same number, a psychological phenomenon known as the assumption of numerical equivalence. Partially this is because cognitive bias affects problem solving and decision making. Sometimes the manner in which numbers are presented can affect how people interpret them. But also some numbers are small on a scale that most people fail to comprehend.

Images taken of the same Galaxy (M100) before and after NASA corrected a mistake on the Hubble telescope of 2 microns. ^[1]

Some small numbers have huge consequences. For instance the Hubble Telescope, when it was first launched, sent back blurry images. It turns out that the primary mirror, an object \(2.4\) m in size, was ground down, along the edge, 2 microns too thin. That's one millionth of one meter, or \(1\times10^{-6}\) m too thin. And that difference caused the telescope to focus only \(10\%-15\%\) of a star's light, when it was supposed to gather \(70\%\) of a star's light into the focal point. NASA had to send another manned mission to space to fix the telescope at an estimated cost of \($1.5\) Billion, that's \($1,500,000,000\) to fix a \(0.000001\) m mistake.

Common Small Units of Measurement
Units of measurement sometimes make small numbers appear larger than they are. For instance, from the problem above, a hydrogen atom is 52.9 pm. That doesn't seem so small. But it is.

Small numbers expressed in Scientific Notation
Numbers like \(0.1 \times 10^{-1}\) and \(0.5 \times 10^{-5}\) might appear very similar. But it's the difference between \(\frac{1}{100}\) and \(\frac{5}{100000}\) or, put another way, if a donor has \(\$100\), and gives each of \(100\) recipients \(0.1 \times 10^{-1}\) of his \(\$100\), then each recipient would have \($1\). That same donor would find it impossible to to divide the \(\$100\) donation \(0.5 \times 10^{-5}\) parts, unless the donor is capable of giving away recipient will get \(\frac{5}{10}\) of one penny. This gets more complicated as the exponents get larger.

Expressing Small Numbers as an Exponent
Small numbers can also get more complicated as exponents compound. For instance: \[2^{-2^{-100^{24^{2*3^{468}}}}}\] Numbers like these can arise when solving for complex numbers or working in small spaces like Quantum Mechanics.

Infinitesimally Small Numbers
Some numbers, particularly one-sided limits of series, can descend towards the infinite. Archimedeas' archimedean property defines a number \(n\) as infinite if \(n\) can satisfy the property, \(|n| > 1, |n| > 1 + 1, |n| > 1 + 1 + 1, ...\) and \(n\) to be an infinitely small number if \(n \neq 0\) and a similar property holds. However, a very small natural number retains properties that infinity does not, or may not.

Medical fields, like Health and preventative medicine are just two areas where small numbers affect the everyday lives of everyone. For instance, a test might be said to be 99% affective, or an antibacterial soap might advertise that it kill 99% of bacteria. While the claim seems impressive and may make customers feel safe, several vital pieces of information are missing, such as: 1. What percentage of bacteria is killed by a competitor, such as regular soap? - I.E. Is this a large difference or a small difference. 2. What sort of bacteria and how much bacteria does it take to make a person sick? - what if the 1% is sufficiently dangerous? Perhaps a customer might be better off with a product that's 99.9% effective?

Cancer offers a good example of these numbers at work. Cancer is a disease where human cells lose their control mechanisms and multiply at an abnormal and inappropriate rate. Most malignant tumors contain about one billion cells before they are detectable. But some late stage cancers could have one trillion cells. This difference, between one billion and one trillion may require vastly different treatments to achieve the same result.

For comparison, an early-stage tumor containing about one billion cells (\((10^9\)) would be reduced to 10,000 cells by the same treatment.

This is one of the reasons why early detection and treatment are key to effectively treating cancer, because the number of tumor cells can easily increase by an order of magnitude requiring either far more affective treatments or leaving an order of magnitude more cancer cells remaining after treatment.

The difference between one very small number and another very small number can actually be very large. For instance the difference between \(0.0000001\) and \(0.0000000001\). But it's easy to gloss over these similar looking numbers. It's also easy because sometimes small adjustments result in huge differences.

Just a few more coin flips all coming up tails changes the odds dramatically. But it doesn't seem all that dramatic, not when stated like that. But the above problem can be reconfigured on the scale of time.

Let's suppose you have a machine to try and hit one-in-a-million odds. It flips twenty coins at once. As said before, the odds of all twenty coming up tails is about \(1 \text{ in a } 1,000,000\). Let's assume that if you run the machine through \(1,000,000\) trials, one of them will come up with twenty tails. (Although, because these are independent events there is still some chance after one million trials that not a single trial would have all twenty coins to be tails). If it took just \(2\) seconds to run and reset each trial of the machine, it would still take \(2,000,000\) seconds or \(23 \text{ days} 3 \text{ hours} 33 \text{ min and } 20 \text{ seconds}\) for all \(1,000,000\) trials to complete.

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Now let's suppose we turn that machine into a one-in-a-billion machine. We add enough arms to it, that it flips more than twenty coins a second, trying to make all of them turn up tails at once. If the only way to guarantee success is to run \(1,000,000,000\) trials on the machine, and each trial again takes only \(2\) seconds to run and reset. It would take \(2,000,000,000\) seconds to run.
The one-in-a-million machine took just \(23 \text{ days } 3 \text{ hours } 33 \text{ min and } 20 \text{ seconds}\) for all \(1,000,000\) trials to complete. Much less than \(1 \text{ month}\).
This one-in-a-billion machine would take \(63 \text{ years } 137 \text{ days } 9 \text{ hours } 33 \text{ min and } 20 \text{ sec}\) (assuming there are \(366\) days in a year every fourth year) or roughly one human lifespan.

The problem below shows a classic example of how a regular number can become a very small number very quickly.

This problem shows how scaling very small numbers can help to show contrasts between two very small numbers.

Some very small numbers are constants or measurements from Quantum physics, Astrology and Cell Biology. Below are examples of a few of these, with the idea for this chart coming from a site by Luke Mastin.^[2]

1. NASA, . Hubble-Images-of-M100-Before-and-After-Mirror-Repair. Retrieved july 13 2016, from http://cdn.worldsciencefestival.com/wp-content/uploads/2014/11/Hubble_Images_of_M100_Before_and_After_Mirror_Repair_-_GPN-2002-00006_750.jpg
2. Mastin, L. The-Physics-of-the-Universe. Retrieved July 13 2016, from http://www.physicsoftheuniverse.com/numbers.html
3. Van-Swinderen-Institute-for-Particle-Physics-and-Gravity, . Capture-of-a-single-Barium-Ion. Retrieved July 13 2016, from http://www.rug.nl/research/vsi/trimp/trimpnews/news-2013/invangst-van-een-barium-ion-in-een-paul-trap?lang=en