Evolution - An Exponentialist View
Famous Exponentialists
Replicators - An Exponentialist View
A New Malthusian Scale
The Scales Of 70
Population Growth Models
Understanding Compound Interest
The Exponential Method
The Myth Of The Exponential Phase
An Exponentialist Glossary
What's New  

Alphabet Option (1024) - New Malthusian Scale - constant rate
Alphabet Option (512) - New Malthusian Scale - constant rate
Alphabet Option (1024) - New Malthusian Scale - variable rate
Alphabet Option (512) - New Malthusian Scale - variable rate

The Mechanism Of Population Doubling

External links
Natural Selection and Differential Reproduction
- from Replicators: Evolutionary Powerhouses

NIST - "Prefixes for Binary Multiples"

Nigel Malthus of Christchurch, New Zealand.

Legal Metrology, Metrication and Bytes - David Coutts

A New Malthusian Scale

As Leonard Adleman (1998), who launched the field of DNA computing,  put it: "But biology and computer science - life and computation - are related. I am confident that at their interface great discoveries await those who seek them."

Malthus - The Principle Of Population

When English clergyman Thomas Robert Malthus wrote "An Essay On The Principle Of Population" in 1798, he was the first to recognise that populations increase exponentially, and the first to attempt to popularise this view. How this exponential growth actually works is explored in my article The Mechanism Of Population Doubling

So, for over 200 years, it has been mathematically proven that any population which experiences a positive sustained annual growth rate will double.

Three Possible Population Counting Systems

Despite this, we do not generally count our own population using the standard exponential population doubling series. Here are the 3 main contenders for counting populations. In this example, the population increases from 1 million to 1024 million:

Scientific notation
(x 10)
20   21    22    23    24    25    26    27    28  29 210
Standard notation
1 2 4 8 16 32 64 128 256 512 1024
linear counting method
1 2 3 4 ... ... ... 1021 1022 1023 1024

Table 1: Counting Systems

Scientific Notation

As pointed out by mathematician John Allen Paulos in his book "Innumeracy", we could use the scientific notation which is ideally suited to handling large numbers or an exponential series. Instead, despite the fact that demographers all accept the fact of population doubling (given a sustained positive growth rate), every web site with a "world population clock" counts people in linear fashion. I suspect that this is because people find numbers off-putting enough, and would "turn-off" any web site that used scientific notation. 

Most descriptions of exponential growth start with base 2 expressed in standard notation (1, 2, 4, 8 etc) and then switch to base 10 expressed in scientific notation.  The New Malthusian Scale allows your explanation of exponential growth to continue using a standard notation.

Standard Notation

The advantage of the standard notation for the exponential doubling series is that all the numbers in the series are easy to comprehend, and the effect of the doubling is obvious. For example, the last double in such a series always accounts for exactly 50% of the growth (in this case, 512). Hence, all the previous doubles only had the equivalent effect of the last double. Furthermore, because the population doubling rate is by definition tied to a passage of time, the standard notation is probably the most useful of the 3 contenders for analysing past population growth and predicting future population growth.

Also, as highlighted in my article The Cassandra Prediction, the Standard Notation is the simplest method for helping us understand the effect of Negative Growth and the extremely slim likelihood of sustaining Zero Population Growth.

The disadvantage of trying to use the standard notation is that it doesn't precisely convey the transition from the thousands to the million and to the billions (and so on). In this sense it is clumsy, but read on...

The Linear Method

Though the linear counting method is clearly precise in stating exactly how many people have been counted, and the relative blur of the figures ticking over does give an impression of rapid increase, the linear system does not convey the exponential nature of population growth. Some people incorrectly assume, because you can count from 1 to 1024 in increments of 1 (the preferred linear counting method), that populations increase in linear fashion (and therefore do not increase exponentially). 

See The Day Of 6 Billion web page from the UNFPA for an example of a linear "world population clock". On this Valentine's Day, 14th February, 2001 at 1.40am Eastern Standard Time in Melbourne, Australia, the estimated world population is now 6,106,594,811. The US Census bureau shows a slightly higher figure on their POPClock

Neither clock helps people understand the exponential nature of the increase that they are witnessing. I find all linear world population clocks entirely inadequate. At the very least, they should display the following:

Two Cautionary Tales

There are many examples of how poorly our minds are equipped to think exponentially. Oh, we can do it - but we often realise our mistake too late...

The story of the game of chess provides one classic and ancient example (from "The Games Treasury" by Merilyn Simonds Mohr). Noticing his men playing games of chance as they often did, King Balhait of India asked his Brahman called Sissa to come up with a more skilful and strategic game. Having come up with the basic idea, the advisor approached his king with the game of Chaturanga (an early form of chess). The king was most impressed with the game - it was exactly what he desired for his men. He asked his advisor to name his reward. The advisor asked for a grain of corn for the first square on the board, two on the next, four on the next and so on (doubling the corn payment for each remaining square...there are 64 squares on a chess board!). The king was bankrupted making his payment, and the advisor became king!

The second cautionary tale is much more recent, and concerns modern medicine's battle against bacteria and viruses. Until recently, it had been hoped that diseases like Tuberculosis had been eradicated. Now, drug-resistant strains of old diseases are re-emerging. What went wrong?

The link between evolution and population doubling is quite strong. After all, Charles Darwin and Alfred Russel Wallace (co-founders of evolutionary theory), were both partly inspired by the writings of Thomas Malthus. But, more directly, it is the rapid doubling of a bacteria population which holds the key - anything from 8 minutes to an hour! Virus populations can double faster still! Then, considering the vast populations attained by bacteria and viruses, it is clear that mutation must become the norm and not the exception. What sized populations are we talking about? Well, bacteria which divide every 8 minutes can multiply in 450 generations into quintillions (think of a number with 18 zeroes after it) of bacteria in just 24 hours! No wonder resistant strains of old scourges are coming back to haunt us! 

Hence, as human populations double anywhere between 30 and 70 years, bacteria manage to mutate much more quickly than the relatively slow-breeding human host populations. This fact has only recently been recognised, and came as a bit of a surprise to humanity...

A New Measurement Scale - Thanks To Computing

Moore's Law states that computer memory and storage capacity will double every 18 months. Various studies have confirmed this to be the case. Both memory and storage capacity are measured in bytes. A byte is a discrete number of bits. A bit is a binary register (On or Off). 

A new measurement scale has come into being to describe both computer memory and storage capacity. Probably due to the binary nature of the bit, this measurement scale embraces the concept of doubling (and so enforces thinking in terms of doubling times). Most significantly, this new measurement scale provides a handy method for precisely equating 1024 of something with 1 of something bigger (see following table).

Bytes 1 2 4 8 16 32 64 128 256 512 1024
Kilobytes 1 2 4 8 16 32 64 128 256 512 1024
Megabytes 1 2 4 8 16 32 64 128 256 512 1024
Gigabytes 1 2 4 8 16 32 64 128 256 512 1024

Table 2 - Bytes. 

Note:  1 Kilobyte = 1024 bytes, 1 Megabyte = 1024KB, 1 Gigabyte = 1024 MB and 1 Terabyte = 1024GB.

At last, a precise and easily understood transition from the vicinity of the thousands (kilobytes are only close to 1,000 bytes, but are precisely 1,024 bytes) to the millions (Megabytes are only close to 1,000 KB, but are precisely 1,024 KB) to the billions (Gigabytes are only close to 1,000 MB, but are precisely 1,024 MB) and so on. 

Hence, the computer upon which I wrote this article has 128MB of memory, and a 13 GB hard disc. Here we see a nice mix of both the decimal system and standard notation for the doubling series. 128MB uses a number from the Megabyte row to declare the exact measurement of Megabytes. This can be easily translated to the linear (decimal) system, where the figure would be 1024 x 1024 x 128 = 134,217,728 bytes. The hard disc capacity measurement features a decimal figure of 13 to state the measurement of Gigabytes for the hard disc. In decimal this is too big for my calculator to state the number of bytes, but it is easy to comprehend what 13GB means in practical terms. 

In science, kilo = 103; Mega = 106; Giga = 109 ; Tera = 1012 ;Peta =  1015 ; Exa = 1018; Zetta = 1021 and Yotta = 1024. In common English usage, "kilo" means 1,000; "mega" means big, "giga" means gigantic. The prefixes used may be clumsy and inaccurate but they have definitely now entered the English language with a new modern meaning. This new meaning, as per Table 2, has allowed the field of computing to pervert the original scientific meaning of these prefixes to provide a crossover terminology which can be understood by scientist and layman alike. 

How can we apply this to humans, or indeed any population of living creatures? Megaman sounds a bit like superman. Gigaman just sounds stupid. Interestingly, this scale might also work for money - kilobuck, megabuck, gigabuck. After all, money in a bank account earns compound interest (hopefully...) which means a doubling rate.

Science To The Rescue - a new Malthusian scale?

Throughout the history of science, people's surnames have been used to fill the gap when a new word for a new idea is needed. Watt, Amp, Volt, Kelvin, Pasteur and Planck are a few that quickly spring to mind. 

Whose name could be better suited to the task than that of Malthus himself? "Malthusian" is already a recognised English word, and he was the first to popularise the idea of exponential population growth (and limited resources). 

Given that Malthus wrote his essay as the global population approached 1 billion, let 1,024 million = 1 Malthus. This begs the question what to use for 1,024 people, or 1,024 billion people and so on. Whilst Malthus' name is familiar, others may not be.

So use "malthus" as a suffix like "byte". Let a kilomalthus, megamalthus and a gigamalthus take the place of (respectively) kilobyte, megabyte and gigabyte (then teramalthus, and so on). OK, so perhaps the "malthus" suffix doesn't roll off the tongue...how's about kilotom, megatom and gigatom (for Thomas)? Apparently he went by his middle name Robert, so perhaps Kilobob, megabob, gigabob? Too irreverent? Plain silly?

Something That Rolls Off The Tongue...

Then I have one last suggestion...kilopop, megapop and gigapop. Actually, this idea has been growing on me...let me know what you think.

Pops (individuals) 1 2 4 8 16 32 64 128 256 512 1024
Kilopops 1 2 4 8 16 32 64 128 256 512 1024
Megapops 1 2 4 8 16 32 64 128 256 512 1024
Gigapops 1 2 4 8 16 32 64 128 256 512 1024

Table 3: Pops

Note:  1KP = 1024 individuals, 1MP = 1024KP, 1GP = 1024 MP and 1 Terapop = 1024GP.

Certainly, this version of the New Malthusian Scale is useful for visualising the demographic history of a nation or our planet.

A Brief Demographic History Of Humanity

Australia In Detail (A Demographic History)

Converting To And From The New Malthusian Scale.

Converting large numbers expressed in scientific notation to numbers in the New Malthusian Scale is simply a matter of using the scientific calculator that comes with your PC's operating system. Convert the scientific notation into standard notation, then repeatedly divide by 1024 until you get a number between 1 and 1024. Keep track of how many times you do the division, and that tells you which row on the New Malthusian Scale your number belongs to. 

A decimal-scale population of 6,000,000,000 (our population on 12th October, 1999 was 6 billion) is 5.59 GP. This nicely demonstrates the use of a dual decimal and Malthusian scale measurement where such precision is necessary. If this sounds awkward, remember that the next Malthusian scale milestone date is when we reach 6 Gigapops which should be quite soon.

Converting from the New Malthusian Scale to the standard decimal linear scale is easy. 6 GP (which is a nice simple number) = 10243 x 6 = 1024 x 1024 x 1024 x 6 = 6,442,450,944 people, a very awkward number if using the linear counting method! 

Alphabet Option For New Malthusian Scale

It would be entirely reasonable to simply draw 26 rows of the New Malthusian Scale, naming each row after a letter of the alphabet (A-pops, B-pops, C-pops etc). As each row represents 10 population doublings, this makes it easy to calculate constant doubling times for extremely large populations and handy for tracking variable doubling times.

Alphabet Option For New Malthusian Scale (constant rate)

Alphabet Option For New Malthusian Scale (variable rate)


You will have noticed that the various versions of the New Malthusian Scale all contain an element of redundancy (e.g. 1 Megapop = 1024 KB, and both values are displayed on Table 3 for Pops). I prefer to retain each number in the doubling series from 1 to 1024 on each row. For me, the 1024 column helps me visualise how the scale works.

If you prefer, here are versions with the 1024 column removed:

Alphabet Option For New Malthusian Scale (constant rate)

Alphabet Option For New Malthusian Scale (variable rate)

Malthusian Wheel

Like a mythical ouroboros which eats its own tail, populations on the Malthusian Wheel cycle endlessly up and down the standard doubling series:

Wheel.gif (11205 bytes)

This is a variant on my New Malthusian Scale. With the wheel, you can track (with tokens) any sized population as it doubles or halves over time. Add a token to your stack (or increment the value of a token by 1) each time you double from 512 to 1024. Your tokens will now be on the 1 space. Remove a token (or decrement the value of a token by 1) each time you drop from 1 to 512. 

Or imagine a single token moving clockwise around the wheel. As it doubles from 512 (to 1024), so it becomes 1 on the next leg of its journey. A simple colour key could denote whether the token represents Pops, Kilopops, Megapops and so on. 


The population in this example starts at 1 and effectively doubles 30 times (the equivalent to increasing along three rows of the New Malthusian Scale.


The concepts of exponential growth, population doubling, and limits to growth were all neatly explored by K. Eric Drexler in Chapter 10 ("Limits To Growth") in his book Engine Of Creation. This is because his book on nanotechnology introduces the concept of the replicator, whereby inanimate matter is programmed at the molecular level to replicate. Life has already mastered this trick, but fully realised artificial replicators could be the equivalent of bacteria in terms of their doubling times (perhaps as little as 20 minutes). 

Kilopops, Megapops and Gigapops could all equally apply to discussions relating to populations of nanotechnology replicators. Before too long it would be probably be necessary to extend the scale to include Terapops (10244), Petapops (10245) and Exapops (10246), depending upon what is being "grown" .

Grey Goo - An Exponentialist Explanation

Population doubling and doubling times are essential elements to understanding how any species (be they nanotech replicators or life's own replicators) evolves and grows over time. Most scientists won't have a problem understanding the message using scientific notation, but the general populace need something easier to understand and express but which is still accurate. My new Malthusian Scale allows people to track population growth over time in a way that everyone can understand.

If the adherents of nanotechnology are going to get the message across where scientific notation and linear counting have failed, then my hope is that one day these adherents will adopt my new Malthusian Scale to impress upon people how nanotechnology will emulate life's ability to grow exponentially. 


Perhaps we could teach our children, along with their multiplication tables, this new Malthusian scale. That might help them become the first generation to believe the truly exponential nature of population growth, and would provide them with a simple measurement scale for analysing past population doublings and predicting future population doublings. The New Malthusian Scale could also be a simple and useful tool to visualise the process of differential reproduction (see Natural Selection and Differential Reproduction - from Replicators: Evolutionary Powerhouses).

Trace your finger from the pale blue 1,024 individuals at the end of the Pops row (which is the equivalent of the 1 kilopop space) through to the 4 Gigapops on the bottom row, and you have traced almost the entire 100,000 year demographic history of Homo Sapiens Sapiens in just 22 doublings. We are now in the process of the 23rd doubling (to 8 Gigapops, or 8,589,934,592 people, which should conclude before 2050AD).

Nigel Malthus, of Christchurch New Zealand. Nigel is a direct descendant of the Reverend Thomas Malthus' older brother, Sydenham. Nigel (2001) made the following comment:

"While TRM grew up being referred to as Robert/Bob, he also came, later, to be called "Pop". I assume it was an abbreviation of "Population," as in "Population Malthus". Maybe it started as a term of abuse. But perhaps because it would also have served as a sort of pun on "pop", meaning "dad", it was adopted within the family as an affectionate nickname: In a letter to Patricia James, which she quotes in her book, my grandfather, writing of the Malthus explosion in NZ, said: 'You will agree that we conform very well to Pop's theory of what tends to happen in a young country where the means of subsistence abound.'

My point is: kilopop, Megapop, Gigapop - works for me!"

When is a Megabyte not a Megabyte?

When it's a Mebibyte! As noted by The National Institute Of Standards and Technology  in their article "Prefixes for binary multiples" under International System Of Units (SI), the current standard so familiar to those in the computing industry is in the process of being changed. In December 1998 the International Electrotechnical Commission approved an international standard which means that kilobyte, Megabyte, Gigabyte etc will now assume the telecommunications industry's decimal  meaning of 1,000 bytes, 1,000 kilobytes and 1,000 Megabytes and so on. The binary multiples of kibibyte, Mebibyte and Gibibyte should now be used for 1,024 bytes, 1,024 kilobytes and 1,024 Mebibytes and so on. 

The IEEE have a Draft Standard Prefixes for Binary Multiples dating from December 1999. Things clearly move slowly in the land of standards (probably for good reason), so I'll change to kibipop, Mebipop, Gibipop etc when the IEEE officially adopt kibibyte, Mebibyte and Gibibyte etc. 

For more on the exponential confusion over definitions relating to bytes, see my article Legal Metrology, Metrication and Bytes.

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Copyright 2001 David A. Coutts
Last modified: 15 December, 2011