The numbers which appear in this tutorial are large ... VERY large. They are of a special type called exponential numbers which look something like this an, which is interpreted as n instances of the number a all multiplied together. When you start to see numbers which look like this, it's time to be careful. They can lead to some very unexpected and implausible circumstances.

Candidate BIG numbers.
  • Numbers which accord with the Collatz conjecture.
    Much computer time has been invested in trying, without success, to find a number which negates the Collatz conjecture. All numbers less than 1018 (That is 1 followed by 18 zeroes) have been so tested. This number sounds considerably more impressive when expressed as one million trillion.

  • Atoms in a grain of salt.
    Just coincidently, the preceding number, 1018, is also a good approximation of the number of atoms in a typical grain of salt.

  • Atoms in the universe.
    Now this is where things become interesting ... the number of atoms in the universe is estimated to be 1080. Have you ever encountered a more counterintuitive state of affairs than this? 1018 atoms in a grain of salt which you can barely see with the naked eye, yet only 1080 atoms in the entire observable universe! Many people simply refuse to believe these numbers, but they are accepted as the most accurate estimates available. It only goes to show how careful you should be when you enter the domain of numbers of this type. Much more on this when we come to consider Economic Growth.

  • The Googol.
    The Googol is a number bigger than you would ever need for any practical purpose. It consists of a 1 followed by 100 zeroes. That is 20 zeroes longer than the number of atoms in the universe, so it is a VERY big number. If you like it is the number of atoms in one hundred million trillion universes! In spite of the size of such numbers, the Rule of 8 program has no difficulty in processing them, and Result 2: Profile of the Signature Chunk data of 100 digit numbers at Hailstone Rule of 8 demonstrates how precisely the numbers obey that rule. Lesser numbers of 1,000 and 10,000 digit numbers have been found to obey the rule equally well.

  • Economic growth.
    Governments the world over insist that their economy, as measured by GDP (Gross Domestic Product), should grow every year, and that an annual growth rate of three percent would do just fine. So, if in any given year the economy is determined to be GDP dollars, then:-

    After 1 year it will be GDP x 1.03
    After 2 years it will be GDP x 1.03 x 1.03 or GDP x 1.032 = GDP x 1.061
    After 3 years it will be GDP x 1.033 = GDP x 1.093
    After 10 years it will be GDP x 1.0310 = GDP x 1.344
    After 23.45 years it will be GDP x 1.0323.45 = GDP x 2
    After 77.9 years it will be GDP x 1.0377.9 = GDP x 10
    And after 980 years it will be GDP x 1.03980 = GDP x 3,806,100,144,451

    A few comments on the above data are in order:-

    • Note that the GDP (or any quantity that grows by 3% in a given period) will double in size after 23.45 periods. A very handy number to remember.

    • Similarly, any quantity that grows by 3% in a given period will increase in size by a factor of 10 after 77.9 periods. Even more useful to remember.

    • The growth factor of 3,806,100,144,451 after 980 years is NOT an error. There will be more to say about this shortly.

    • If you would like to check the accuracy of any of these figures, you can do it easily using the scientific mode of the calculator on your Windows or Apple computer. To check the figure for 77.9 years, click buttons on your calculator in the following order:-

      1    .    0    3    xy    7    7    .    9    =

    • It is strongly recommended that you also check the figure for 980 years. Just replace the 7 7 . 9 clicks with clicks for 980. You will probably have guessed that this is the number of years till the year 3000.
  • There ARE limits to growth!
    And so to the burning question! Does the growth factor of 3,806,100,144,451 after 980 years represent a significant threat to the possibility of Economic Growth continuing beyond the year 3000? The simplest way of approaching this question is by means of a thought experiment. In case you aren't familiar with this technique, here is a definition provided by Google:-

    "A thought experiment is a device with which one performs an intentional, structured process of intellectual deliberation in order to speculate, within a specifiable problem domain, about potential consequents (or antecedents) for a designated antecedent (or consequent)" (Yeates, 2004, p. 150).

    In our thought experiment, instead of considering Australia to be a single economic zone, we consider it to be divided up into many smaller zones with each zone supporting a GDP valued at G dollars, where G dollars is the 2020 value of the Australian GDP. In the year 2020 there is just one zone, but as time passes, the number of such zones increases, reaching the number 3,806,100,144,451 when the year 3000 finally arrives. This is an almighty number of zones, and the very natural question to ask is “How big will each zone be”. To find out, simply take the area of Australia and divide by the said number.

    Recourse to Google with the string Area of Australia yields the answer 7,692,000 square kilometres. Now one square kilometre contains one million square metres, so the area of Australia is simply 7,692,000,000,000 square metres. Dividing this area into 3,806,100,144,451 equal zones yields a zone area of 7,692,000,000,000 / 3,806,100,144,451 = 2.02 square metres.

    Make no mistake about the interpretation of this result! It means that every one of the more than 3.8 trillion 2.02 square metre zones covering Australia would be required to support an economy equal to the 2020 economy, regardless of where they are located, be it in deserts, dry salt lakes, roadways, national parks or even private homes. Now ask yourself the question:- Is that even remotely plausible?

    I won't insult your intelligence by stating the obvious answer.

  • Conclusion to be drawn from this thought experiment.
    Most economists (with very few exceptions *) and probably all politicians will fail to draw any conclusions whatever. They have, for so long, held the view that economic growth is the natural state of affairs, they will find it impossible to accept that growth must cease.

    However, the only possible valid conclusion that can be drawn is that the economic growth we have seen since the end of WWII (or perhaps if you like since the beginning of the Industrial Revolution) is nothing but an accidental anomaly, caused by the low population and low living standards (in comparison with today's situation) that were present during those times. It cannot continue, and must cease at some unspecified time in the future, and be replaced by a Steady State Economy (accompanied also by a Steady State Population).

    So what will happen? Politicians, with the encouragement of Economists will continue to stimulate the economy with all of the tried and failed measures of the past, but the time will come when the economy will stubbornly refuse to respond. We have already seen the beginnings of this during the several years prior to COVID 19. Leaders of the IMF, the US Federal Reserve, the Australian Reserve Bank etc, etc, could be seen on the six o'clock news wearing worried expressions, completely unable to comprehend why the world economies refused to grow. When the time finally comes that even they have to admit their failure, the world economies, along with the environment, will be on the brink of collapse. This will be the worst possible time to have to transition to a Steady State Economy. 1960 might have been a good time to start thinking about it. Anytime in the past would have been better than now for that task. But we didn't do anything about it! Now is a better time to do it than any time in the future, but will anything be done?

    Here is another question to which you know the answer as well as I do!

    * A notable exception was the English-born American economist Kenneth E. Boulding who's most famous quote is Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist.