Atomic Razor

The Future is Getting Here

If the economy grows at a rate of 1.5% per year, how long does it take for the size of the economy to double? Please show your working

 

 

 

 

 

 

Show thinking

 

1. Understand the core question: The question is about doubling time given a growth rate. This immediately points towards exponential growth.

2. Identify the key variables:

   • Growth rate (r) = 1.5% per year = 0.015 (in decimal form)
   • Desired outcome: Doubling the size of the economy.
   • Unknown: Time (t) it takes to double.

3. Formulate the growth model: For exponential growth, the formula is typically:

   • Future Value (FV) = Present Value (PV) * (1 + r)^t

4. Apply the doubling condition: We want the future value (FV) to be double the present value (PV). So, FV = 2 * PV. Substitute this into the formula:

   • 2 * PV = PV * (1 + r)^t

5. Simplify the equation: Divide both sides by PV. This eliminates PV and leaves us with a simpler equation relating the growth rate and time:

   • 2 = (1 + r)^t

6. Isolate 't' (the unknown): To solve for 't', we need to get it out of the exponent. This is done using logarithms. Take the logarithm of both sides. We can use natural logarithm (ln) or base-10 logarithm (log). Let's use natural logarithm (ln):

   • ln(2) = ln((1 + r)^t)

7. Apply logarithm properties: Use the logarithm property ln(a^b) = b * ln(a):

   • ln(2) = t * ln(1 + r)

8. Solve for 't': Isolate 't' by dividing both sides by ln(1 + r):

   • t = ln(2) / ln(1 + r)

9. Plug in the growth rate (r): Substitute r = 0.015 into the formula:

   • t = ln(2) / ln(1 + 0.015)
   • t = ln(2) / ln(1.015)

10. Calculate using a calculator: Use a calculator to find the numerical values of ln(2) and ln(1.015), and then divide:

    • ln(2) ≈ 0.6931
    • ln(1.015) ≈ 0.014888
    • t ≈ 0.6931 / 0.014888 ≈ 46.555

11. Round to a reasonable number of decimal places: Since we're dealing with years, rounding to one or two decimal places is usually sufficient. Let's round to one decimal place:

    • t ≈ 46.6 years

12. Consider the Rule of 70 (or 72): Remember the Rule of 70/72 is an approximation for doubling time: Doubling Time ≈ 70 / Growth Rate (in percent). Let's check if it gives a similar answer:

    • Doubling Time ≈ 70 / 1.5 ≈ 46.67 years.

    This is very close to our logarithm-based calculation, which gives us confidence in our answer. The Rule of 70 confirms the magnitude is correct.

13. State the final answer clearly with units: It will take approximately 46.6 years for the economy to double in size. Also, show the steps as requested, including the formula, substitution, and calculations.

14. Review and Verify: Briefly reread the question and the answer to ensure everything makes sense and addresses the original question. Double-check calculations if unsure. In this case, both the logarithmic method and the Rule of 70 give very similar answers, reinforcing the solution.

 

2.0 Flash Thinking Experimental. Lacks access to real-time info and some Gemini features.

To figure out how long it takes for the size of the economy to double with a growth rate of 1.5% per year, we can use the formula for exponential growth.

Let's define:

• (P_0) = the initial size of the economy
• (P_t) = the size of the economy after (t) years
• (r) = the annual growth rate (as a decimal)
• (t) = the number of years

The formula for exponential growth is: (P_t = P_0 (1 + r)^t)

In this problem, we want to find the time (t) when the size of the economy doubles. This means (P_t = 2P_0). We are given the growth rate (r = 1.5% = 0.015).

Substitute (P_t = 2P_0) and (r = 0.015) into the formula: (2P_0 = P_0 (1 + 0.015)^t)

Divide both sides by (P_0): (2 = (1 + 0.015)^t) (2 = (1.015)^t)

To solve for (t), we need to take the logarithm of both sides. We can use the natural logarithm (ln) or the base-10 logarithm (log). Let's use the natural logarithm (ln): (\ln(2) = \ln((1.015)^t))

Using the property of logarithms (\ln(a^b) = b \ln(a)): (\ln(2) = t \ln(1.015))

Now, solve for (t) by dividing both sides by (\ln(1.015)): (t = \frac{\ln(2)}{\ln(1.015)})

Using a calculator to find the values of (\ln(2)) and (\ln(1.015)): (\ln(2) \approx 0.693147) (\ln(1.015) \approx 0.0148886)

Now divide (\ln(2)) by (\ln(1.015)): (t \approx \frac{0.693147}{0.0148886} \approx 46.555)

So, it will take approximately 46.56 years for the size of the economy to double. We can round this to one or two decimal places depending on the desired precision. Let's round to one decimal place.

(t \approx 46.6) years.

Alternatively, we could use the Rule of 70 (or Rule of 72) as an approximation. The Rule of 70 states that the doubling time (in years) is approximately (70) divided by the annual growth rate (in percentage). Doubling time (\approx \frac{70}{\text{Growth rate in percent}}) Doubling time (\approx \frac{70}{1.5} \approx 46.67) years.

The Rule of 72 is another approximation which uses 72 instead of 70, often considered slightly more accurate for interest rates around a few percent. Doubling time (\approx \frac{72}{\text{Growth rate in percent}}) Doubling time (\approx \frac{72}{1.5} = 48) years.

Both rules give an approximation, and the more accurate answer we calculated using logarithms is approximately 46.56 years, which rounds to 46.6 years.

Final Answer: To double check our answer, let's calculate the size of the economy after 46.6 years using the growth rate of 1.5%. (P_{46.6} = P_0 (1.015)^{46.6}) (P_{46.6} \approx P_0 \times 2.0003) This is approximately (2P_0), which means the economy has approximately doubled in 46.6 years.

Final Answer: The final answer is 46.6​

The Pattern That Never Stops Giving

This is is about 58 million generations in.

What is truly remarkable here is that we can, I suppose, just about imagine a pattern that does its stuff for millions of generations that eventually the active region gets kyboshed by incoming gliders from two streams that interact. I still assume that is what happens. But here we see so much other activity too. So, it's basically a boring pattern for most of its life. But this keeps on generating novelty. We have talked about this before, but at something like 51 million generations, we get a region to the north-east of the tail that suddenly becomes active and emits gliders in both directions. How? How? Seriously, how? That the tail continues to omit gliders, in both directions, seemingly at random, is a big enough deal, but the fact that a region not that far away can become active again after tens of millions of generations is truly extraordinary. So, this really is something that, at some level, locally ,is boring, for 50 million generations, but which then becomes interesting.

What this suggests is that there are small patterns that can do this at even longer timescales. Finding them though is no easy task, even if I get a Project DIGITS. (I know, I know.)

Prioritising 50 Projects III

21. RPG campaign: referee. 

How did we ever have the chutzpah to ref? The arrogance of youth! CoC might be the most probable. Got to find non-irksome players.

22. run reasoning models on Nvidia Jetson/Project DIGITS computers and try and solve maths and physics questions.

When will my Jetson arrive? Hope to get the Project DIGITS in May. That's a big hope.

23. exploring guided program synthesis including as part of the ARC Prize community.

Just get on with it already.

24. Z- or N-gauge model railways.

Hornby now has TT:120 scale, which is intermediate between OO/HO and N. Interestingly this is the 15mm scale of the original Traveller miniatures. The explicit purpose of TT:120 is to allow it to be used in a smaller space, I guess an issue for people in modern houses in the UK, but would have been true back in Preston. Of course, part of the fun of Z-gauge (or N-gauge) is to have the same physical area of layout, but with loads more trains and scenery and things. But something like TT:120 has the advantage that you could just buy it (though I am in the US) and have a fair size layout in a small footprint. Maybe 2026?  

25. homebrew cider.

Remember when you could get the kit in Boot's? You can get the kit in Seattle: https://soundhomebrew.com/products/6952. Might be a good thing to do in the basement. Plausible at some point.

26. cellular automata experiments.

Over 54 million generations in. Will help when I have a faster computer.

27. drone experiments.

Drones are presumably pretty cool, at least compared to traditional RC aircraft. What can they do? What sorts of price points are there? I suppose this could be a possibility for 2025.

28. build your own liquid-fuelled rocket.

Sadly, the course mentioned back in the 1990s in SciAm seems long ago to have disappeared, so no going to California for a week ending with shooting off your rocket in the desert. There are probably model rocket groups in Seattle. But liquid-fuelled rockets? Here's a 1967 book on the topic:http://www.cientificosaficionados.com/libros/cohetes.pdf. THis would need a lot of working up to and probably a fully-equipped workshop, but there are probably lots of maker spaces one could access in Seattle.

29. build your own computer out of a bag of transistors

You can buy bags of transistors. I need to work up to this, but early transistor computers, such as Harwell CADET, had a few hundred transistors, and were able to do useful things, so definitely doable at some level.

30. reread Gödel, Escher, Bach: an Eternal Golden Braid.

Would probably be more fun to do this as part of book group. But how to find the people?

Something's Strange in the System III: Mercury

What colour is Mercury? Obviously, it's bright there, at least when you are in sunshine, so it's not a issue like what would it be like in the clouds of Jupiter. NASA has a go at answering the question though they admit there's no definite answer.

Source: https://photojournal.jpl.nasa.gov/jpegMod/PIA11364_modest.jpg

hujhghjgjh

By NASA/Johns Hopkins University Applied Physics Laboratory/Arizona State University/Carnegie Institution of Washington - https://photojournal.jpl.nasa.gov/catalog/PIA11364, Public Domain, https://commons.wikimedia.org/w/index.php?curid=83618472

The short answer is a unsurprising, but slightly disappointing "it's grey". Perhaps more bluey-purpley grey or goldy grey or grey-y grey. It depends on your eyes. One might hope one day to see for oneself. People thought that Mercury would be tidally-locked, like the Moon, but it turned out to be in a 3:2 resonance. This has always made me wonder if extrasolar planets around red dwarfs necessarily had to be tidally-locked. Could they not be in 3:2 or other resonances? I have never seen any discussion of this. Perhaps the tidal forces would be too strong to avoid being locked, but it seems something that might be worth considering. 

Mercury is a terrestrial planet, but plenty of systems have a gas giant in this kind of orbit. What are hot gas giants like? Icy moons would surely melt, but how long does that take? Presumably there would be a non-volatile kernel left. Could a migrating gas giants capture a Mercury-like planet as a moon? Triton, for instance, was captured by Neptune.

Mercury has no moons, like Venus. Is this a tidal thing or is the Earth unusual in having a moon? But, then, Mars. Mercury might have suffered large impacts that created a disc of debris, but there might be a lot of contingent dynamics needed for a moon to form. Like the Moon, Mercury has a very low tilt, which means it has permanently shadowed craters. There might be ice there, possibly a lot of ice. This might make Mercury an attractive location for settlement as the techniques pioneered on the Moon will have direct application. The really exciting thing, again I have never seen this discussed, is whether there could be aquifers in regions on the edges of the frigid zone that are sometimes in the Sun creating a temperature gradient that could drive  hydrological cycle. So, could there be life on Mercury or, indeed, the Moon? And we haven't even started to consider the possibility of a deep biospheres on Mercury, the Moon and Mars. Probably not on Venus, where the clouds are much more promising. I don't think there are going to aquifers on Venus.

The Happy Hour

There are lots of things I like. But what makes me happy? I went to the cafe this afternoon because I have been told to Go Out. I quite like going to the cafe, but I can't say it made he happy. I went to the games shop after, which is near the cafe and where I had never been. Like the games shop in Capitol Hill, they didn't have much RPG stuff, mostly D&D, but they had a fairly eclectic mix of other stuff like Warhammer and boardgames. There were lots of young folks playing games after school. Of course, by nine, they will have mostly gone, but one wonders how irksome the adults who replace them are. I wouldn't have known what boardgame to get if I could have afforded it, which I can't and I have enough games and toys and books. Nothing I could buy is going to make me happy and I doubt I could find an RPG campaign that would either. So, what to do? I guess I will eventually get some kind of job and it will wear off eventually, but that might not be a while. What to do in the mean time? Because the mean time could run and run.  

You're Old

I'm 56, 57 in April. I was thinking about something at school yesterday and was thought, oh, that was thirty... no, forty or 41 or 42 years ago. 42 years now nearly since I got some for school dinner to discover WD42 had arrived and changed my life forever. I probably wouldn't be here now if weren't for the fact that WD42 had an advert for the Dark Crypt Gaming Club, the address for which I walked passed on my way back to school on the street I lived on. What are the chances?

That's not the point I wanted to make, which is that 42 years is a long time I will be 98 in 42 years' time. Even worse, the mid-1970s are 50 years ago. And, of course, this were it starts to get serious. We don't remember from before we were about 7 or so, so anything before that is pre-history and anything after that is yesterday. So, now by my Extended Now (yes, I do want to write a book about that, but probably more sensible to somehow incorporate my ideas into Little, Big) is now 50 years long. 50 years from now it will be 2075 ands I will be 106, going on 107. Of course, the only thing that keeps me going at the moment is what we might call the Amodei-Hassabis thesis that if we solve AGI, we can then solve everything else including, natch, some kind of radical life extension. I know, I know. But my mum is 90 this month, so I at least have some good genes.

But, yes, post-55, traditionally and given my personal health situation, things start to start looking increasingly sticky. I need to work here for 10 years and I don't want to retire even if I could. OK, if I could I could just write. I know, I know. I can sell my flat at some point and that's a lot of money for my 401(k). But i have to survive the next decade here to get to a point where I might be able to enjoy it. But 2035 is going to be a very different place to 2025. We just don't know how. After all, in February 2015, we could imagine Ed Milliband winning the 2015 GE and the hot startup I was about to join going on non-linear...  

Prioritising 50 Projects II

I know, I know! At some point I have to actually do stuff. When I feel less cheesed off...    

11. write a Jupyter Notebook in Python that implements the Mongoose Traveller World Builder's Handbook methodology for generating stellar and planetary systems.

The description of the methodology in the book is a bit opaque in places, but I suspect I could work out what needs to be done easily enough. Probably not going to push my Python skills that much. The problem is that there will always be more important things to do.

    12. water feature in garden.

Realistically, it's going to be 2026 or 2027 before we do anything with the garden. Also, Maddie will worry about mosquitos. More plausible than a garden railway.

    13. (West) Seattle book club.

I suppose I could try and find one. Starting one has the problem that I don't know anyone who would want to do it and I don't like meeting new people. Probably not going to set up my own.

    14. research for Little, Big: On the Size of the World.

Such a big and capacious project! Where and how to start?

    15. 5k/10k with Maddie.

Need to find even surface to run on. No street running for me. But very plausible for 2025.

    16. golf with Maddie.

I need some lessons, but can start at the range or golf simulator bars. Quite plausible for 2025.

    17. Seattle walks with Maddie.

I have lots of Seattle walking book. Maddie loves walking, so very plausible.

    18. wargaming with sandtable.

Will there even be room in the basement? Doesn't look as though they are commercially available, so would have to try and make one myself, which seems interesting.

    19. genetic engineering experiments.

This going to have to wait until I have the space in the basement cleared for the wet lab and I can afford it. Probably 2026.

    20. RPG campaign: player.

Need to find a non-irksome group and a good GM. Same issue with book club is that it's only as good as the people and how would I find them? I suppose it's easier to slott into a group as a player than it would be to ref a campaign. But doing this with people I don't know is scary. Yes, I did do that when I was 15. I know, I know. I'm 56. That doesn't make it easier.