Exponential growth and order of magnitude reasoning

In Unit 7 we’re covering the exponential and logarithm functions. One of the features of exponential functions that takes the most time to get used to is how fast they grow.

A famous illustration of the surprisingly fast rate of exponential growth is the Wheat and chessboard problem has been popularized in several children’s books which you may have encountered before; for example One Grain Of Rice: A Mathematical Folktale. The premise of the story is that a wise man is offered a gift by a king and the wise man asks to be given rice for 64 days, one grain on the first day, two grains on the second day, four grains on the first day, and so on. The king readily agrees as the amount of rice as it initially seems like the wise man is asking for a few grains of rice but as the days go on the king discovers that the wise man was implicitly asking for far more rice than the kingdom could possibly produce.

One day in 4th period precalculus I commented that the number of terms in the formula for the determinant of a 100 by 100 matrix is 100! which is bigger than the number of atoms in the observable universe. Many students were incredulous; claiming that the number of atoms in the universe was so big this couldn’t possibly be true. I used Wolfram Alpha to compute that 100! is approximately 10^(158) and those students who were incredulous remained so.

I think that the source of confusion is not a lack of intuition about chemistry or physics or astronomy but rather a lack of understanding how big a number 10^(158) is. Hopefully I can give some sense of this via a plausibility argument for my claim from class via a Fermi Calculation.

[Disclaimer: I have no subject matter expertise in astronomy or astrophysics, the numbers that I give below are just the numbers that I found with a quick internet search from apparently reputable sources; it’s possible that the expert consensus for some of these numbers is very different from the numbers quoted or that I’ve misunderstood something crucial. Despite this, I think that the analysis below will carry some pedagogical value.]

Number of Atoms in the Observable Universe: Many of you are in chemistry and so are familiar with the fact that Avogadro number is the number of particles in a mole of a substance. So, e.g. there are Avogadro number of hydrogen atoms in 1 grams of hydrogen. The sun is mostly made up of hydrogen. The more massive the elements are that compose the sun, the fewer atoms there are in the sun, so if we assume that the sun is made up entirely of hydrogen we’ll get an upper bound on the number of atoms in the sun of the sun. The mass of the sun is less 10^(34) grams and Avogadro’s number is less than 10^(24) so the number of atoms in the sun is less than (10^(24))(10^(34)) = 10^(58).

We can use this as an estimate of the mass of the solar system. In class some of you commented that the calculation above neglects the masses of other objects other than stars like Earth. But the mass of Earth is a billionth the mass of the sun, and indeed the mass of Jupiter (the largest planet in the universe) is a thousandth of the mass of the sun. The mass of the asteroid belt is only 4% of that of the moon which is in turn less than a 10 billionth the mass of the sun. So the sun is by far the dominant mass in the solar system; everything else combined is less than 1% of the mass of the sun. Intuitively this makes sense; if other objects were of comparable mass then the dynamics of the solar system would be more complicated than what they are; everything wouldn’t rotate around the sun in such a clean fashion.

Now, I claimed that 100! is bigger than the number of atoms the observable universe, not just in the solar system. One issue is that there are stars that are bigger than the sun. The most massive stars known stars seem to have ~ 300 times the mass of the sun. But suns aren’t the most massive stellar objects in the universe, that honor belongs to supermassive black holes which are “on the order of hundreds of thousands to billions of solar masses.” To be conservative let’s imagine that every star is a supermassive black hole of a trillion (10^12) solar masses. Assuming that such a black hole is made out of hydrogen atoms; (I actually don’t know what kind of matter a black hole is considered to be!) the number of atoms is no more than:

(10^(58))(10^(12)) = 10^(70)

Okay, now the universe doesn’t just have one stellar object, it has many stellar objects. Nevertheless, there are no more than a trillion (10^(12)) stars in the Milky Way Galaxy. Even the most massive galaxy discovered has no more than a quadrillion (10^(15)) stars. So the largest number of atoms that a galaxy could have is bounded above by

(10^(70))(10^15) = 10^(85).

Now, the current best estimate for the number of galaxies in the observable universe is about 500 billion, to be conservative let’s use the number of ten trillion (10^16). Then an upper bound on the number of atoms that the observable universe could have is

(10^(85))(10^(16)) = 10^(101).

This number is less than 100! ~ 10^(158); not only a little bit smaller but an unimaginable amount.

I’ll note that National Solar Observatory webpage allegedly has an article estimating the number of atoms in the universe as 10^(78).

Conclusion: Our natural intuitions are not well calibrated to thinking accurately about exponentially large or exponentially small quantities; accurately thinking about them requires a combination of care and re-calibrated intuition gained from experience with working with such quantities.


7 thoughts on “Exponential growth and order of magnitude reasoning

  1. Dr. Sinick, black holes are not matter. They are anomalies in space-time. The actual matter in them is a singularity that causes them to exist in the first place. These singularities tend to be very dense, usually many solar masses compacted into a space a bit smaller than the moon.

    • Right, so I don’t know general relativity and am out of my depth here.

      My impression is that black holes gain mass from absorbing surrounding particles and if this is the case one has the question of how the particles switch from being “matter” to not being “matter.” However, the real difficulty here is one of definitions: (a) what is meant by “matter”? (b) what is meant by “space-time” ? (c) What is meant by “mass”? This gets into thorny theoretical issues.

      Anyway, these issues aside, the post establishes a plausible upper bound on the number of atoms in the observable universe.

  2. Well, matter is anything that occupies space. Mass is a property of matter, and space-time is the field that matter occupies. In a mathematical perspective, it is a model in which space and time occupy the same continuum. And it is not really known what happens when matter enters a black hole, though many plausible theories arise. You must know about Einstein’s E=mc^2 equation. This brought forth the theory that as matter enters a black hole it oscillates from matter to energy, as the equation shows is possible. There are 2 types of black hole according to general relativity: the non-rotating, or Schwarzschild black holes, and Kerr black holes that do in fact rotate. In a non-rotating black hole, there is nothing stopping matter from the black hole’s accretion disk from meeting the singularity. At this point, it is unknown what happens to the matter, seeing as mathematically, gravity’s value is infinity. Some speculate the matter is sent into another dimension, but physicists say this is unlikely since the high gravitational radiation at a singularity drastically changes the geometry of space time, preventing such things from happening. Rotating black holes, on another hand, can rotate at speeds approaching the speed of light. In these holes, depending on the trajectory of the matter, some say it is possible that matter can enter the hole and be shot back out, like a sort of gravitational sling shot. What is interesting is that the matter could be shot out thousands of years and millions of light years from where it entered, considering anomalies in space-time in the black hole.

  3. Dear Dr. Sinick,

    While this is a good article, I noticed one small inaccuracy – technically, everything else in the solar system is made from about 3% of the nebula that formed the entire system; the sun is made up of about 97%. Therefore, everything else in the solar system has about 3.1% of the mass of the sun, not 1%. It’s still a negligible error, though, since the point is that the mass of everything else is irrelevant compared to the sun, and that is still true.

  4. Dear Dr. Sinick,
    That was a fascinating article, and I agree that we often cannot comprehend how big numbers really can be. I’d just like to share a link I found that explains how the determinants and terms in matrices work. I found it pretty helpful in understanding this concept, and it puts what we’ve learned in class (the determinant formulas) into context as well.

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