Powers Of Two

[ Posted Tuesday, October 16th, 2018 – 16:18 UTC ]

At a first glance, this may resemble a political column, but it's not. Instead, it is a digression to examine the awesome progression of exponential mathematics. No, really. I'm taking a slight break from the political world to opine on the powers of two.

Of course, this idea didn't pop into existence out of a vacuum. There is indeed an event in the political world that gave rise to it. But I already spent yesterday talking about Elizabeth Warren's DNA test, so today I'd like to address a common misperception among anyone claiming any specific ancestry in the far and distant past. Again, Warren's situation provoked this column, but this column is not about her specific case.

I've actually had this entire conversation before, in my private life. A good friend of mine is proud that her family traced her lineage all the way back to the Mayflower. While not dismissing her claim in any way, I pointed out that we all love to trace one single branch out of our family tree, but usually in doing so we discount the thousands upon thousands of others on all the other branches.

Her example is a good one to use, mostly because it goes so far back in time. The Mayflower arrived in 1620, and we're now roughly 400 years away from that point in time. It takes two people to make a baby. Each generation therefore expands the number of ancestors by a power of two. You have two parents. You have four grandparents, which is two squared. The third generation back, your great-grandparents, has two cubed (two to the power of three), or eight ancestors. The fourth generation is two to the power of four -- sixteen. And so on. It took sixteen great-great-grandparents to make you. Your are one-sixteenth of each of them, genetically. It doesn't really matter whether any one of them was a prince or a pauper, because either way they're still only 1/16th of your makeup.

Now, to really identify how many ancestors you'd have from the time of the Mayflower, you'd have to know the exact age each mother was when she had the baby who is in your direct lineage. That's tough to do, so instead let's just generalize and say that all the mothers were 25 years old. This is a relatively generous estimate -- 25 years to each generation -- because the further you go back in time, the more common it was to have children at a much earlier age than 25 years old. But then you've got to weigh that against which child is in your lineage -- it could have been the fifth of seven children, not the first. So 25 years is a conservative estimate, but it's good enough for our purposes.

Let's say my friend with the Pilgrim lineage is now 25 years old. She's not, but it works better for the example, so let's just go with it. Her own life is generation zero -- because she is one person, and two to the power of zero is one. Her parents' life is generation one, because there are two of them (two to the first power is two). Using our 25-year rule of thumb, her parents were born 50 years ago. A century ago, her great-grandparents were born. Since they are the third generation back, there were eight of them. That puts us roughly at 1920.

The Mayflower landed 300 years earlier than this. Each 100 years is another four generations. That means the baby that was born right after the Mayflower landed would have been in the fifteenth generation back, and his or her parents would have been in the sixteenth generation back. Thus the baby is only one person out of 32,768 (two to the fifteenth power) whose genetic makeup contributed to my friend's DNA. The parents -- the ones who actually travelled on the Mayflower -- would have been two people our of a staggering 65,536 individuals who were necessary to produce my friend.

The moral of this story is that the powers of two are pretty awesome. They stack up fast. Another classic example is of doubling a penny. Let's say Billy was walking down the street and saw a shiny penny on the sidewalk. He picked it up, and as he did so a genie popped out of thin air and granted him a guaranteed string of luck -- starting with only the shiny penny, he could place one 50/50 bet at the local casino's roulette table each and every day for the next month, and he'd always win. This would only work once a day, for a single bet, though.

Let's posit that it was the first of the month in a month with 30 days. How much money would Billy have at the end of the month? The answer is pretty staggering. Billy rushes off to the casino, puts his penny down on red, and the ball rolls into a red slot. That's day one -- he has doubled his money and now has two cents. The next day, Billy bets his two cents and turns it into four. Each day, Billy faithfully returns and places his money on either red or black and he wins each time, which doubles his money.

The process begins slowly -- it takes Billy a whole week to get over a single dollar. On the seventh day, Billy leaves the table with a paltry $1.28. It takes him another three days to get past ten bucks, and by the fifteenth of the month, Billy has only $327.68. That doesn't seem like that much, but then again he started with only a penny.

The second half of the month is where things start to really build up. By the 20th of the month, Billy has over $10,000. By the 24th, he's got over $150,000. And here's the surprisingly large answer to the example -- on the 30th day of the month, Billy walks away from the table with a staggering $10,737,418.24. When you remove the decimal point, you get 1,073,741,824 -- or two to the thirtieth power. Starting with one and doubling it every day for a month adds up to over a billion in thirty days' time.

Computer programmers are familiar with powers of two, because just about everything in the computer world is measured this way. Consider the beginning of the series:

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096...

Anyone who has ever bought a memory card will recognize most of these numbers. A "kilo" in the computer world is not 1,000, but instead 1,024.

Getting back to the real-life example of my friend's ancestry, though, she was pretty astonished to realize that the one branch her family had painstakingly traced was merely a tiny, tiny fraction of her entire family tree. She's rightfully proud of the two people in her line that came over on the Mayflower, but they were but two of 65,536 people alive during that generation who had to come together and make babies in order for her to exist.

My overall point, and the one that certainly has relevance to the debate currently raging in the political world, is that we all tend to focus on the most impressive (or most notorious) branches of our own ancestry. Claiming that a famous (or infamous) person is a direct ancestor is a big point of pride for many people, but human nature is such that we focus on the notable and tend to ignore the mundane. Family lore is usually a collection of stories about the exceptional, in other words, and not all the others who weren't as interesting. Once you get more than a few generations back, any individual ancestor is merely one out of hundreds (or thousands, or even ten-thousands) of your ancestors. This is the way genetics works, and will go on working as long as it takes precisely two parents to produce a child.

-- Chris Weigant


Follow Chris on Twitter: @ChrisWeigant


17 Comments on “Powers Of Two”

  1. [1] 
    Paula wrote:

    Interesting column Chris!

  2. [2] 
    Don Harris wrote:

    It's like how I remember my previous life when I was King but not the other lives when I wasn't, like now.

  3. [3] 
    Mezzomamma wrote:

    Not to quarrel with your main point--

    Geneticist Adam Rutherford points out in his book, A brief history of everyone who ever lived, that as you go back, these numbers start to collapse, because the same individual will appear at different points in a family tree. He talked to a statistician (Joseph Chang of Yale) about how far back, statistically, all Europeans would have a common ancestor: 600 years ago. And of course some of a 'founding' population have no descendants after a certain time. So in a sense, everyone of European ancestry has Charlemagne among their ancestors, just as everyone of some Asian ancestries has Genghis Khan among their ancestors. The statistics for a common ancestor of everyone alive today suggest only 3400 years ago. Yes, 'just' statistics, but a useful corrective to people who take the family tree a bit too seriously.

    The book is well-written, Rutherford presenting a regular science program on BBC Radio 4, and I recommend it.

    Rutherford is not a fan of commercial DNA testing, it should be said. In any case, we all have more similarities than differences among our DNA, which brings me back to the point you made in the previous column about splitting hairs about race.

  4. [4] 
    neilm wrote:

    Well I'm going to boycott T.I.

    Not that he'll notice as I'd never heard of him before yesterday and have never bought a rap album in my life.

    Seriously, after "Citizens United" the right can't get too worked up about this and be taken seriously.

  5. [5] 
    nypoet22 wrote:


    power of two? what about the power of pie? why haven't you responded to my request for a comment about pie? pie often has honey and rarely if ever has vinegar.

  6. [6] 
    Paula wrote:

    [5] nypoet: I am loving the pie posts!

  7. [7] 
    MyVoice wrote:

    I like pie as well, but today I'd like to talk about closed systems. All of the following were cribbed from Wikipedia, that bastion of the people's knowledge:

    A closed system is a physical system that does not allow certain types of transfers (such as transfer of mass and energy transfer) in or out of the system.

    In nonrelativistic classical mechanics, a closed system is a physical system that doesn't exchange any matter with its surroundings, and isn't subject to any net force whose source is external to the system.

    In quantum physics ... an isolated or closed quantum system, that is, by definition, a system which does not interchange information (i.e. energy and/or matter) with another system.

    In chemistry, a closed system is where no reactants or products can escape, only heat can be exchanged freely (e.g. an ice cooler).

    In an engineering context, a closed system is a bound system, i.e. defined, in which every input is known and every resultant is known (or can be known) within a specific time.

    Basic science.

  8. [8] 
    Kick wrote:


    power of two? what about the power of pie? why haven't you responded to my request for a comment about pie?

    I know why! Because he's too busy writing about my idea for lemonade and the power of writing in "Don Lemon" on your ballot. :p

  9. [9] 
    Kick wrote:


    Well I'm going to boycott T.I.

    What did he do that upset you?

    Not that he'll notice as I'd never heard of him before yesterday and have never bought a rap album in my life.

    Oh... nevermind. :)

  10. [10] 
    nypoet22 wrote:


    your lemonade stand idea is total moosepoop because it doesn't have pie!


  11. [11] 
    nypoet22 wrote:

    but i suppose if don lemon wanted to, he could become a pie candidate. voters could choose him by writing in "lemon pie" - that would make him an acceptable pie candidate in my book.

  12. [12] 
    TheStig wrote:

    CW- Mitochondrial DNA and Y chromosome DNA pass directly from mother to daughter and father to son, respectively. The two squared law doesn't apply if you look at ancestory strictly along male and female lines, instead of marriage to marriage.

  13. [13] 
    Don Harris wrote:

    The power of two has been replaced with the behavior of two year olds.

  14. [14] 
    nypoet22 wrote:


    thank you for your support, and please make sure to sign up with the american pie council to let them know you support pie.

  15. [15] 
    Kick wrote:


    your lemonade stand idea is total moosepoop because it doesn't have pie!

    We can get pie, but it has to be lemon!

    but i suppose if don lemon wanted to, he could become a pie candidate. voters could choose him by writing in "lemon pie" - that would make him an acceptable pie candidate in my book.

    Don Lemon is not a candidate, JL. He's the guy whose name you write in on your ballot instead of not voting at all. Oh, sure, in the majority of states, you're still not voting at all because they don't even count your write-in vote, and in many of the other states, write-in candidates have to register in order to be considered valid at all, so all things considered, writing in Don Lemon's name will really show them. :)

  16. [16] 
    Balthasar wrote:

    Math is one of my least favorite subjects - algebra and geometry are just nice Arabic and Greek words to me, to be honest. But since it is the topic, I think it's time to have a serious discussion about squares.

    Everyone used to know what squares were: they were conformists, people afraid to try new things. The term apparently originated in jazz culture during the 1920's - 1940's (although my 86-year-old mom can't remember it being used in conversation until the 1950's).

    Its use was sometimes accompanied by the tracing of a square in mid-air, which younger folks might recognize from the movie Pulp Fiction when Mia Wallace tells Vincent Vega, “Don’t be a…” and draws a square in the air, where the square magically appears. As with everything Tarantino, the visual gag is derivative, also dating back to the 1950's as this handy video demonstrates.

    The prototypical 'square' probably wore horn-rimmed glasses, shunned mainstream culture, liked old-fashioned things, dressed in out-of-date clothes, and despised anything called 'popular'. They were often confused with folks with scientific or atypical intellectual interests, who were later termed 'nerds', though they might might self-identify as members of those social groups.

    The opposite of the word 'square' in the 1950's was the word 'hip', which derives from 'hep', another bit of 20th-century jazz slang, meaning roughly 'in the know to popular culture', and had another contemporaneous synonym 'cool', meaning 'good', which use has lasted into the present.

    'Hip' as a descriptor however, has had a mixed history. The terms 'hipster' and 'hepster' appeared in the late 40's as a term for white fans of jazz. The derivative term 'hippies' was originally used as a derogatory term for pseudo-beatniks, as used by Shel Silverstein in Playboy in 1960: "Then the hippies, they come on cool – they will let me make it if I dig too! Ha!!!!" This was also the sense of the word used in the 1963 Orlons song, 'South Street' when they sing "where do all the hippies meet?".

    The first use of 'Hippies' in the counterculture/flower child sense appears to have originated in a series of articles by Michael Fallon in the San Francisco Examiner in 1965 in which he uses the term interchangeably with 'beatniks' and 'heads'.

    But the term 'hip' in the positive sense persisted, even as the term 'square' dropped out of popular use. One of my favorite uses was in the 1973 song 'What is Hip?' by Tower of Power, about which co-writer Emilio Castillo later said, "Well, what I mean is being hip is so short lived. You can be hip by wearing your hair a certain way today and then in three months that style's gone and you're as unhip as you could possibly be. I want to write a song about that.' It's saying what's hip today might become passé."

    About a decade later, the word hip was folded into a new term for danceable rap music: Hip Hop, coined, according to some sources, by Disco Fever club DJ Lovebug Starski.

    In the modern age, the word hip morphed into the word 'hipster', a term that was first applied to affluent millennials living in gentrified neighborhoods.

    The prototypical 'hipster' probably wears horn-rimmed glasses, shuns mainstream culture, likes old-fashioned things, dresses in out-of-date clothes, and despises anything called 'popular'. They are often confused with folks with scientific or atypical intellectual interests, known as 'nerds', though they might might self-identify as members of that social group.

    But that's essentially the same description applied to squares in the 1950's. So these days you have hipsters that are squares being called hipsters by squares that used to be called hippies by hepsters.

    Right. A circle. Told you I was bad at geometry.

  17. [17] 
    Paula wrote:

    [14] Balthasar: So these days you have hipsters that are squares being called hipsters by squares that used to be called hippies by hepsters.


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