Category Archives: Mathematics

How many gallons of human blood are there?

For this we need to know how many people there are and the average amount of blood in each one. Unfortunately, I will be assuming that all humans are adult sized with an adult amount of blood, which I’m fairly sure is not the case, so the estimated value will be higher than the actual value by a fairly substantial amount.
Now then. As of January 3rd, 2014, the date which this was written, the Worldometers world population clock [1] put the total world population at about 7.204 billion people.
As for blood, hypertextbook [2] cites several scientific studies which have found that the amount of blood in the average adult is right around 5 liters (or about 5 quarts). This means that the average person has about 1.25 gallons of blood.
By multiplying the two results we find that the total amount of human blood is right around 9 billion gallons.
Using data from a 2010 Pew poll of world religions [3] we find that there are 2.8 billion gallons of Christian blood, 2.1 billion gallons of Muslim blood, 1.4 gallons of Hindu blood and only 18 million gallons of Jewish blood. (These numbers don’t add up to the total because of other religions and the 1.5 gallons of unaffiliated blood)

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How rich do I have to be in order to fill an Olympic swimming pool with pennies?

We need to know a few things for this answer, the volume of an Olympic swimming pool and the volume of the penny (US coin).

First of all, the FINA lists the dimensions of an Olympic swimming pool as such:

50 meters long by 25 meters wide by at least 2.0 meters deep [1]

Since the depth could conceivably be unlimited, we will use a depth of 2.0 meters, meaning that the total volume of an Olympic swimming pool is 2,500 cubic meters.

The US Mint gives the following specifications for a penny:

A diameter of 19.05 millimeters and a thickness of 1.52 millimeters [2]

This means that the US penny has a volume of 4.3323E-7 cubic meters

Using these results, we find that 5.771 billion pennies are needed to fill the pool, which all together are worth 57.7 million US dollars.

One more thing:

Forbes lists the worlds billionaires here, the richest people on Earth. Currently, Carlos Slim Helu and his family are listed at #1 with a net worth of about 73 billion US dollars. [3]

If they so desired, they could fill over 1200 Olympic swimming pools with pennies. So yeah.

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Leap Years, Days and Seconds

2012 was a leap year, 2013 wasn’t, 2016 will be, 2014 won’t be, 2000 was and 1900 was not. Years are about 365 days long, but not exactly (measuring years is kind of hard to do and there are different ways of doing it, but a sidereal year is about 365.256363004 days long, and a tropical year is about 365.24219 days long, the difference is in the way the two are measured), and this difference between clean lengths of time like 365 days and the awkward 365.24219 days is enough to throw seasons off so that summer occurs in December. The Gregorian calendar uses a trick to adjust for this, adding a day to February. Normally, a year in the Gregorian calendar has 365 days, except for every fourth year (4, 8, 12, 16, …2004, 2008, 2012, 2016) which have 366 days. However, every 100th year (100, 200, 300, …2100, 2200, 2300) will not have a leap day, and thus have 365 days. Finally, every 400th year (400, 800, 1200, 1600, 2000, 2400) will have a leap day, and be 366 days long. This brings the average Gregorian calendar year to 365.2425 days long, not terribly different from the 365.24219 day long tropical year. Finally, leap seconds. The length of a day varies slightly, caused by gravitational forces on the Earth by the moon, sun, and other planets and usually only changes by a second, over the course of time. To keep the average length of a day as close to 86400 seconds as possible. Like leap years, leap seconds keep the rigid 86400 second day from drifting so 12-noon was sunset.

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Tetration

As mentioned in the last post, 2↑↑2=4, but what does the ↑↑ symbol mean? ↑↑ is a symbol (there are many notation systems, like ^^) for the operation known as tetration. Anyone with a basic knowledge of mathematics will know that multiplication is just repeated addition, and that a*b is the same as adding a to itself b times. Exponentiation is similar, a^b is just a multiplied by itself b times. Tetration, now, is an extension of these. a↑↑b is a to the power of itself b times, so 3↑↑3=3^3^3. As you can probably tell, tetration makes big numbers quickly, and isn’t supported by many calculating devises, or at least not for very long. Here’s some tetration involving 2:

2^^1=2

2^^2=4

2^^3=16

2^^4=65536

2^^5=2^65536≈2*10^19728

And some involving 3:

3^^1=3

3^^2=27

3^^3=7625597484987

3^^4=3^7625597484987≈10^10^10^1.01

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Why Two is My Favorite Number

Many math geeks will pick a favorite number of theirs because of a certain property of said number. Some people choose e, the base of the natural logarithm, some choose π, the ratio between a circles circumference and its diameter, some clever few choose i, the square root of -1, and some people choose a natural number, that is, a positive integer, that has some significance to them, be it 17, 300, 132, or any other number. I choose two (2) as my favorite number and here’s why:

  • 2 defines what numbers are even, and what numbers are odd.
  • 2 is the first prime number, the smallest prime number and only even prime number. 2 and 3 are the only two consecutive primes.
  • The decimal expansion of any simple fraction where the denominator is 2 will always terminate in a 5 or a 0, depending on whether the numerator is even or odd.
  • 2 is the base of binary, the number system with the smallest base in which numbers can be written (relatively) easily.
  • 2+2=2×2=2^2=2↑↑2=…=4. x↑↑y is called tetration, I’ll go over it later.
  • 2 is its own factorial.
  • 2 is part of a bunch of other special prime categories, like Fibonacci primes, Lucas primes, factorial primes.
  • 2 is a highly composite number, meaning it has more positive whole number factors than any number less than it.
  • There are two characters for two in Chinese, 二 and 两.
  • 2 is the atomic number of helium, the first noble gas.
  • Mersenne primes are found using powers of two.
  • The square root of 2 (about 1.414) was the first irrational number to be discovered.
  • 2 is the fewest number of dimensions needed for polygons to exist in geometry.
  • And finally, a superficial one, my birthday is in February, the second month in the Gregorian calendar.

 

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Countable and Uncountable Infinity

Hold on to your pants, this might get interesting here. What if I told you that some types of infinity are bigger than other types of infinity? It’s true. Just think about it for a second. The number of positive whole numbers, or natural numbers, is really big, infinite actually, because for the largest number you can think of I can just add one and there’s a bigger number. The set of natural numbers is what’s called “countably infinite” and any set of numbers that can be paired with the set of natural numbers is also considered “countably infinite”. Theoretically, any countable set can have its entire contents listed if the counter were given enough time. However, the set of all real numbers is not countably infinite because for every two values in the set there are infinitely many values between them. In fact, the set of all numbers between 0 and 1 is greater than the set of all natural numbers, that’s what uncountability does.

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Double Post: Subsidies and the Prisoner’s Dilemma

Again, I’m sorry for missing a day, but I’ll do some related topics in today’s double post: Subsidies and the Prisoner’s Dilemma.

Subsidies are support income provided by the government to certain companies, namely agriculture and energy providers. In the past, these sectors were weak and needed support from their companies otherwise life in general would collapse (no food, no electricity, no fuel, etc.). I’m not going to argue that subsidies are not needed, I’m no expert in economics and it’s not my place to do so, but some subsidies, such as those that pay farmers not to farm (which I’ll get to in a bit because it has a little to do with the next topic) are counter-intuitive.

So what does paying farmers not to farm have to do with the Prisoner’s Dilemma? First we have to know what the Prisoner’s Dilemma is. Say we take two prisoners, Alvin and Bruce, who have together committed a crime (the details aren’t exactly important). If either one sells out the other, and the other confesses, the one who confessed gets one year in prison while the one who betrayed gets off free, if both betray each other, both will get three months in prison, and if both confess, both get only one month in prison. It’s, individually, in both Alvin’s and Bruce’s to betray, but it’s in their collective interest if they both confess. We see something similar in raw food prices. If a farmer produce large amounts of food, that food will be worth less due to inflation, than it would be if they produced less, but if farmer produce less food they are at the whim of the markets and could get in trouble due to other farmers potentially producing more food and driving prices down. That’s the farmer’s dilemma, and that’s where subsidies come in.

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