Tag Archives: Math

Wrap Your Head & Hands Around Density

Heat rises, boats float, clouds hang listlessly in the sky, and icebergs aimlessly bob around the ocean. All of these things happen because of the same scientific property: density!

Density is a rather simple property. It depends on two things, mass and volume. Mass is basically how much something weighs, but something still has the same amount of mass weather it is on Earth, Jupiter, or out in space where its weight would change dramatically. Volume doesn’t have anything to do with how loud something is, but rather with how much space it takes up. Mathematically, an object’s density is equal to its mass divided by its volume.

The demonstration here using the pipette in a bottle of water is a great way to buoy your own understanding of density. The pipette is carefully balanced to barely float in the water. However, by squeezing the bottle, water is forced up into the pipette. This makes the pipette heavier, which raises its density. Releasing the bottle down the opposite. For another cool example of density, click here!

By simple comparing the density of different things, we can tell if they would float or sink in water or air or anything else that has a known density. This is incredibly useful and can lead to some fun facts. For example, Saturn is less dense than water. That is to say it would float in a bathtub, but it might leave some rings! Red giant stars, like the famous Betelgeuse, are actually less dense than air! They are incredibly heavy–Betelgeuse weighs in at about 18 suns worth of mass–but they take up so much space that they are essentially GIANT balloons!

The sun would not float like a balloon now, in fact it is currently much more dense than water. However, given 5 billion years it will enter the red giant phase of its own life and grow in volume by about 800 million times! This will decrease its density so much that it goes from 40% higher than water to lower than air! As you can clearly see, density is not necessarily a constant property of something, it is actually something that can change.

One of the most commonly accepted consequences of density is that heat rises, but that isn’t technically true. It would be more correct to say that heat rises on Earth. This is because as things warm up, their volume increases which lowers their density. However, if we leave the friendly confines of Earth, this is no longer true! In the microgravity environment aboard the International Space Station, there isn’t really an “up” or “down”, gravitationally speaking. That makes it hard to decide what “heat rises” even means! Fire gets equally confused. Check out this comparison between fire on Earth and aboard the ISS from NASA.

Looking for Order in March Madness

Why is it so hard to pick the perfect bracket? Let’s take a look at the mathematics of March Madness!!
BracketThis year’s complete printable March Madness bracket by ESPN.

It all comes down to probability. Let’s start by making things simple before we truly delve into the madness. Imagine two perfectly even teams are about to play a single basketball game. Each team is equally likely to win. Who is going to win? Quite clearly in this scenario, anyone would have a one in two shot to get the winner right. In this simple one game bracket, you would have a 50% chance of having a perfect bracket!

Coin Flip crop

Credit: Joseph Smith

 
In the NCAA tournament, there is a field of 68 teams. Only one team can win, and one team is eliminated in each game, meaning there 67 games must be played. If each team were evenly matched (we’ll deal with the alternative in a moment), then the odds to pick the winner of every game correctly is a paltry one in 267, or one in 147,573,952,589,676,412,928. That’s over 147 quintillion! Not coincidentally, these are the same odds as picking heads or tails on a flipped coin 67 times in a row1.

Fortunately, most brackets don’t deal with the play in games, meaning they consider the field to be 64 teams, requiring 63 correct picks for the elusive perfect bracket. In addition, the top team in each region has never lost to the bottom team, giving four more games that stand as pillars of stability amongst the madness of March. Assuming this holds true, our odds have just improved tremendously to one in 259 or one in 576,460,752,303,423,488! This number also looks humongous, but the odds of a perfect bracket have improved by a factor of 256 (or 28)!

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Probability has not been kind to President Barack Obama in his quest for the perfect bracket either. Photo credit ESPN

Of course, not all of the teams in March Madness are the same. Each one has its own players, strategies, strengths, and weaknesses. If someone really followed basketball, they might be able to use these to their advantage. Actually, that’s kind of the point! After all, knowledge is power, so let’s see how that knowledge holds up against the mighty bracket! If in each matchup, one team held an edge raising its win probability to 60% and someone knew enough to identify every one of these teams, the odds of each favored team winning is as follows:

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mega-millionsThat’s still only one in 12 trillion–about 100 times worse than your odds of winning the Mega Millions jackpot! Here is where some of the struggle of the perfect bracket comes from. Even if each game has a favorite, the odds of the favorite always winning are not favored. The far more likely scenario is that some teams slip…but how can you tell which ones? So far, the answer has been resounding:

“You can’t”

  1. While certainly close enough for our purposes, flipping a coin is actually not truly random. Despite often being held up as the paragon of probability, a flipped coin shows a very slight preference towards the side that begins up.
  2. There is a very cool chart that takes this analysis further by FiveThirtyEight that uses each team’s actual probability to calculate their chances of winning or losing each game. Check it out here!

Love Pie? That’s Not Irrational! Happy Pi Day!

???? is the ratio between the diameter and the circumference of a circle. Any circle. It doesn’t matter how big or small a circle is, this ratio is always the same. This may not seem incredibly important, but many things are circular or spherical. This results in ???? showing up in all kinds of equations in science and math.

pi ratio

One especially weird thing about this number is that it is irrational. Most numbers can be written as a fraction, like ⅗ or ⅔, but irrational numbers just don’t fit not matter how big the top and bottom numbers become. Computers have calculated ???? to billions of digits, and even in that expanse of numbers, no pattern seems to emerge.

???? isn’t the only irrational number. Another famous one is e which is approximately 2.71828 but similarly continues on forever with no pattern in its decimal places.  Unfortunately, “e day” is a much less celebrated hlid, probably because it doesn’t share its phonetic name with a delicious dessert. It also doesn’t have such a nice definition as ????’s simple ratio. Instead, e is the result of the the following series:

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Where things get especially weird is when this numbers are put together with another famous number, i. As you may recall, i is an imaginary number, which has a value of sqrt(-1). In school, many students ask how this number can possibly be important if it is imaginary. Well, look at this:

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Woah! This happens because both e and ???? are important for the mathematics of waves. To learn more, check out Euler’s formula, preferably over some pie!

pie

As noted previously, ???? is important because of its relationship with circles. It seems appropriate that we consume some pleasantly round delicious phonetic namesake to celebrate!

Written by Scott Alton

Thomas Edison and the Bright Idea

Thomas Edison famously invented the incandescent light bulb. These light bulbs are pretty simple things. They are really just a glass casing around a tiny metal resistor called the filament. As electricity runs through it, the filament heats up until it produces light. But how much does it have to heat up? It turns out to be a pretty interesting question.

Britannica Light Bulb

Incandescent light bulb. The filament is the horizontal piece in the center of the bulb. Image credit: Encyclopædia Britannica

Everything gives off light based on its temperature, but not all light is the kind we can see. That part, known as the visible spectrum, is a tiny part of the larger electromagnetic spectrum. These different kinds of light are separated by their energies, which can be described with wavelength.

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Where is a simple equation that allows us to find out what wavelength is given off by an object at a given temperature. It’s called Wien’s Displacement Law.

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Humans are about 90 degrees Fahrenheit, which is about 305K. To find the wavelength emitted by humans, all we have to do is divide .002898 by 305. This gives about 9500 nanometers (nm) which is what we measure wavelength in for visible light. Visible light is between 400nm and 700nm, so 9500nm is pretty far away from that. This puts it squarely in the infrared.

NASA EM

The full electromagnetic spectrum, including the very small visible light portion. Credit NASA

So how hot does that filament need to be to give off visible light? Lets aim for 500nm, which is definitely in the visible light range. By using algebra, Wien’s Displacement Law can be rearranged to T=b/. Dividing b by 500nm gives a temperature of 9973℉, which is about the temperature of the surface of the sun!

This actually makes total sense too! This means that the sun’s primary light emission is smack dab in the middle of our visible spectrum. It definitely makes sense that if we were to gain the ability to see, it would be to see the kind of light that was the most common. However, pretty much everything in the sun is a plasma due to the immense heat. There is no way that a light bulb could survive such insane temperatures.

Light bulb filament comes in at a relatively cool to the sun but still insanely hot 4500℉. Throwing this back into the equation, we get 1052nm, which is much closer to the 400nm to 700nm range. The primary emission is still in the infrared. However, this only gives the peak wavelength of light given off. The actual light is a mix of light more like this:

Screen Shot 2015-10-21 at 12.39.16 PM

For a light bulb, only about 5% of the energy is given off as light that we can see. The rest of it is mostly given off as infrared light, or heat which is why these bulbs feel hot. The reason they don’t feel like 4500℉ is that the inside of the light bulb is a vacuum–meaning it contains no air–and the heat thankfully doesn’t conduct through it very well. They actually use this vacuum insulation for water bottles, thermoses, and even the Dewars that hold things like liquid nitrogen!

This inefficiency is the reason behind moving to much more eco-friendly fluorescent light bulbs. These bulbs don’t use Wien’s Displacement Law to produce light, but instead are filled with atoms that naturally release light when hit with a bit of electricity. Even many new bulbs that look like incandescent bulbs actually contain several small Light Emitting Diodes, or LED’s inside which share the energy efficient qualities of fluorescent bulbs. What a bright idea!

LED Bulb

How Many Stars are Out There?

The numbers used in Astronomy are truly staggering.  For starters, the Earth is about 25,000 miles around. The nearest star to us is–obviously–the sun, which is 93 million miles away. To travel that distance, you would have to circle the Earth nearly 4000 times! The larger the numbers get, the harder it gets to understand what they mean.

For example, if someone is a millionaire, they have at least a million dollars. If someone is a billionaire, they have at least a billion dollars. What is the difference between that million and billion? A factor of one thousand! That means that to be a billionaire, you have to make a million dollars one thousand times! Getting to trillions is similarly outrageous. To be a trillionaire, you would have to make a million dollars ONE MILLION TIMES!

Moving back to astronomy, the numbers naturally get even more difficult to understand! We learned from the video that our galaxy has around 300 billion stars! Remember how big a billion was!? Even when Max was typing 3,050,374 zeroes per day, he still had to go on for 270 years to type that many zeroes! Take into account that there are an estimated 100 billion galaxies in our universe, and things really start to get out of hand.

We estimate that the number of stars in the universe is around 70,000,000,000,000,000,000,000. Thats 70 sextillion, or 70 thousand million million million if that helps! For Max to type out that many zeroes would take 62,871,248,000,000 years (62 trillion!). Keep in mind that the accepted age of the universe is only 13.8 billion years. It would take Max over 1,000 times the age of the universe, just to type out the number of zeroes that there are stars in the universe!

Perhaps Neil Degrasse Tyson said it best: “There are more stars in the universe than grains of sand on any beach, more stars than seconds have passed since Earth formed, more stars than words and sounds ever uttered by all the humans who ever lived”.

And it isn’t particularly close.  http://www.naturalhistorymag.com/universe/201367/cosmic-perspective?page=2

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