Tag: Blog

The Enormous Kuiper belt

Super fact 55 : The enormous Kuiper belt.

The Kuiper Belt is a vast torus/donut shaped region of space beyond Neptune, filled with icy, rocky bodies, including dwarf planets like Pluto. It shares a lot of similarities with the Asteroid belt, but it is much larger, and further out. The Kuiper belt is 20 times wider than the Asteroid belt, 1,000 larger by volume, and 20 to 200 times more massive than the Asteroid belt. It extends from roughly 30 to 50 astronomical units (AU) from the Sun.

I can add that one Astronomical Unit (AU) is the distance from the sun to Earth.

In the middle of the picture is the sun and around it is Mercury, Venus, Earth and Mars. Then there is a grey circular band representing the asteroid belt. Further out is Jupiter, Saturn, Uranus, Neptune and Pluto and a large circular band representing the Kuiper belt | The Enormous Kuiper belt
I drew this illustration of the solar system and the Kuiper belt. It is not entirely to scale, and in reality, Mercury and Venus are not attached to the sun.

The Kuiper belt is like a giant Asteroid belt located further out, beyond Neptune. The Kuiper Object Pluto, formerly known as the Planet Pluto, is the most admired, the cutest and most beloved of all planets, and it was the first Kuiper object discovered in 1930. However, we did not know of the existence of the Kuiper belt at the time. The Kuiper belt was discovered in 1992 and predicted to possibly exist by Astronomer Gerard Kuiper in 1951. The discovery of the Kuiper belt was one of the reasons Pluto was demoted from its planet status in 2006. There are other dwarf planets in the Kuiper belt similar Pluto, including Makemake, Haumea, and Eris. However, there could be hundreds. Ceres is a dwarf planet located in the Asteroid belt. To read more about the Kuiper belt and verify the facts above, click here, or here, or here.

This picture features the photo of Pluto taken by NASA’s New Horizons spacecraft in 2015 plus some text. The text says : This is Pluto! In 2006, the International Astronomical Union declared that Pluto is no longer a planet. Despite that, it keeps revolving around the Sun the same way it has been doing for billions of years. Pluto doesn't care what others think about it! Be Like Pluto!
Pluto and its moon Charon from NASA/JHUAPL/SwRI. NASA / Johns Hopkins University Applied Physics Laboratory / Southwest Research Institute, Public domain, via Wikimedia Commons. NASA’s New Horizons spacecraft captured this high-resolution enhanced color view of Pluto in 2015.

I selected this to be a super fact because the existence of the Kuiper belt drastically changed our view of our Solar system, so it is important, we know it exists, so it is a true fact, and despite its enormous size the Kuiper belt is much less known than the Asteroid belt, and its existence often comes as a surprise to people.

The Kuiper Belt Resides in Darkness

You may wonder why the Kuiper belt was discovered so late whilst the Asteroid belt has been known since the beginning of the 19th century (Ceres 1801, Pallas 1802, Vesta 1807, etc.) The reason is that the Kuiper belt resides in darkness. The Asteroid belt is 2.2AU to 3.2AU from the sun whereas the Kuiper belt is between 30 to 50AU from the sun.

Let’s say you take an object that is 2.5AU from the sun and place it at a distance that is 40AU from the sun. Due to the spreading of the light the object will now receive 16 X 16 = 256 times less sunlight. This is called Geometric dilution. In addition, this light needs to be reflected back to earth for us to see the object, and once again the light will  spread resulting in 256 X 256 = 65,536 times less light reaching our telescopes. The Kuiper belt is huge, but it resides in darkness. Despite this fact, we have now discovered and catalogued more than 2,000 Kuiper belt objects. However, it is estimated that there are hundreds of thousands of Kuiper belt objects wider than 100 kilometers.

What is a Dwarf Planet?

A planet as well as dwarf planet is a celestial body that orbits the Sun and is nearly round due to its own gravity. Basically, it must be large enough to have compressed itself to a near spherical shape. To be classified as a planet and not a dwarf planet it must also have cleared its orbit of debris. So, a dwarf planet is therefore a celestial body that orbits the Sun, is nearly round due to its own gravity, but has not cleared the neighborhood around its orbit. Obviously, a planet in the Asteroid belt or the Kuiper belt is a dwarf planet. Just to make this complicated Astronomers have found giant exoplanets that have not cleared their orbit of debris . I wonder, are these exo-planets giant dwarf planets?

Oort Cloud

Astronomer and Author David Lee Summers (blog here) reminded me of the Oort cloud, which could be interesting to bring up in this context. The Oort Cloud is a vast spherical cloud of icy bodies, which is hypothesized to surround the solar system, extending from about 2,000 to 200,000 AU. It is thus thousands of times further out and wide than the Kuiper belt. I say hypothesized because the objects are so small, there’s really no direct observation of them and there’s some variation in numbers for its distance and extent, meaning it’s still not well defined yet. Still, its outer edge is believed to be the boundary between where the sun’s gravity dominates and the galaxy’s gravity dominates.

The Oort cloud is generally considered to be the outer edge of the solar system and believed to be the origin of most long period comets. The Oort cloud is thought to encompass two regions: a disc-shaped inner Oort cloud aligned with the solar ecliptic (also called its Hills cloud) and a spherical outer Oort cloud enclosing the entire Solar System.

The picture is of the Oort cloud with an inset picture of the Kuiper belt at the top. The inset picture is an enlargement of the dot in the middle corresponding to the Kuiper belt.
NASA This SVG image was created by Medium69.Cette image SVG a été créée par Medium69.Please credit this : William Crochot, Public domain, via Wikimedia Commons.

Other Astronomy Related Super Facts

To see the other Super Facts click here

Digging Up Super Facts

Daily writing prompt
What change, big or small, would you like your blog to make in the world?
View all responses

I have two blogs, Leonberger Life and this one called Superfactful.

Leonberger Life feature amusing and heartwarming stories about our late Leonberger dog Bronco, as well as other Leonbergers. It also has a lot of information about the Leonberger breed, the history, care, training, Leonberger organizations, etc. I also wrote a Leonberger book, which I am featuring in the sidebar. With my Leonberger Life blog I want to spread information about the Leonberger breed, a rare, large, furry, friendly and fun dog breed and also bring attention to my book The Life and Times of Le Bronco von der Löwenhöhle, stories and tips from thirteen years with a Leonberger.

The goal of Superfactful, which is this blog, is to create a long list of facts that are important, not trivia, and that are known to be true and yet are either disputed by large segments of the public or highly surprising or misunderstood by many. I call these kinds of facts Super facts because they could potentially be very impactful on how we view the world.

Humans have accumulated an enormous amount of knowledge. Science does not know everything, but it knows a lot. The same cannot be said for us as individuals. We know next to nothing and harbor a lot of false beliefs. I think that is pretty much true for all of us, including me, but we may not know it. With this blog I am trying to correct some of that, at least regarding important facts. In addition, along the way, I am hoping to learn a lot myself and have some of my own false beliefs corrected.

In short, the change I would like to make in the world with my blog is to correct as many false beliefs as possible and educate my readers, and myself, about facts that are both important and mind blowing.

A blue brain is splitting up into pieces.
Smash your old beliefs with new surprising facts, super facts. Expand your mind. Shutterstock ID: 1685660680 by MattL_Images

What is a Super fact?

A super fact is:

  • An important fact but it can be simply stated.
  • Very surprising, shocking, widely disputed, misunderstood, or mind-blowing.
  • Yet it is true with a very high degree of certainty.

The first two criteria are subjective. The last criteria can be determined from longstanding scientific consensus, my own expertise and education (valid for just a few topics), agreement between multiple reliable online or offline sources such as agreement among research papers, reputable scientific organizations, NASA, NOAA, Pew Research Center, Our World in Data, etc. I should say I also link to less academic sources such as Wikipedia, but I do not solely rely on them.

You can read more about how I choose super facts here. Also, I am open to challenges based on good data (not opinions), as well as questions. With that I don’t mean that you cannot give your opinion in a comment. I just won’t update or remove a super fact based on an opinion. In addition, I am happy to receive suggestions for super facts. I am trying to collect a few hundred super facts and need all the help I can get. In the end I want to pick the 100 best ones. I might use a poll for that.

Sometimes a super fact involves doing myth busting of a popular myth and sometimes a super fact is stating something that is well known but disputed by many. In these cases, the evidence is likely to be conclusive, but the fact is still disputed by those who don’t know much about the evidence, or don’t want it to be true. In this case, I will include a substantial amount of evidence, and it might be lengthy. People get bamboozled all the time, and that includes me. It is not easy to admit that you have been bamboozled. You can read about my own experiences with that here.

Picture shows a scale held by a pointing finger. Fact is on the left shown as a bright light bulb. Myth is shown on the right as grey ball | Digging Up Super Facts
Fact or myth. Shutterstock Asset id: 2327968607

Examples of Super Facts

At the time of posting this I have made plans for 150 super facts and so far, I have posted 54. I will post a lot more. Below I am listing a few of my first 54 super facts.

Superfact 5 : Two events may be simultaneous for some but not for others. Click to visit post.

Two events may be simultaneous for some but not for others. This means that two events that are simultaneous to an observer may happen at different times to other observers. If two lamps A and B turn on at the same time according to observer #1, lamp A may turn on first for observer #2, and lamp B may turn on first for observer #3. All three observers are correct because time is relative.

The picture shows three pairs of lamps and three people. The setup is used to show three situations.
Amy is traveling at a high speed to the left compared to two lamps A and B. Alan is standing still compared to the lamps. Adam is traveling at a high speed to the right compared to two lamps A and B. Alan turns on the lamps at the same time. After considering the travel time of the light she sees, Amy concludes that lamp B turned on first. After considering the travel time of the light he sees, Adam concludes that lamp A turned on first. I should add this non-simultaneity can only happen if the lamps are separated by a distance.
Superfact 7 : Poverty and child mortality has been sharply reduced worldwide. Click to visit post.

Extreme poverty as well as child mortality has been sharply reduced the world over. The countries that are the worst-off today are still better off than the countries that were doing the best at the beginning of the 19th century. Over the last 20 years extreme poverty and child mortality have continued to decline sharply.

This graphics contain two graphs one for the world (blue) and for the United States (red) | Digging Up Super Facts
Child mortality in in the world since 1950. The spike you see around the end of 1950 to 1960 is the great leap forward famine in China. In 1950 the child mortality rate was 22.7% and in 2023 the child mortality rate was 3.6%.
Superfact 25: Global Warming is Happening and is Caused by us – click to visit post.

Global warming is happening. Or if you call it Climate Change or Climate Disruption is indeed happening. And it is happening very fast. We also know that it is caused by us primarily as a result of our burning of fossil fuels. There is a long-standing scientific consensus on these two facts because the evidence is conclusive. Check the evidence in the post.

Superfact 28: That Earth is round was well known long before Columbus – click to visit post.

That Earth is round, or spherical (or closely spherical) had been known for at least a couple of thousand years by the time Columbus set sail. Columbus did not set sail to prove that earth was round, and he knew it was round.

On the left a wheel with spokes. On the right there is a sphere and sun rays that hit in two places. One without a shadow and one with a shadow | Digging Up Super Facts
Illustration of the measurement of the Earth circumference by Eratosthenes (2,300 years ago). On June 21st there is no shadow in Syene/Aswan but there is one in Alexandria. Asset id: 2319651251 by Javier Jaime
To see a list of the Super Facts click here

Small Microscopic Subatomic and Strings

Esther’s writing prompt: 6th August : Small

Click here or Here to join in

Small Things

Fire ants are small. They average 1/8 inch to 1/4 inch in length, or 3 to 6 millimeters. Mites are very small arachnids that are less than 1 millimeters. They are so small that they are difficult to see with the naked eye unless they are on a white sheet. However, amoebas are typically even smaller than mites. Most amoebas range from 10 to 500 micrometers in diameter. 500 micrometers is the same as half a millimeter. You typically need a microscope to see an amoeba. I should say that there are some large amoebas that are 2 millimeters.

The photo shows six different types of amoebas | Small Microscopic Subatomic and Strings
Amoebas from Wikimedia commons. Attribution Respectively: NIAID, Cymothoa exigua, ja:User:NEON / User:NEON_ja, Jacob Lorenzo-Morales, Naveed A. Khan and Julia Walochnik, ja:User:NEON / User:NEON_ja, ja:User:NEON / User:NEON_ja, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0&gt;, via Wikimedia Commons

Microscopic Things

If you want to go even smaller, much smaller, we can enter the microscopic world. Bacteria are microscopic, single-celled organisms with sizes typically ranging from 0.5 to 5 micrometers in length and 0.2 to 1 micrometer in width. That means that bacteria are around 100 times smaller than amoebas. Well, if you consider length. If you consider the volume that is a million times smaller. Comparing an amoeba to a bacterium is like comparing a horse to a small cicada. You certainly need a microscope to see bacteria.

If you think bacteria are small, I can tell you that viruses are even smaller. Viruses typically range in size from 20 to 300 nanometers in diameter. 1000 nanometers is 1 micrometer. A small corona virus (SARS-CoV-2) is 50 nanometers, which is 20 times smaller (in diameter) than a bacterium that is 1 micrometer in size and 100 times smaller (in diameter) than a bacterium that is 5 micrometers. Again, a horse to a medium size insect.

Illustration of Covid-19 Virus
Illustration of Covid-19 Virus (SARS-CoV-2) from Wikimedia Commons. Attribution: SPQR10, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0&gt;, via Wikimedia Commons

Atoms Are Very Small

Atoms are much smaller than viruses. This reddit user calculated that there are roughly 52 million atoms in a normal sized covid virus (100 nanometers). Also keep in mind that there is a lot of space between atoms. The size of a hydrogen atom is 0.1 nanometer or 100 picometer. Comparing a hydrogen atom to a normal sized covid virus is like comparing a flea to a horse. If you consider volume, you could fill a normal sized covid virus with 1 billion hydrogen atoms.

You cannot see an atom using a regular microscope. You must use specialized microscopes that don’t rely on visible light to see atoms, such as scanning tunneling microscopes and electron microscopes. So, in summary, a hydrogen atom is to a normal sized covid virus like a flea is to a horse, and a normal sized covid virus is to a 100 micrometers amoeba (small sized amoeba) like a flea is to a horse.

Below is an illustration of a Helium atom, which is the next element after Hydrogen. A Hydrogen atom has one electron and one proton and possibly one or two neutrons. A stable Helium atom has two electrons and two protons and one or two neutrons.

Illustration of a Helium atom. A nucleus with protons and neutrons is surrounded by a grey fuzzy electrons cloud | Small Microscopic Subatomic and Strings
Illustration of a Helium atom. It has two electrons and a nucleus with two protons and two neutrons in the middle. The two electrons are depicted as clouds because they don’t have an exact position. Attribution : User:Yzmo, CC BY-SA 3.0 <http://creativecommons.org/licenses/by-sa/3.0/&gt;, via Wikimedia Commons

Subatomic Things

But let’s go smaller, much smaller. A hydrogen atom is gigantic in comparison to subatomic particles. Most of the mass in an atom is concentrated in the nucleus, which consists of protons, neutrons, quarks and gluons, and quark pairs called mesons. The size of an atomic nucleus varies, but it typically ranges from 1.6 femtometers (1.6 x 10⁻¹⁵ meters) for a proton to about 15 femtometers for the heaviest atoms.

I should say this is difficult to estimate so take this with a grain of salt. In any case that makes the hydrogen atom about 100,000 times wider than the nucleus in its middle. If the hydrogen atom was 100-meter giant ball the nucleus in the middle would be just 1 millimeter (half the size of a flea). That is despite the fact that the vast majority (+99.95%)  of the mass of the atom is in the nucleus. In this case, we are not comparing a flea to a horse, but a flea to a mountain. A mountain of mostly empty space with a super massive flea at its center. The YouTube video below explains the details.

Strings Are Extremely Small

However, the smallest things there are, might be strings. Strings, in the context of physics, are one-dimensional, extended objects that are thought to be the fundamental building blocks of the universe. These strings vibrate at different frequencies giving rise to elementary subatomic particles. Strings are thought to be about 10^-35 meters, which is 100,000,000,000,000,000,000 times smaller than the atomic nucleus described above. Comparing a string to a nucleus would be like comparing the hydrogen atom to a ball, or a giant star, containing one billion planet earths. I should mention that string theory has not been experimentally confirmed.

That is small, very small, extremely small, as small as it can get.

This post is not a super fact since it features a lot of facts and not all of them confirmed or exact.

To see the Super Facts click here

Satellites handle a very small amount of global internet traffic

Super fact 54 : Satellites currently handle a very small percentage of global internet traffic, estimated at about 1%. The vast majority of internet traffic is carried by undersea cables.

I consider this a super fact because it is surprising, true and not trivia. It is an important fact since most of us use internet every day. It is a very common belief that satellites handle most of the global internet traffic, or at least a very large portion of it. I should say that a few years ago I thought so myself.

Space satellite orbiting the Earth. 3D rendering | Satellites handle a very small amount of global internet traffic
Stock Illustration ID: 1372134458 by Boris Rabtsevich
Multiple layers inside a black cable. Optical fibers emerge from the cut end, and they shine.
Submarine underwater communication fiber optic cable on deep seabed. Asset id: 2175977719 by Dragon Claws

How Much Internet Traffic is Handled by Satellites

I should say that I did not find a lot of sites that answered this question, but all of the sites that I found gave similar answers such as, less than 1%, 1.5%, 1-2%, very little, etc.

I started out by asking ChatGPT this question “How much of internet communication does Satellites handle?” The answer I got was that Satellites handle a relatively small percentage of global internet traffic — typically less than 1–2% — with most of the world’s internet communication carried through undersea fiber optic cables and terrestrial infrastructure (like cell towers and wired broadband). Wikipedia states that satellites handle less than 5% – to an estimate of even 0.5%. I should add I do not rely on ChatGPT, SGE or Gemini, or Wikipedia for this blog but I take hints from them.

According to NOAA, over 95 percent of international data and voice transfers are currently routed through the many fiber optic cables that crisscross the world’s seafloors, whilst satellites currently carry just about 1% of global internet traffic according to Research Outreach. Operations Forces Report, Space Voyage Ventures, Neterra, and Newsweek made similar claims .

The reason satellite internet is used less is because satellite internet is significantly more expensive to use than traditional wired connections. In addition, satellites have higher latency (delay) compared to fiber optic cables, and less bandwidth (data capacity). Satellite internet is primarily used in remote areas where other forms of internet access are unavailable. Satellite internet is also used for military and government operations, as well as maritime and aviation connectivity. However, satellite internet is improving so this may change in the future. This is a comparison between satellite internet and optical fiber.

Satellites handle a very small amount of global internet traffic
Photo by SpaceX on Pexels.com
Eight layers in various colors are shown. See below for indicators.
This is a cross section of submarine fiber optic cable. Picture by Oona Räisänen (User:Mysid), Public domain, via Wikimedia Commons.

The layers in the picture of the submarine communications cable above are (from outside to inside): (1) Polyethylene (2) Mylar tape (3) Stranded steel wires (4) Aluminum water barrier (5) Polycarbonate (6) Copper or aluminum tube (7) Petroleum jelly (8) Optical fibers.

Related Post

A related post and super fact is: GPS uses relativity for accuracy

To see the other Super Facts click here

The Euler Number Math Magic

Super fact 53 : The Euler number denoted e, is an irrational number, which like the number pi is extremely important in mathematics. In addition, the relationship between the Euler number and pi; seemingly unrelated numbers, is quite amazing, especially if you throw the imaginary number: i = square root of -1 into the mix. Euler’s formula e^ix = cos(x) + isin(x), where x is degrees expressed in radians, is mind blowing to say the least. Radians means that 180 degrees is replaced by pi, and 90 degrees is replaced by pi/2, etc. A simpler special case, but equally amazing is Euler’s identity e^ix = -1, or e^ix + 1 =0. This is amazing math assuming you understand it.

As I said all this is amazing, mind blowing if you will, if you understand it, which is why I will try to explain it. Why I consider this a super fact is because when you first encounter the Euler number and the Euler formula, and you somewhat understand what it means, it is likely to be a mind-blowing experience. Those among you who have studied higher math, AP math classes in high school, or college level math are probably familiar with what I am about to describe, so your mind may not be blown. By the way you pronounce Euler like “Oiler”.

The formula e^ix + 1 =0 shown on a blue and black background | The Euler Number Math Magic
Euler’s formula in cyber space with grid 3d illustration, Asset id: 1636161301 by Giggle2000

Euler’s Number and Pi Two Irrational Numbers

Pi is the number you get when you divide the distance around a circle (the circumference) by the distance across the middle (the diameter). The Euler number is a bit more complicated to explain. I will do that next. Both pi and the Euler number are irrational numbers, which means that when written as a decimal, the number neither terminates nor repeats. As I mentioned, both pi and the Euler number are extremely important numbers in math. Perhaps the Euler should have its own day, just like pi has its own day (March 14). Maybe we should start celebrating Euler number day on February 7.

The picture shows a circle and a brief explanation of what pi is. The 20 first decimals of pi as well as the 20 first decimals of the Euler number are shown.
The first 20 decimals of pi and of the Euler number.

Exponents

Before I explain what, the Euler number is, I need to explain what an exponent is. If you multiply a number by itself x number of times, then x is the exponent. If you multiply two by itself four times 2*2*2*2, called 2 raised to 4, then 4 is the exponent. By the way the answer is 2^4 = 16 (called 2 raised to 4 is 16). I hope the illustration below will explain it.

This picture explains what an exponent is.
Overview of exponents.

And finally, before explaining what the Euler number is I should also mention what a factorial is. The factorial of a number is the product of all positive integers less than or equal to that number. The factorial of 5 is denoted 5! and is 1*2*3*4*5 = 120. Also, the factorial of 0 or 0! = 1 (per definition).

Definition of the Euler Number

One more thing I need to explain before I go ahead with the definition for the Euler number is what is meant by allowing a number n in a formula to go towards infinity (limit –> infinity). Let’s say you have a formula that contains the number n. If the value of the formula does not change much as n becomes very large than it might be approaching a specific number as n approaches infinity. You say that it approaches a limit. I am trying to illustrate this in the picture below.

The formula (1 + 1/n)^n is given for a lot of different numbers n. You can see that a number, a limit, is reached as n approaches infinity | The Euler Number Math Magic
As the number n gets bigger the formula stops getting bigger and instead approaches a limit. When n approaches infinity that will be a very specific number. Which number do you think it is?
The picture features the definition of Euler’s Number as well as another formula consisting of an infinite sum that is also Euler’s number.
The definition of Euler’s number plus an infinite series sum that is also the same as Euler’s number.
Two formulas equal to Euler’s Number. One is an exponent approaching infinity and the other is a sum from 0 to infinity | The Euler Number Math Magic
Definition of the Euler’s constant in two different ways, Asset id: 1227561829, by benjaminec.

Euler’s Number in Calculus

As I mentioned, Euler’s number shows up in mathematics in a lot of places. It is an extremely useful number with some amazing properties and that includes calculus. However, explaining functions and calculus may be going a bit too far, so I am just going to simply state that the derivate of e^x is just e^x and the indefinite integral, or the anti-derivative of e^x is e^x. In other words, differentiation / integration does not change this function. It also means that the slope of the curve is the same as the curve itself. Among all the infinite number of functions this is only true for e^x.

Differentiation and integration formulas for the exponential function.
Differentiation and integration does not change the function e^x.

Trigonometric Functions

Next, I would like to launch into Euler’s formula. However, before I do that, I need to explain what trigonometric functions and imaginary numbers are. The trigonometric function sin(x) is the ratio of the length of the side opposite to a given angle to the length of the hypotenuse. In other words, if the hypotenuse is equal to 1, then sin(x) is the length of the opposite side to the given angle. The trigonometric function cos(x) is the ratio of the side of the triangle adjacent to the angle divided by the hypotenuse. In other words, if the hypotenuse is equal to 1, then cos(x) is the length of the adjacent side to the given angle.

Sin and cos are always between 1 and -1. ‘x’ is often expressed in degrees going from 0 to 360 (or 0 to 90 in a right-angled triangle). However, there is another way to express angles in triangles and that is radians. In this case the number pi corresponds to 180 degrees, pi/2 corresponds to 90 degrees, pi/4 corresponds to 45 degrees, etc. Euler’s formula uses trigonometric functions, but it only works if you use pi instead of degrees. Pi and Euler’s number have a special relationship. Sin and cos are illustrated in the picture below.

The picture shows a right-angled triangle with the sides being the hypotenuse set to 1 and the two other sides sin(x) and cos(x) respectively | The Euler Number Math Magic
Illustration of the trigonometric functions sin(x) and cos(x).

Imaginary Numbers

The last thing I need to explain before demonstrating Euler’s formula is imaginary numbers. The square root of a number is another number that, when multiplied by itself, equals the original number. For example, the square root of 4 is 2, because 2 * 2 = 4. The square root of 9 is 3, because 3 * 3 = 9. As long as you deal with real numbers, square roots must be positive numbers because you cannot multiply two numbers and get a negative number. -2 * -2 is 4, not -4.

However, that did not stop some mathematicians from making up a square root that was negative. This imaginary number is the square root of -1 and is referred to as i, yes just i, for imaginary. So, what’s the point of making up numbers that can’t exist? Well, it turned out to be quite useful and you can manipulate imaginary numbers to result in real numbers. For example, if you multiply the imaginary number i by itself i*i you get -1. If you multiply i by itself four times, in other words i^4, or i raised to 4, you get 1. Even more impressively, i raised to i, or i^i, is a real number. i^i = 0.207879… This is illustrated in the picture below.

The picture features the definition of the imaginary number and an explanation for what imaginary numbers are, as well as examples.
Imaginary numbers illustrated

Eulers Formula

Without giving the proof, or any detailed explanations, below is Euler’s identity and Euler’s Formula (e^ix = cos(x) + isin(x)). Notice the mix of Euler’s number, pi, the trigonometric functions using radians (based on pi), and the imaginary number. Well, likely mind-blown, if you have not seen it already and you understood this post up to here.

Euler’s identity and Euler’s formula | The Euler Number Math Magic
Euler’s identity and Euler’s formula.
The picture shows Euler’s formula illustrated in the complex plane.
Euler’s formula illustrated in the complex plane. Asset id: 2345669209 by Sasha701

If you want to see how you prove Euler’s Formula check out this youTube video.

If you want to learn more about the importance of Euler’s number in sommon and useful mathematics, check out this youTube video.

To see the other Super Facts click here