How Imaginary Numbers Were Invented

It contains a chapter on the cubic equation, which today we can write as ax3 + bx2 + cx + d = 0. For at least 4,000 years, people have been trying to find a general solution to the cubic equation... but the civilizations that They had to deal with it...the Babylonians, Greeks, Chinese, Indians, Egyptians and Persians...they all didn't succeed. According to Pacioli, the cubic equation has no solution. This should be at least a little surprising, because without the x3 term... it's just a quadratic equation. And many ancient civilizations had solved quadratic equations thousands of years before. Today, after the fourth grade, everyone knows the general solution: But most use this formula without knowing anything about geometry... mathematicians once applied it to derive it.

Back then, mathematics was not yet written in equations. It was written with words and images. For example, take the equation x2 + 26x = 27. In ancient mathematics, the term x2 was interpreted as a literal square with sides of length x. And 26x then becomes a rectangle with one side of length 26 and the other side of length x. And those two areas together are 27. So how do we know what x is? We can take this 26x rectangle and cut it into 2. Now I have two 13x rectangles and I can position them so that the new shape is almost a square.

Only one piece missing at the bottom. But I know the dimensions of this piece. It's just 13 times 13. Since I added 132 or 169 to the left side of the equation... I also need to add 169 to the right side of the equation to maintain equality. Now I have this larger square with sides of length x + 13 and it is equal to 196. The square root of 196 is 14. So I know that the length of the sides of this square is 14, which means x = 1. This is a nice visual way to solve a quadratic equation,... but it is not complete. If we look at our original equation, x = 1 is a solution, but so is -27.

For thousands of years, mathematicians were unaware... of the negative solutions to their equations... because they were talking about things in the real world, lengths, areas and volumes. What would it mean to have a square with sides of length -27? That makes no sense. So, for those mathematicians, negative numbers did not exist. You could subtract, so calculate the difference between 2 positive values... But you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers... that there wasn't just one quadratic equation. Instead, there were 6 different versions arranged so that the coefficients were always positive.

The same approach was followed for the cubic comparison. In the 11th century, Persian mathematician Omar Khayyam discovered 19 different cubic equations. Also in this case all the coefficients remained positive. He found numerical solutions to some of them by looking at... the intersections of shapes, such as hyperbolas and circles, but failed in his ultimate goal: a general solution to the cubic equation. He wrote: “Maybe someone coming after us will manage to find them.” 400 years later and 4,000 kilometers away, the solution is beginning to take shape. Scipione del Ferro is a professor of mathematics at the University of Bologna. Around 1510 he finds a way to solve cubic equations without x2.

This is a subset of cubic equations without x². What do you do after solving a problem that has baffled mathematicians for millennia? A problem considered impossible by Leonardo da Vinci's mathematics teacher? He doesn't tell anyone. It wasn't easy being a mathematician in the 1500s. Your work is constantly undermined by other mathematicians who... can appear at any time and challenge you for your position. It's similar to a math duel. Each participant asks the other a series of questions. The one who answers the most questions correctly gets the job while the loser is publicly humiliated. As far as Del Ferro knows, no one else in the world can solve the cubic equation without x2.

So by keeping your solutions secret, you ensure your own job security. Del Ferro kept the secret from him for almost two decades. Only on his deathbed in 1526 did he tell his student Antonio Fior Fior that he is not as talented a mathematician as his mentor, but he is young and ambitious. And after Del Ferro's death, he boasted about his own mathematical prowess. More specifically, his ability to solve cubic equations without a x2 factor. On February 12, 1535, Fior challenged the mathematician Niccolo Fontana Tartaglia... who had recently moved to Venice, Fior's hometown. Niccolò Fontana knows adversity. When he was a child, a French soldier cut open his face and he has stuttered ever since.

That's why he is known as Tartaglia, which means stutterer in Italian. Tartaglia grew up poor and has largely been self-taught. He worked his way through Italian society to become a respected mathematician. Now all that is at stake. As usual, Tartaglia gives Fior a diverse set of 30 problems in the challenge. Fior gives Tartaglia 30 problems, all cubic equations without x2. Each mathematician has 40 days to solve the 30 problems assigned to him. Fior cannot solve any problem. Tartaglia solves Fior's 30 cubic equations without x2 in 2 hours. It seems that Fior's arrogance was his downfall. Before the challenge, Tartaglia learned that Fior claimed to have solved the cubic equation...without x2, but he is skeptical. “I did not consider him capable of finding that rule for himself,” Tartaglia writes.

But there was a rumor that a great mathematician had revealed the secret to Fior,... which was much more credible. Now that he knows there is a solution to the cubic equation... and with his income on the line... Tartaglia sets out to solve the cubic equation without x2. To do this, he extends the idea of ​​completing the square to three dimensions. He takes the equation x3 + 9x = 26. You can think of x3 as the volume of a cube with sides of length x. If you add a volume of 9x, you get 26. So, just like completing the square, we need to add to the cube... to increase the volume by 9x.

Suppose you extend 3 sides of this cube a distance y... creating a new, larger cube with sides of length called z. Z is simply x + y. The original cube has been expanded and we can divide the extra volume into 7 ways. There are 3 rectangular prisms of dimensions x by x by y, ... and another 3 narrower prisms of dimensions x by y by y. And there is a cube with a volume of y3. Tartaglia rearranges the six rectangular prisms into a single block. One side has length 3y, the other has length x + y. That is z and the height is x.

So the volume of this shape is the base, 3yz times the height x. Tartaglia realizes that this volume can perfectly represent the 9x in the equation, if the base is 9. So he sets 3yz = 9. When we put the cube back together, we see that we are missing the small block y3. We can then complete the cube by adding y3 to both sides of the equation. Now we have z3, the largest cube = 26 + y3. We have two equations and two unknowns. If we solve for z in the first equation and substitute it into the second, we get y6 + 26y3 = 27.

At first glance it seems that we are now in a worse position than at the beginning. The variable is now raised to the sixth power, instead of the third. But if you consider y3 as a new variable, the equation is actually a quadratic equation... the same quadratic equation we solved by completing the cube. Then we know that y3 = 1, which means y = 1, and z = 3/y so z = 3. And since x + y = z, x = 2, ... which is in fact a solution of the original comparison . And with that, Tartaglia becomes the second person on earth to solve the negative cubic equation.

To save himself the geometry work of each new cube... Tartaglia summarizes his method in an algorithm. A set of instructions. He does not write this as a series of equations as we would today. Because modern mathematical notation would not exist for another 100 years. He writes this like a poem. Tartaglia's victory made him famous. Mathematicians are eager to know how he solved the cubic equation... especially Gerolamo Cardano, a scientist from Milan. As you can imagine, Tartaglia is having none of that. He refuses to reveal a single quiz question. But Cardano is persistent. He writes a series of letters in which he alternates flattery with hostile attacks.

With the promise of introducing him to his rich patron... Cardano still manages to lure Tartaglia to Milan. And there, on March 25, 1539, Tartaglia revealed his method. But only after forcing Cardano to take a solemn oath... that he would not tell anyone about the method, he would not publish it and would only write it in code. “So that after my death no one will understand.” Cardano is delighted and immediately starts working with Tartaglia's algorithm. But he has a bigger goal in mind: a solution to the entire cubic equation, including x2. And miraculously he finds the solution. If he replaces x with x – b/3a, all x2 disappears.

This is the way to convert any general cubic equation... into a cubic equation without x2. This can then be solved with Tartaglia's formula. Cardano is so excited to have solved the problem... that top mathematicians have struggled with for thousands of years... that he wants to publish it. Unlike his colleagues, Cardano does not have to keep the solution a secret. He makes his living not as a mathematician, but as a doctor and famous intellectual. For him, credit is more valuable than secrecy. The only problem is the oath he swore to Tartaglia, which he can't break. And you would think this would end here.

But in 1542, Cardano travels to Bologna to see a mathematician... who turns out to be Scipione del Ferro's son-in-law. The man who, before dying, had revealed the solution of the cubic equation sin x2... to Antonio Fior. Cardano finds the solution in Del Ferro's old notebook, which he showed him during the visit. This solution is decades older than Tartaglia's. Cardano can now publish the complete solution to the cubic equation... without violating the oath to Tartaglia. Three years later, Cardano Ars Magna publishes De Grote Kunst,... an updated compendium of mathematics. “Written in 5 years, to last 500 years.” Cardano writes a chapter with a unique geometric proof... for the 13 orders of the cubic equation.

Although he recognizes the contributions of Tartaglia, del Ferro and Fior, Tartaglia is, to say the least, dissatisfied. He writes insulting letters to Cardano and sends copies to much of the mathematics community. and the he has a point. To date, the general solution of the cubic equation... is often called Cardano's method. But

Cardano asks Tartaglia about it. He avoids it and claims that Cardano... is not smart enough to use his formula correctly. In reality, Tartaglia has no idea what to do either. Cardano steps back through the geometric derivation of a similar problem... to see what exactly goes wrong. Although cutting and rearranging the 3D cube works well... the final quadratic equation to complete the square... results in a geometric paradox. Cardano finds a part of a square that must have an area of ​​30,... but also sides with a length of 5. Since the entire square has an area of ​​25... to complete the square, Cardano must somehow add a negative area.

That's where the square roots of negatives come from, the idea of ​​negative areas. This is not the first time square roots of negatives have arisen in mathematics. Previously on

Mathematicians understood that mathematics with square roots of negative numbers... was intended to indicate that there is no solution. But this cubic equation is different. With a little guessing and checking you can discover that x = 4 is a solution. Why doesn't the method that works for any other cubic equation... provide a solution?logic to this problem? Cardano sees no way out and avoids it in

He wants to find a way to find a solution through the chaos. He points out that the square root of a negative element… “cannot be called either positive or negative”… and makes it his new type of number. Bombelli argues that two terms in Cardano's solution can be represented as a combination... of an ordinary number and this new type of number, which includes the square root -1. He thus discovers that the two cube roots of the Cardano equation are equal to 2... or square root -1 approximately. So when he adds them up in the last step, these square roots cancel out...leaving the correct answer, 4.

This seems like a miracle. Cardano's method works, but you have to ignore the geometric proof... of it. The negative aspects, which don't really make sense... should serve as an intermediate step to reach a solution. Modern mathematics will take shape over the next hundred years. In the 17th century, François Viète introduced modern symbolic notation for algebra. In this way he puts an end to the ancient tradition of mathematical problems in drawings... and descriptions in words. Geometry is no longer the source of truth. René Descartes makes extensive use of the square roots of negatives...and thus he popularizes them.

And although he understands his point, he calls them

imaginary

numbers, a name that sticks. Therefore, Euler later introduced the letter i to represent the square root -1. When combined with regular numbers they form complex numbers. The cubic equation led to the invention of these new numbers and separated algebra from geometry. By leaving aside what seems to be the best description of reality, the geometry that you can see and touch... you get a much more powerful and complete mathematics that can solve real problems. And the cubic equation turns out to be just the beginning. In 1925, Erwin Schrödinger searches for a wave equation that governs the behavior of quantum particles... based on de Broglie's idea that matter is made up of waves.

He comes up with one of the most important and famous physics equations: ...the Schrödinger equation. And it highlights i, the square root -1. While mathematicians have become accustomed to imaginary numbers, physicists have not. They are not comfortable seeing them appear in such a fundamental theory. Schrödinger writes: "What is unpleasant and objectionable is the use of complex numbers. The Psi wave function is, without a doubt, fundamentally a real function." This seems like a valid objection... but why does an imaginary number arise from the solution of cubic equations in fundamental physics? This is due to some unique properties of imaginary numbers.

Imaginary numbers exist in a dimension perpendicular to the real number line. Together they form the complex plane. Look what happens when we multiply by i several times. We start at 1. 1 x i = i i x i = -1, by definition. -1 times i = -i and -i x i = 1 We are back at the beginning and if we continue multiplying by i we will continue turning the point. So if you multiply by i, you're basically rotating 90° in a complex plane. There is a function that multiplies by i several times as you go down. X axis. And that is eix. It creates a spiral by essentially extending these rotations over the entire X axis.

If you look at the actual part of the spiral, it is a cosine wave. If we look at the imaginary part, it is a sine wave. The two characteristic functions that describe waves are contained in eix. When Schrödinger writes a wave equation, he assumes that the solutions will look something like eix... more specifically ei(kx-ωt). You might be wondering why he would use that formulation and not simply a sine wave... but the exponential function has some useful properties. If you take the derivative with respect to position or time, that derivative is proportional to the original function.

And that is not correct if we use the sine function, whose derivative is the cosine. And since the Schrödinger equation is linear, you can add any number of solutions that way... creating any waveform you want, which will also be a solution to the Schrödinger equation. Physicist Freeman Dyson wrote: “Schrödinger added the square root -1 to the equation and suddenly it became clear. It suddenly became a wave equation instead of a heat conduction equation. Schrödinger discovered that the equation has solutions that fit the orbits quantized in Bohr's atomic model. Now it turns out that Schrödinger's equation correctly describes everything we know about the behavior of atoms.

It is the basis of all chemistry and most of physics. And that square root -1 shows that nature works with complex numbers and not real numbers. “This discovery came as a complete surprise to both Schrödinger and everyone else.” So imaginary numbers, which were discovered as a strange intermediate step... in solving the cubic equation, turn out to be fundamental to our description of reality. Only by giving up the connection between mathematics and reality... can we discover the truth about how the universe works. And you learn by doing.

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