basic_tools:imaginary_numbers
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basic_tools:imaginary_numbers [2017/12/16 12:43] jakobadmin [Why is it interesting?] |
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| ====== Imaginary Numbers ====== | ====== Imaginary Numbers ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | Imaginary numbers are essential in modern theories of physics like [[theories:quantum_theory:quantum_mechanics|quantum mechanics]] and [[theories:quantum_theory:quantum_field_theory|quantum field theory]]. In these theories, we describe a physical system using complex functions, which means functions that contain combinations of imaginary numbers as arguments. | ||
| - | Moreover, imaginary numbers can often be used to make calculations simpler. | + | <tabbox Intuitive> |
| + | In the beginning mathematicians only used natural numbers: $1,2,3, \ldots$. Then, negative numbers were invented to represent things like debt. For example, $+5$ means a profit of $5$ units, while $-5$ means $5$ units of debt. | ||
| + | The invention of negative numbers then lead to a surprising observation: | ||
| - | <blockquote>The shortest path between two truths in the real domain passes through the complex domain. <cite>J. Hadamard</cite></blockquote> | + | * Multiplying a __positive__ number with a negative number yields another __negative__ number: $ -3 \times 5 = -15$. |
| + | * Multiplying a __negative__ number with another negative number yields a __positive__ number: $ -3 \times -5 = 15$. | ||
| + | * However, there is no number that yields a negative number when multiplied by itself. Therefore, with positive and negative numbers alone, simple equations like $x^2 = -1$ can't be solved. | ||
| - | * See also https://math.stackexchange.com/a/168/120960 | + | For this reason, the __imaginary number__ $i$ was invented. This new number is defined by the property: $i \times i=-1$. In this sense, $i$ was introduced to fill the gap that we discovered above since now we have a number that yields something negative when multiplied by itself. |
| - | ---- | + | In the beginning, $i$ was merely a convenient tool to help simplify calculations and its introduction was criticised by many mathematicians. This early criticism is also where the name "imaginary numbers" comes from, which was meant as a derogatory term. |
| - | **Important Related Concepts:** | + | Nowadays, imaginary numbers are an essential tool. Combinations of real and imaginary numbers like, for example, $ 4+3i $, are known as [[basic_tools:complex_analysis|complex numbers]]. Complex numbers are the standard number system that physicists use. |
| - | * [[basic_tools:complex_analysis]] | + | --- |
| - | <tabbox Layman> | + | |
| - | <note tip> | + | [{{ :basic_tools:imaginarynumbers.jpg?800 |https://mobile.twitter.com/elzr/status/1254478112223637507/photo/1}}] |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
| - | </note> | + | |
| - | + | <tabbox Concrete> | |
| - | <tabbox Student> | + | |
| * The best introduction is [[https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/|A Visual, Intuitive Guide to Imaginary Numbers]] by Kalid Azad | * The best introduction is [[https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/|A Visual, Intuitive Guide to Imaginary Numbers]] by Kalid Azad | ||
| * See also https://math.stackexchange.com/questions/199676/what-are-imaginary-numbers | * See also https://math.stackexchange.com/questions/199676/what-are-imaginary-numbers | ||
| + | |||
| + | ---- | ||
| + | |||
| + | <note tip>There are further expanded number systems like, for example, quaternions and octonions where multiple "complex units" are introduced. | ||
| + | |||
| + | There is a [[https://eudml.org/doc/58420|theorem due to Hurwitz]] that the only "normed division algebras" are the real numbers, the complex numbers, the quaternions, and the octonions. | ||
| + | </note> | ||
| - | <tabbox Researcher> | + | <tabbox Abstract> |
| <note tip> | <note tip> | ||
| Line 34: | Line 43: | ||
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| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| + | Imaginary numbers are essential in modern theories of physics like [[theories:quantum_mechanics:canonical|quantum mechanics]] and [[theories:quantum_field_theory:canonical|quantum field theory]]. In these theories, we describe a physical system using complex functions, which means functions that contain combinations of imaginary numbers as arguments. | ||
| - | --> Example1# | + | Moreover, imaginary numbers can often be used to make calculations simpler. |
| - | + | For some further motivation, see the nice list [[https://math.stackexchange.com/a/168/120960|here]]. | |
| - | <-- | + | |
| - | --> Example2:# | + | ---- |
| - | |||
| - | <-- | ||
| - | <tabbox FAQ> | + | <blockquote>The shortest path between two truths in the real domain passes through the complex domain. <cite>J. Hadamard</cite></blockquote> |
| + | |||
| + | ---- | ||
| + | |||
| + | **Important Related Concepts:** | ||
| + | |||
| + | * [[basic_tools:complex_analysis]] | ||
| | | ||
| <tabbox History> | <tabbox History> | ||
| + | The name "imaginary numbers" was introduced by Descartes as a derogatory term. | ||
| </tabbox> | </tabbox> | ||
| + | {{tag>theories:quantum_theory:quantum_mechanics theories:quantum_theory:quantum_field_theory theories:classical_theories:electrodynamics}} | ||
basic_tools/imaginary_numbers.1513424585.txt.gz · Last modified: 2017/12/16 11:43 (external edit)
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