Complex number

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A complex number is a number, but is different from normal numbers in many ways. A complex number is made up using two numbers combined together. The first part is a real number. The second part of a complex number is an imaginary number. The most important imaginary number is called Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle i} , defined as a number that will be -1 when squared ("squared" means "multiplied by itself"): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle i^2 = i \times i = -1\ } . All the other imaginary numbers are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle i} multiplied by a real number, in the same way that all real numbers can be thought of as 1 multiplied by another number. Arithmetic functions such as, addition, subtraction, multiplication, and division can be used with complex numbers. They also follow commutative, associative and distributive properties, just like real numbers.

Complex numbers were invented to answer special equations that have exponents in them. These began to pose real problems for mathematicians. As a comparison, using negative numbers, it is possible to find the x in the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a + x = b} for all real values of a and b.

With exponentiation, there is a problem.<ref name="wa">Template:Cite web</ref> There is no real number that gives −1 when it is squared. In other words, −1 (or any other negative number) has no real square root. To solve this problem, mathematicians introduced an imaginary number called i. That imaginary number will give −1 when it is squared.

The first mathematicians to have thought of this were probably Gerolamo Cardano and Raffaele Bombelli. They lived in the 16th century. It was probably Leonhard Euler who introduced writing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathrm i} for that number.

All complex numbers can be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a + bi} <ref name="wa" /> (or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a + b \cdot i} ), where a is called the real part of the number, and b is called the imaginary part. We write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \Re (z)} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \operatorname{Re}(z)} for the real part of a complex number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z = a + bi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a = \Re (z) = \operatorname{Re} (z)} . Similarly, we write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \Im (z)} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \operatorname{Im} (z)} for the imaginary part of a complex number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z} ; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle b = \Im (z) = \operatorname{Im} (z)} .

The complex number can also be written as a set Template:Math. Both a and b are real numbers. Any real number can simply be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a + 0 \cdot i} or as the set Template:Math.<ref name="wa" />

The set of all complex numbers is usually written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbb{C}} .

Operations over complex numbers

Addition, subtraction, multiplication, division as long as the divisor is not zero, and exponentiation (raising numbers to exponents) are all possible with complex numbers. Some other calculations are also possible with complex numbers.

The rule for addition of complex numbers is pretty simple: let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z = (a + bi), w = (c + di)} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z + w = (a + bi) + (c + di) = (a + c) + (b + d)i} . Multiplication is a bit different: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z \cdot w = (a + bi)(c + di) = ac + bci + adi + bdi^2 = (ac - bd) + (bc + ad)i.}

Another notable operation for complex numbers is conjugation. A complex conjugate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \overline{z}} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z = a + bi} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a - bi} . It is pretty simple, but is important for calculations, because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z \times \overline{z}} belongs to the real numbers for all complex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z} .

Other forms of describing complex numbers

Complex numbers can be shown on a so-called complex plane. If you have a number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z = a + bi} , you can go to a point on the real axis and to b on the imaginary and draw a vector from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle (0, 0)} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle (a, b)} . You can calculate the length of this vector using the Pythagorean theorem and angle between the real axis and this vector, counting counterclockwise. The length of a vector for a number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z} is called modulus (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle |z|} ), and the angle is argument (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \arg z} ).

File:Complex number.svg
A complex number can be visually shown as two numbers which form a vector on an Argand diagram, representing the complex plane.

This leads to the trigonometrical form of describing complex numbers: by the definitions of sine and cosine, for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z} stands that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle z = |z|(\cos \arg z + i \sin \arg z).}

This is closely connected to De Moivre's formula.

There exists even another form, called exponential.

Conclusion

With the addition of complex numbers to math, every root of a polynomial with complex coefficients is a complex number. The successful addition of the complex numbers to math also help to open a path to the creation of another sorts of numbers that could resolve and help to explain many different problems, for example the: hypercomplex numbers, sedenion, hyperreal numbers, surreal numbers and many others. See types of numbers.

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References

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