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Advanced Math/Scientific Notation for Complex Numbers

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QUESTION: Dear Prof Randy

http://en.wikipedia.org/wiki/Scientific_notation
http://en.wikipedia.org/wiki/Complex_number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, where i2 = −1.

Is it possible to also have a scientific notation for complex numbers ?.

Examples :


6720000000 + 4i
1991230000000 + 6i
18.7 + 2i


Awaiting your reply,

Thanks & Regards,
Prashant S Akerkar

ANSWER: First note that a complex number, z = a + ib, can be ( and often is) written

z = |z|exp(iθ)

where |z| = sqrt(a^2+b^2) = modulus of z, and θ = tan^-1(b/a) = phase ange of z. The modulus |z| is a real number and could be written using scientific notation, although I've never seen it done so. The angle θ can be arbitrarily large, but the function exp(iθ) is periodic with period 2π, and so just keeps wrapping around the unit circle. This means that there is no real reason to write θ in scientific notation since any large angle is equivalent to an angle between 0 and 2π. Again, I've never seen it represented in scientific notation.

---------- FOLLOW-UP ----------

QUESTION: Dear Prof Randy

Thank you.

1. Can this also be taken as a valid example and written
with scientific notations ?.

Here a & b are large numbers

6720000000 + 1991230000000i

Similarly example of small numbers

0.0000000000345 + 0.34562222222222i

i.e. a+bi complex form where a,b are real numbers.

2. The mathematical computations viz Multiplication, Division of two
complex numbers where a & b for very large or very small numbers
with use of scientific notation (10^9 OR E9) can become complex ?.

Awaiting your reply,

Thanks & Regards,
Prashant S Akerkar

Answer
There is no technical reason why the real numbers a and b can't be written in scientific notation. However, the notation could be a little awkward. If both a and b are both really big or really small, and thus candidates for scientific notation, it would be very useful to write the complex number as given in the previous answer since then you would just need to represent the modulus in scientific notation, with the phase angle beng dependent on their ratio, which would be a more manageable number.

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randy patton

Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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Also an Expert in Oceanography

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