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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

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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Comment | Dear Prof Randy Thank you. Thanks & Regards, Prashant S Akerkar |

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