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Among other things, complex numbers play an important role:* In electrical circuits - quantities in AC circuits are described by complex numbers.

* In quantum mechanics - the "probability amplitude" is an important concept in quantum mechanics, and it is described by a complex number.

* In art - for example, the Mandelbrot set is based on calculations with complex numbers.

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Imaginary and complex numbers are simply extensions of the real number system in much the same way as negative integers were an extension to the set of natural numbers, or rationals were an extension to integers. Negatives provided closure for subtraction, while rationals gave closure for division of integers by (non-zero) integers. Imaginary and complex numbers give closure to taking roots of any real number.

Q: How do we use complex and imaginary numbers?

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No difference. The set of complex numbers includes the set of imaginary numbers.

Yes, imaginary numbers are a subset of complex numbers.

A complex number has a real part and a (purely) imaginary part, So imaginary numbers are a subset of complex numbers. But the converse is not true. A real number is also a member of the complex domain but it is not an imaginary number.

An imaginary number has the form b*i, where b is any real number, and i is the imaginary unit number sqrt(-1). A complex number is in the form a + b*i, where a and b are any real number. Complex numbers are called complex, because they combine numbers from the real domain and the imaginary domain.I thought the Wikipedia article was very good. Also, Answers.com has a video about complex and imaginary numbers.

Imaginary number is a number that consist of only Imaginary part. Such as i, 40i, 1/2i, etc. While the difference between the imaginary numbers and the complex numbers are that complex number also contains Real numbers, and can be written as a + bi. For example, 30+i, 1/2+1/2i, etc.

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No difference. The set of complex numbers includes the set of imaginary numbers.

Complex math covers how to do operations on complex numbers. Complex numbers include real numbers, imaginary numbers, and the combination of real+imaginary numbers.

Yes, imaginary numbers are a subset of complex numbers.

A complex number has a real part and a (purely) imaginary part, So imaginary numbers are a subset of complex numbers. But the converse is not true. A real number is also a member of the complex domain but it is not an imaginary number.

No.

The imaginary axis is used in the definition of the complex numbers. Complex numbers are used in many fields in engineering, in particular - electric engineering, aerodynamics, acoustics etc.

No. All Complex Numbers are of the form a + bi where a and b are Real Numbers and i is the square root of -1. So only ones where a = 0 are pure Imaginary Numbers.

An imaginary number has the form b*i, where b is any real number, and i is the imaginary unit number sqrt(-1). A complex number is in the form a + b*i, where a and b are any real number. Complex numbers are called complex, because they combine numbers from the real domain and the imaginary domain.I thought the Wikipedia article was very good. Also, Answers.com has a video about complex and imaginary numbers.

Always. The set of imaginary numbers is a subset of complex numbers. Think of complex numbers as a plane (2 dimensional). The real numbers exist on the horizontal axis. The pure imaginary are the vertical axis. All other points on the plane are combinations of real and imaginary. All points on the plane (including imaginary axis and real axis) are complex numbers.

Physics (e.g., quantum mechanics, relativity, other subfields) makes use of imaginary numbers. "Complex analysis" (i.e., calculus that includes imaginary numbers) can also be used to evaluate difficult integrals and to perform other mathematical tricks. Engineering, especially Electrical Engineering makes use of complex and imaginary numbers to simplify analysis of some circuits and waveforms.

The real numbers together with the imaginary numbers form a sort of vector. What these complex numbers (complex means the combination of real and imaginary numbers) represent depends on the specific situation. Just as there are situations where it doesn't make sense to use negative numbers, or fractional numbers, in many common situations it doesn't make sense to use complex numbers. In an electrical circuit (specifically, AC), the real numbers might represent resistance, while the imaginary number represent reactance - and voltages and currents are also represented by complex numbers, with the angle of the complex number representing how much one quantity is ahead or behind another quantity (the "phase angle"). In quantum mechanics, the complex numbers might not represent anything (perhaps they don't, I am not sure...), but they are useful for calculations.

Imaginary number is a number that consist of only Imaginary part. Such as i, 40i, 1/2i, etc. While the difference between the imaginary numbers and the complex numbers are that complex number also contains Real numbers, and can be written as a + bi. For example, 30+i, 1/2+1/2i, etc.

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