What is the condition on ##a## and ##b## for ##Z## to be real?alijan kk said:does it tell from this expresssion that the complex number is a pure real ?
b should be zero if the complex number is pure ruleDrClaude said:What is the condition on ##a## and ##b## for ##Z## to be real?
We can set b = 0, but we don't have to, so this is not the answer to the problem.alijan kk said:b should be zero if the complex number is pure rule
can we put b= 0
so this a wrong question in my book ?DrClaude said:We can set b = 0, but we don't have to, so this is not the answer to the problem.
a unit circle ,mjc123 said:No. What does a2 + b2 = 1 mean geometrically? Think of a graph in the complex plane.
DrClaude said:##a^2+b^2## will always be real. That doesn't tell you anything about ##z##.
alijan kk said:options are:
z is purely imaginary
z is any complex number
z is real
none of these
Complex numbers are numbers that have a real and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit equal to the square root of -1.
To add or subtract complex numbers, you simply combine the real parts and the imaginary parts separately. For example, (4 + 2i) + (3 + 5i) = (4+3) + (2i+5i) = 7 + 7i.
To multiply complex numbers, you use the FOIL method, which stands for First, Outer, Inner, and Last. You multiply the first terms, then the outer terms, inner terms, and last terms, and finally combine like terms. For example, (4 + 2i)(3 + 5i) = 12 + 20i + 6i + 10i^2 = (12 - 10) + (20 + 6)i = 2 + 26i.
To divide complex numbers, you use the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. You multiply the numerator and denominator by the conjugate of the denominator, and then simplify. For example, (4 + 2i) / (3 + 5i) = (4 + 2i)(3 - 5i) / (3 + 5i)(3 - 5i) = (12 + 8i - 20i - 10i^2) / (9 + 25) = (2 - 12i) / 34.
Complex numbers are used in many fields, including engineering, physics, and economics. They are particularly useful in solving problems involving alternating currents, such as in electric circuits. They are also used in signal processing, control systems, and many other applications in science and technology.