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Complex Numbers

Complex Numbers. C.A-1.5. Imaginary numbers. i represents the square root of – 1. Complex Numbers. A Complex Number is a number that can be written in the form:. Where a and b are real numbers and. The a is the Real part , and b is called the imaginary part.

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Complex Numbers

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  1. Complex Numbers C.A-1.5

  2. Imaginary numbers i represents the square root of – 1.

  3. Complex Numbers A Complex Number is a number that can be written in the form: Where a and b are real numbers and

  4. The a is the Real part, and b is called the imaginary part.

  5. Equality of Complex Numbers • Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. • If a + bi and c + di are two complex numbers, then a + bi = c + diif and only if a = c and b = d

  6. Addition and Subtraction of Complex Numbers • Two complex numbers such as a + bi and c + di are added and subtracted as if they were binomials: • (a + bi) + (c + di) = (a + c) + (b + d )i • (a + bi) - (c + di) = (a - c) + (b - d )i

  7. Multiplication of Complex Numbers • The numbers a + bi and c + di are multiplied as if they were binomials, with i2 = -1: • (a + bi)(c + di) = (ac – bd) + (ad + bc)i

  8. Errors involving signs • To avoid errors in determining the sign of the result, always express numbers in a + bi form before attempting any algebraic manipulations.

  9. Complex Conjugates The complex numbers and Are complex conjugates of each other.

  10. Powers of i

  11. Powers of i If n is a natural number that has a remainder of r when divided by 4, then When n is divisible by 4, the remainder r is 0 and i0= 1.

  12. Absolute Value of a Complex Number If a + bi is a complex number, then

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