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# Complex Numbers ## What are Complex Numbers? - Combination of a **Real Number** and an **Imaginary Number** ### Real Number - The type of number we use on a daily basis. - Can be positive, negative, big, small, fractions, decimals etc. - Written as a normal number: - 1, 12.38, -0.666, $\sq...

# Complex Numbers ## What are Complex Numbers? - Combination of a **Real Number** and an **Imaginary Number** ### Real Number - The type of number we use on a daily basis. - Can be positive, negative, big, small, fractions, decimals etc. - Written as a normal number: - 1, 12.38, -0.666, $\sqrt{2}$, 99/13 ### Imaginary Number - When squared, gives a negative result - Written with "i" next to it. - i, 4i, -0.2i, $\sqrt{3}$i - Can be added to real numbers: - 1 + i, 12.38 - 0.2i, -0.666 + $\sqrt{3}$i ## The number "i" - Also known as "the unit imaginary number" - Defined as $i = \sqrt{-1}$ ### Squaring i - $i^2 = -1$ - Therefore: $\sqrt{-1}$ * $\sqrt{-1}$ = -1 ### i Raised to Other Powers powers of i cycle through a pattern: - $i^1 = i$ - $i^2 = -1$ - $i^3 = -i$ - $i^4 = 1$ - $i^5 = i$ and so on... ### Example $\sqrt{-9} = \sqrt{9}$ * $\sqrt{-1} = 3i$ ## Complex Number ### General Form z = a + bi Where: - a is the real part - b is the imaginary part ### Example z = 3 + 2i - Real Part = 3 - Imaginary Part = 2 ## Plotting Complex Numbers ### The Complex Plane - Complex numbers are plotted on a complex plane - The complex plane is similar to the cartesian plane, but: - The x-axis represents the real part - The y-axis represents the imaginary part ### Example Plot the complex number z = 3 + 2i - Move 3 units along the real (x) axis - Move 2 units along the imaginary (y) axis ### Argand Diagram - Another name for a complex plane ## Adding Complex Numbers ### Adding - Add the real parts together & add the imaginary parts together: $(a + bi) + (c + di) = (a + c) + (b + d)i$ ### Example $(3 + 2i) + (1 - i) =$ $(3 + 1) + (2 - 1)i =$ $4 + i$ ## Subtracting Complex Numbers ### Subtracting - Subtract the real parts & subtract the imaginary parts: $(a + bi) - (c + di) = (a - c) + (b - d)i$ ### Example $(5 + 3i) - (2 + i) =$ $(5 - 2) + (3 - 1)i =$ $3 + 2i$ ## Multiplying Complex Numbers ### Multiplying - Expand the brackets like in algebra, and simplify $i^2 = -1$ $(a + bi)(c + di) = ac + adi + bci + bdi^2$ $= ac + (ad + bc)i - bd$ $= (ac - bd) + (ad + bc)i$ ### Example $(3 + 2i)(1 - i) =$ $3 -3i + 2i -2i^2 =$ $3 - i + 2 =$ $5 - i$ ## Complex Conjugates ### Complex Conjugate - The complex conjugate of a + bi is a - bi - The sign of the imaginary part is changed - Denoted with a star: $z^*$ is the complex conjugate of z ### Multiplying by Conjugate $z z^* = (a + bi)(a - bi) =$ $a^2 - abi + abi - b^2i^2 =$ $a^2 + b^2$ ### Example z = 3 + 2i $z^* = 3 - 2i$ $zz^* = (3 + 2i)(3 - 2i) =$ $9 + 4 = 13$ ## Dividing Complex Numbers ### Dividing - Multiply both the numerator and denominator by the complex conjugate of the denominator. ### Example $\frac{3 + 2i}{1 - i} =$ $\frac{(3 + 2i)}{(1 - i)} * \frac{(1 + i)}{(1 + i)} =$ $\frac{3 + 3i + 2i + 2i^2}{1 + 1} =$ $\frac{1 + 5i}{2} =$ $\frac{1}{2} + \frac{5}{2}i$ ## Modulus of a Complex Number ### Modulus - Distance from the origin to the complex number in the complex plane. - Found using Pythagoras' Theorem: $|z| = \sqrt{a^2 + b^2}$ ### Example z = 3 + 4i $|z| = \sqrt{3^2 + 4^2} =$ $\sqrt{9 + 16} =$ $\sqrt{25} = 5$ ## Argument of a Complex Number ### Argument - The angle between the positive real axis and the line joining the complex number to the origin in the complex plane. - Measured anticlockwise from the positive real axis - Usually given in radians ### Calculating the Argument - $\tan{\theta} = \frac{b}{a}$ - $\theta = \arctan{\frac{b}{a}}$ ### Example z = 1 + i $\theta = \arctan{\frac{1}{1}} =$ $\frac{\pi}{4}$