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The imaginary unit is defined by the property that its square equals negative one. This definition is the foundation for all arithmetic with complex numbers, since it lets us replace i squared with a real number whenever it appears.
Addition of complex numbers is performed componentwise: add the real parts and add the imaginary parts separately. Here 3 + 4 = 7 for the real part and 2 + (-5) = -3 for the imaginary part, giving 7 - 3i.
Distribute each term of the first factor over the second: 2(1) + 2(-i) + 3i(1) + 3i(-i). This yields 2 - 2i + 3i - 3i squared, and since i squared equals -1, the last term becomes +3, giving 5 + i.
The complex conjugate keeps the real part unchanged and reverses the sign of the imaginary part. Applying that rule to -4 + 7i leaves -4 fixed and changes +7i to -7i.
The modulus of a + bi is the square root of a squared plus b squared, which is the distance from the origin in the complex plane. Here 36 + 64 = 100, and the square root of 100 is 10.
Since i squared equals -1, multiplying by i squared again gives (-1)(-1) = 1. The powers of i therefore cycle through i, -1, -i, and 1 with period four.
Multiply numerator and denominator by the conjugate 2 + i of the denominator. The denominator becomes 4 + 1 = 5, and the numerator becomes 10 + 5i + 2i + i squared, which simplifies to 9 + 7i.
Rearranging gives x squared = -9, so x is a square root of negative nine. Since the square root of 9 is 3 and the square root of -1 is i, the solutions are 3i and -3i.
Expanding gives 1 + 2i + i squared, and since i squared equals -1, the real terms 1 and -1 cancel. The result is purely imaginary, equal to 2i.
Multiplying a complex number by its conjugate always gives the square of its modulus, a nonnegative real number. Here 2 squared plus 3 squared equals 13, and no imaginary part remains.
Every real number is a complex number of the form a + bi with b equal to zero. Thus 7 equals 7 + 0i, so its real part is 7 and its imaginary part is 0.
The powers of i repeat with period four, so reduce the exponent modulo 4. Since 23 divided by 4 leaves remainder 3, i to the 23rd equals i cubed, which is -i.
The additive inverse of a complex number changes the sign of both the real and imaginary parts so that the sum is zero. Negating -5 + 2i gives 5 - 2i, and adding the two yields 0.
The quadratic formula gives x = (4 plus or minus the square root of 16 - 52)/2. The discriminant is -36, whose square root is 6i, so the solutions are 2 plus or minus 3i.
The modulus is the square root of a squared plus b squared. Here a squared is 1 and b squared is 3, so the sum is 4 and the modulus is 2.
The conjugate of i is -i, and their product is -i squared. Because i squared equals -1, the product equals 1, which agrees with the rule that a number times its conjugate equals the square of the modulus.
Cube the coefficient and cube the imaginary unit separately: 2 cubed is 8 and i cubed is -i. Multiplying gives -8i, a purely imaginary result.
Multiply numerator and denominator by the conjugate -i of i, whose product with i is 1. The numerator becomes -3i - 4i squared, which equals 4 - 3i after using i squared equals -1.
A double real root occurs when the discriminant b squared minus 4ac equals zero. Here k squared minus 100 must be zero, so k squared equals 100 and k equals 10 in magnitude.
Conjugate pairs a plus or minus bi have sum 2a and product a squared plus b squared. Setting 2a equal to 6 gives a = 3, and then a squared plus b squared equals 25 forces b squared to be 16, so b = 4.
Multiply numerator and denominator by the conjugate 3 + 4i. The denominator becomes 9 + 16 = 25, and the numerator expands to 6 + 8i + 3i + 4i squared, which simplifies to 2 + 11i.
De Moivre's theorem states that raising a complex number in trigonometric form to a power n multiplies the angle by n and leaves the modulus unchanged. With modulus 1, the cube is cos(3theta) + i sin(3theta).
The equation asks for the fourth roots of unity, and a nonzero complex number has exactly n distinct nth roots. They are equally spaced around the unit circle, here at angles 0, 90, 180, and 270 degrees, namely 1, i, -1, and -i.
The modulus is the square root of 1 squared plus 1 squared, which is the square root of 2. The argument is the angle whose tangent is 1/1, and since both components are positive the angle lies in the first quadrant at 45 degrees.
The product of a complex number with its conjugate equals the square of its modulus. A sum of two squares of real numbers is zero only when both components are zero, so z must be the number 0.
The imaginary unit is defined by the relation i squared equals -1, so any power of i reduces by replacing pairs of factors of i with -1.
The conjugate of a + bi is a - bi, so only the sign of the imaginary part changes while the real part stays -4.
Addition of complex numbers is done componentwise: add the real parts 3 + 2 = 5 and the imaginary parts 5 + (-8) = -3.
Subtracting means subtracting real parts and imaginary parts separately: 4 - 1 = 3 and -2 - (-6) = 4, giving 3 + 4i.
A pure imaginary number has the form 0 + bi, so its real part is 0 even though the imaginary part is -6.
The third quadrant contains points whose real part is negative and whose imaginary part is negative, which matches -3 - 4i.
Multiplying conjugates gives a squared plus b squared, here 4 + 9 = 13, and the imaginary terms cancel.
Powers of i repeat with period 4, and 30 divided by 4 leaves remainder 2, so i to the 30th equals i squared, which is -1.
The reciprocal is 1/(2 + i); multiplying numerator and denominator by the conjugate 2 - i gives (2 - i)/(4 + 1) = (2 - i)/5.
Expanding gives 3 - i + 6i - 2i squared, and since i squared equals -1 this becomes 3 + 5i + 2 = 5 + 5i.
The modulus is the square root of a squared plus b squared, so the square root of 25 + 144 equals the square root of 169, which is 13.
Multiplying by the conjugate gives a squared plus b squared, so 9 + 16 = 25, which is the squared modulus of z.
The additive inverse negates both components so that the sum is zero, giving -4 + 9i.
Since -16 = 16 times i squared, its square roots are plus or minus 4i, because (4i) squared equals -16.
The sum of the roots is 4 and the product is 5, so the equation is x squared minus 4x plus 5 equals zero.
Here z squared equals 2i and 2z equals 2 + 2i, so subtracting gives 2i - 2 - 2i = -2.
The distance is the modulus of the difference (4 - 1) + (6 - 2)i = 3 + 4i, whose modulus is 5.
A double root requires the discriminant k squared minus 64 to equal zero, so k squared equals 64 and k is 8 or -8.
The conjugate pair gives 1 + 1 = 2, and multiplying 2 by 2 + 3i gives 4 + 6i.
Squaring gives 4 - 4i + i squared, and since i squared equals -1 the result is 3 - 4i.
The modulus is the square root of 4 + 4 = 2 sqrt 2, and the point lies in the second quadrant at angle 135 degrees.
Since i to the fourth is 1, i to the -3 equals i to the first, because -3 is congruent to 1 modulo 4.
The expression |z - z0| measures distance from z to z0, so setting it equal to 2 gives all points at distance 2 from the point 3.
The modulus of 1 + i is sqrt 2 and its angle is 45 degrees, so the eighth power has modulus (sqrt 2) to the 8th = 16 and angle 360 degrees, giving 16.
Factoring gives (z + 2)(z squared - 2z + 4) = 0; the quadratic factor has discriminant -12, so its roots form a nonreal conjugate pair.