5+2i 4-i -5+2i -3-3√2i
Every real number is a complex number with b=0.
Pure imaginary number: a=0. 2i -3i 3√2i
A superset of the Real numbers.
To accommodate square root of negative number.
So to fully solve and factor polynomial (incl. quadratic) equations.
Imaginary unit i = √-1
Just like with negative numbers and irrational numbers, you get used to them.
i2 = -1
Simplify: √-4 = √4√-1 = 2i
√-18 = √9√-2 = 3√2√-1 = 3√2i
Complex number: a + bi, a and b real numbers.
a is the real part, bi the imaginary part.
5+2i 4-i -5+2i -3-3√2i
Every real number is a complex number with b=0.
Pure imaginary number: a=0. 2i -3i 3√2i
Complex addition/subtraction: add/subtract the real part, add/subtract the imaginary part
(5+2i) + (3+4i) = 8+6i
Complex multiplication: FOIL the two complex numbers as if they were binomials. NB i2=-1
(5+2i) * (3+4i) = 15+20i+6i+8i2 = 15+26i-8 = 7+26i
Complex conjugate: of a+bi is a-bi
Each is each other's conjugate.
Multiplying a pair of complex conjugates yields a real number:
(a+bi) * (a-bi) = a2-abi+abi-b2i2
= a2+b2
(5+2i) * (5-2i) = 25-10i+10i-2i2 = 25+4 = 29
Complex division: first multiply numerator and denominator by denominator's conjugate
to clear the denominator of i's then simplify.
(5+2i)/(3+4i) = (5+2i)(3-4i) / (3+4i)(3-4i) = (15-14i-8i2) / 9+16 = (23-14i)/25 = 23/25-14/25i
Multiplying or dividing two pure imaginary numbers yields a real number.
bi*ci = bci2 = -bc
bi/ci = b/c
MP understands i and can do complex arithmetic. but cannot solve equations having complex solutions.
Powers of i:
i0 = 1
i1 = i
i2 = -1
i3 = -i
All higher powers are one of the four above, cycling through them. Remainder after integer dividing by 4.
i73 [73 DIV 4 = 1] → i1 = i
i74 [73 DIV 4 = 2] → i2 = -1 [MP can do these]
(MP: factors only over integers, e.g. x2-9 → (x+3)(x-3) and Solve =0.
not over rationals, e.g. x2-9/4 → "" but does Solve =0.
nor over irrationals, e.g. x2-3 → "Doesn't factor" but does Solve =0.
Does not Solve over complexs, e.g. x2+1=0 → "No real solutions")
MP can partially calculate with x and i: (x+1)(x-i)(x+i) but Wolfram does.
A quadratic equation that has no real number solutions is called irreducible.
The discriminant b2-4ac (the part under the radical in the quadratic formula) is negative.
The quadratic expression can not be factored.
The corresponding quadratic function has no x-intercepts.
BUT now with complex numbers we can solve ALL quadratic equations and factor them.
Ex. x2+1=0 has no real solutions: x2=-1. But go complex: x=±√-1=±i
x2+1 = (x-i)(x--i)
fully factored. 2 solutions: a pair of complex conjugates: i and -i
Ex. x2-4x+8=0 has no real solutions: QF: x=(4±√-16)/2.
But go complex: x= (4±4i)/2 = 2±2i
x2-4x+8 = (x-(2+2i))(x-(2-2i))
fully factored. 2 solutions: a pair of complex conjugates: 2+2i and 2-2i
Ex. x2+x+1=0 has no real solutions but does have a pair of complex conjugates solutions:
Using quadratic formula, instead of stopping at negative discriminant x=(-1±√-3)/2
go complex:
x = (-1±√3i)/2 = -1/2 + √3i/2, -1/2 - √3i/2
x2+x+1 = (x-(-1/2 + √3i/2))(x-(-1/2 - √3i/2))
fully factored. 2 solutions, a pair of complex conjugates..
Higher-degree polynomial functions/equations factor into linear factors (x-c)
and irreducible quadratic factors each of which can be factored into a pair of complex conjugates.
For a total of n factors and n solutions for an n-degree polynomial.
Ex. x3+x2+x+1 has one x-intercept. MP factors it: (x+1)(x2+1).
We know from above about complex factors of x2+1, so: (x+1)(x+i)(x-i)
fully factored. 3 solutions: one a real number (-1) and a pair of complex conjugates.
Wolfram: does all
i is in the equations of quantum mechanics: Dirac's equation and QED which describe the fundamental stuff of the universe: light, electrons