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, i.e. MOD 4.
i73 [73 MOD 4 = 1] → i1 = i
i74 [74 MOD 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 over the real numbers.
The graph of 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
The complex plane:
Analogous to the real number line for real numbers.
i is in the equations of quantum mechanics: Dirac's equation and QED which describe the fundamental stuff of the universe: light, electrons