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1 3-4i 3-4i
3+4i * 3-4i = 9+16
3 4
25 - 25
1+4i 5+12i 5+12i +20i + 48i²
5-12i *5+12i = 25 +144
-43 +32i
169
-43 +32
169 169
i1 = i i2 = –1
i3 = –i i4 = 1,
i5 = i i6 = –1
i7 = –i i8 = 1
i9 = i i10 = –1,
i11 = –i i12 = 1
i27 = i24*i3 = (i4)^6 *i3=1^6 *i3=-i
i101 = i100 *i1= (i4)^25*i = 1^25 *i=i
Product of Complex Numbers
(a+bi)*(c+di) = (ac-bd) + (ad+bc)i
Example: (2i)(2i) = 4i² = -4
(2+i)(1-i) = 2-2i +i-i²
3-i
Conjugates
z = a+bi= a-bi
z= 3+4i
z=3-4i
zz=(3+4i)(3-4i)
9-12i+12i-16i²
9+16
Example: 2+3i= 2-3i
-6-2i=-6+2i
25
When a complex number is written in the form a+bi, where a and b are real numbers, we say it is in standard form.
A pure imaginary number is an imaginary number of the form a+bi where A is 0.
Example
1
3+4i
Equality of Complex Numbers
a+bi=c+di if and only if a=c and b=d
i
Sum of Complex Numbers
(a+bi) + (c+di) = (a+c) + (b+d)i
Difference of Complex Numbers
(a+bi) - (c+di) = (a-c) + (b-d)i
Product of Complex Numbers
Example
1+4i
5-12i
-N
Example: (3+5i) + (-2 +3i)
[3 +(-2)] + (5+3)i
=
Ni
1+8i
Example: (5+3i)*(2+7i)
5*(2+7i) +3i(2+7i)
10+35i+6i+21i^2
10+41i+21(-1)
i
-11+41i
Example
Write (2+i)^3 in standard form.
(2+i)^3= 2^3 +3*i*2^2 +i^3
=8+12i+6(-1) + (-i)
=2+11i