Chapter 3 · Complex Numbers and Equations · Concept 1
3-1 Complex Numbers
An ECG trace and an MRI scan are both waves, and tracking a wave means tracking two things at once — how big it is, and when it peaks. One ordinary number cannot hold two facts. So mathematics builds a number that can.
Guiding question
What if a number could carry two pieces of information at once?
Learning outcome. Identify the complex number z = a + bi and its two components, the real part and the imaginary part.
One new number
No ordinary number squares to give a negative. So we name one:
i2 = −1
that single line is the whole definition
A number written a + bi, with a and b both ordinary real numbers, is called a complex number.
a is the Real Part
the ordinary bit
b is the Imaginary Part
the number in front of the i — not including the i itself
If b = 0 the number is just an ordinary real number. If b ≠ 0 it is called an imaginary number. So the reals have not been replaced — they are the complex numbers whose imaginary part happens to be zero.
And when are two of them equal?
a + bi = c + di ⟹ a = c and b = d
a + bi = 0 ⟹ a = 0 and b = 0
One complex equation is two real ones. That is the single most useful fact in this lesson, and it is what every exercise on page 108 is really testing.
A real number can never equal an imaginary one, so the two parts cannot help each other out. Tidy both sides into a + bi, split them, and compare like with like.
Warm Up 2
(2+i)x + (1+i)y = 2 + 3i
−8
write it as −8 + 0i
Real −8, Imaginary 0
A plain number is complex too — its imaginary part is zero.
2i
write it as 0 + 2i
Real 0, Imaginary 2
And a pure imaginary has real part zero.
(4 − √5 i)/3
write it as 4/3 − (√5/3)i
Real 4/3, Imaginary −√5/3
Split the fraction first, then read it off.
The imaginary part does NOT include the i.
For 3 − 2i the imaginary part is −2, not −2i. The book puts this in its own margin note, and it costs marks every year.
2. Find the real numbers x and y with (2 + i)x + (1 + i)y = 2 + 3i.
- 1
Multiply out the left side, treating i as an ordinary letter.
2x + ix + y + iyx and y are REAL, so they behave normally. Only the i is new.
Work with a partner. For question 2, one of you writes the real equation and the other writes the imaginary one. Then solve them together.
1. State the real and imaginary parts.
2. Find the real numbers x and y.
A sound engineer balances two audio paths — a left speaker with volume multiplier x and a right one with multiplier y. To centre the sound the paths must satisfy
(1 + 2i)(x + i) = x(y + i)
Find the exact values of the real numbers x and y.
Why does a + bi = c + di give TWO equations rather than one? Could a real part ever be made equal to an imaginary part?
Ask what it would mean for a real number to equal a number with an i in it.
Putting the i into the imaginary part.
For 3 − 2i the imaginary part is −2, not −2i. The i is a label on the account, not part of the amount.
Comparing before tidying into a + bi.
You cannot split (2+i)x + (1+i)y until it has been multiplied out and gathered. Do that first, every time.
Forgetting that 0 means 0 + 0i.
An equation ending "= 0" gives TWO equations, not one. Both parts have to vanish.
Losing a sign on i × i.
In (1 − i)(x + yi) the last term is −i × yi = −yi² = +y. Two minuses and an i² — that is where the sign goes.
Checking only one of the two equations.
A pair of values can satisfy the real part and fail the imaginary one. Test both.
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