Imaginary numbers
- has no real answer, so we define a brand-new number by .
- That one rule does everything. Treat like any letter in algebra. Every time you see an , swap it for .
is the only new thing in this chapter. Everything else is the normal algebra and geometry you already do. You just change to whenever it turns up.
i² = −1, and √(−b) = i√b
- To square-root a negative number, pull out an : (for ). Now the part left under the root sign is positive.
- Powers of repeat every four:
- To find any power of , take out the fours: .
- A complex number is a real part plus an imaginary part, written . It is just two real numbers joined by .
Why we needed a whole new number, not just a symbol
No real number, squared, gives a negative. So instead of saying “no solution” forever, we make up with . Every algebra rule you already use still works. The payoff: now every polynomial has all of its roots (the Fundamental Theorem of Algebra), and these “imaginary” numbers even describe real waves and rotations.
Worked examples
Worked example
Simplify and
- Split off the negative as , then simplify the surd: .
- Square term by term, then use : — a real number.
Answer
Worked example
Evaluate and
- Powers of cycle in fours, so .
- , and .
- Add the real and imaginary parts: .
Answer
Where the marks go
Common mistake
Don't forget : , not . And simplify the surd all the way: , not .
Common mistake
The rule breaks when both are negative: , not . Always turn each into first, then multiply.
Now you try
Simplify . (9709-style)
Your turn— tap to reveal the worked answer (9709-style)
Pull out the , then use : , and :
Answer