Number Systems
Understanding the Imaginary Unit i
The imaginary unit i is defined by i^2=-1. It extends real arithmetic so equations such as x^2+1=0 have solutions.
Reviewed
Worked examples
Simplify i^37
Divide the exponent by 4 and use the remainder.
37=4 x 9 + 1i^37=(i^4)^9 ii^4=1
Result: i^37=i
Solve x^2+16=0
Isolate the square.
x^2=-16x=+/-sqrt(-16)sqrt(-16)=4i
Result: x=+/-4i
Why the real numbers are not enough
Every real square is nonnegative, so no real number solves x^2=-1. Introducing i provides a consistent solution and creates the complex number system a+bi.
This extension preserves ordinary addition, multiplication, and distributive laws.
i^2=-1; sqrt(-a)=i sqrt(a) for a>0 under the principal-root conventionThe four-step power cycle
Successive powers repeat every four exponents: i, -1, -i, 1. Reduce a large positive integer exponent modulo 4 to find its value quickly.
An exponent divisible by 4 gives 1. Remainders 1, 2, and 3 give i, -1, and -i respectively.
i^(n+4)=i^nSquare roots and branch choice
The equation z^2=-9 has two solutions, 3i and -3i. The principal square-root symbol sqrt(-9) conventionally returns 3i.
As with positive square roots, distinguish a principal function value from all solutions of an equation.
Common mistakes
- Replacing i^2 with i or with 1.
- Assuming sqrt(-a)sqrt(-b)=sqrt(ab) under unrestricted real radical rules.
- Reporting only one solution to x^2=-a.
Where this idea is used
- Expressing nonreal roots of polynomial equations.
- Building rectangular complex-number notation.
- Representing quarter-turn rotations through multiplication by i.
Questions about the imaginary unit i
Is i the same as sqrt(-1)?
It is the designated principal square root of -1, defined so that i^2=-1.
Why do powers of i repeat?
Multiplying by i four times gives i^4=(i^2)^2=(-1)^2=1.
Can i appear in a final engineering answer?
Yes. Engineering often uses j instead of i when i already denotes electric current.