Lesson 2 of 12 · 7 min
The imaginary unit i and complex numbers
NCERT §4.2
The rover's control screen shows its position as a single number rather than two. Standing 3 m east and 4 m north of the base, it reads 3 + 4i. What does that i mean, and how are two such numbers compared?
The lesson in notes
In short
The symbol i stands for √−1, so i² = −1 and i is a root of x² + 1 = 0.
A complex number is any number a + ib in which a and b are both real, for example 3 + 4i, −2 + √5 i, 7i and 6. The letter z is used for a general complex number.
For z = a + ib, the real part is Re z = a and the imaginary part is Im z = b. For z = −3 + 7i, Re z = −3 and Im z = 7: the imaginary part is the real number 7, not 7i.
Every real number a is the complex number a + i0, so the real numbers sit inside the complex numbers. A number 0 + ib with b ≠ 0, such as 5i, is called purely imaginary.
Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal: a + ib = c + id means a = c and b = d.
So one equation between complex numbers gives two equations between real numbers. If x and y are real and 3x + i(2x − y) = 6 − i, then 3x = 6 and 2x − y = −1, so x = 2 and y = 5.