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The Imaginary Number i

i

The symbol used to represent the imaginary number Square root of negative 1.

 

 

See also

Complex numbers

Key Formula

i=−1equivalently,i2=−1i = \sqrt{-1} \quad \text{equivalently,} \quad i^2 = -1
Where:
  • ii = The imaginary unit, defined so that its square equals −1

Worked Example

Problem: Simplify √(−16) using the imaginary number i.
Step 1: Separate the negative sign from the radicand using the definition of i.
−16=−1⋅16\sqrt{-16} = \sqrt{-1 \cdot 16}
Step 2: Rewrite √(−1) as i and simplify √16.
−1⋅16=i⋅4\sqrt{-1} \cdot \sqrt{16} = i \cdot 4
Step 3: Write the result in standard form.
=4i= 4i
Answer: √(−16) = 4i

Why It Matters

Without ii, equations like x2+1=0x^2 + 1 = 0 would have no solution, limiting what algebra can describe. The imaginary unit is the foundation of complex numbers (a+bia + bi), which are essential in electrical engineering, signal processing, and quantum physics. It also guarantees that every polynomial equation of degree nn has exactly nn roots (counted with multiplicity), a result known as the Fundamental Theorem of Algebra.

Common Mistakes

Mistake: Forgetting the cyclic pattern of powers of i and incorrectly computing i³ or i⁴.
Correction: The powers of i cycle every four: i1=ii^1 = i, i2=−1i^2 = -1, i3=−ii^3 = -i, i4=1i^4 = 1, then the pattern repeats. To find ini^n, divide nn by 4 and use the remainder to determine the value.

Related Terms