Prerequisites

Imaginary Number Definition | \(j = \pm \sqrt{-1}\)

Description

This equation shows the relationship between the most fundamental imaginary number, \( \htmlId{tooltip-imag}{j} \), and the real numbers.

Equation

\[\htmlId{tooltip-imag}{j} = \sqrt{-1}\]

Symbols Used

\(j\)

This symbol represents the imaginary unit, which is defined as the square root of \(-1\). \( j = \sqrt{-1}\). It is the most fundamental unit in the field of complex numbers, allowing for the expression of numbers that cannot be represented on the real number line.

Derivation

Consider the definition of an Imaginary Number
\[\htmlId{tooltip-imag}{j} = \pm \sqrt{-1}\]
From here, we can square both sides, to get...

\[(\sqrt{-1})^2 = (\pm \htmlId{tooltip-imag}{j})^2\]
From here, we can square both sides, to get...

\[(\sqrt{-1})^2 = (\pm \htmlId{tooltip-imag}{j})^2\]
which simplifies to...
\[-1 = \htmlId{tooltip-imag}{j}^2\]
which is equivalent to...
\[\htmlId{tooltip-imag}{j}^2 = -1\]
as required.

Example

Find the value of \( (5 + \htmlId{tooltip-imag}{j})^2\)

\( (5 + \htmlId{tooltip-imag}{j})^2 \) can be rewritten to be

\[ (5 +\htmlId{tooltip-imag}{j})(5 + \htmlId{tooltip-imag}{j}) \]

This is equivalent to...

\[ 5 \cdot 5 + 5 \cdot \htmlId{tooltip-imag}{j} + 5 \cdot \htmlId{tooltip-imag}{j} + \htmlId{tooltip-imag}{j}^2 \]

This simplifies to...

\[25 + 10 \htmlId{tooltip-imag}{j} + \htmlId{tooltip-imag}{j}^2\].

We can now use our identity, to get...

\[25 + 10 \htmlId{tooltip-imag}{j} - 1\]

which simplifies to...

\[24 + 10 \htmlId{tooltip-imag}{j}\]