Algebra

Imaginary Expression

The following are conventions: That √(−𝑎²) is equivalent to 𝑎√(−1); that 𝑎√(−1) vanishes when 𝑎 vanishes; that the symbol 𝑎√(−1) is subject to the ordinary rules of Algebra. √(−1) is denoted by 𝑖.223 If 𝛼+𝑖𝛽=𝛾+𝑖𝛿; then 𝛼=𝛾 and 𝛽=𝛿.224 𝛼+𝑖𝛽 and 𝛼−𝑖𝛽 are conjugate expressions; their product=𝛼²+𝛽².225 The sum and product of two conjugate expressions are both real, but their difference is imaginary.226 The modulus is +𝛼²+𝛽².227 If the modulus vanishes, 𝛼 and 𝛽 must vanish.228 If two imaginary expressions are equal, their moduli are equal, by (224).229 The modulus of the product of two imaginary expressions is equal to the product of their moduli.230 Also the modulus of the quotient is equal to the quotient of their moduli.231

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive