082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

9.2 Complex numbers 325
9.2 COMPLEXNUMBERS
We have already examined quadratic equations such as
x 2 −x−6=0 (9.1)
andhavemettechniquesforfindingtherootsofsuchequations.Theformulaforobtainingthe
roots of a quadratic equationax 2 +bx +c = 0 is
x = −b ± √ b 2 −4ac
2a
Applying thisformulatoEquation (9.1), wefind
(9.2)
x = +1 ± √ (−1) 2 −4(1)(−6)
2
= 1 ± √ 25
2
= 1 ±5
2
so thatx = 3 andx = −2 are the two roots. However, if we try to apply the formula to
the equation
we find
2x 2 +2x+5=0
x = −2 ± √ −36
4
A problem now arises in that we need to find the square root of a negative number. We
knowfromexperiencethatsquaringbothpositiveandnegativenumbersyieldsapositive
result; thus,
6 2 = 36 and (−6) 2 = 36
so that there is no real number whose square is −36. In the general case, if
ax 2 +bx +c = 0, we see by examining the square root in Equation (9.2) that this problemwillalwaysarisewheneverb
2 −4ac < 0.Nevertheless,itturnsouttobeveryuseful
to invent a technique for dealing with such situations, leading to the theory of complex
numbers.
To make progress weintroduce a number, denoted j,with the property that
j 2 =−1
We have already seen that using the real number system we cannot obtain a negative
numberbysquaringareal numbersothenumberjisnotreal-- wesay itisimaginary.
Thisimaginarynumberhasaveryusefulroletoplayinengineeringmathematics.Using
it we can now formally write down an expression for the square root of any negative
number. Thus,
√
−36 =
√
36 × (−1)
= √ 36×j 2
= 6j