Introduction to imaginary numbers?
As earlier stated, Euler is attributed with the notation of i in 1777 but the idea and struggle is much older. One of the earliest
records of an encounter with √(-1) is the mathematician-engineer
Heron of Alexandria in the first century AD.
In his calculations the Heron just replaced the √(-1) with √ 1 (Nahin, 1998). It would be interesting to know how that
affected his calculation but that is not recorded.
Diophantus , a mathematician two centuries after the
Heron, simply wrote that the equation was not possible since
"they could see no way physically to interpret a number that is 'less than nothing'" (Nahin, 1998). This idea of leaving this mystery
alone lasted for a number of centuries. George Gamow wrote a nice little limerick:
"There was a young fellow from Trinity
Who took √∞.
But the number of digits
Gave him the fidgets;
He dropped Math and took up Divinty." (Nahin, 1998)
Where did the ancients call imaginary numbers?
In the those years of discovery many derogatory names given to imaginary numbers. Some of these names were "impossible numbers",
"improbable numbers" and "nonsense numbers" (Bragg, Yusotoi, Stewart, & Sirius, 2010).
Cardano , who will be important later called
them "mental tortures" and "fictitious" solutions. (Mazur, 2003). The symbol √(-1) was an "uninterpretable symbol" that it
should be "void of meaning, or rather self-contradictory and absurd." (Nahin, 1998)
Gottfried Leibniz said "[imaginary numbers]
are almost an amphibian between
being and not being." (Prochazka, 2006) However, similar things were said about negative numbers when they were being discovered.
Advances were obviously eventually made. It was mathematicians investigating roots to cubic equations that lead to the acceptance
of the imaginary number i.
Rafael Bombelli provided a structure for
multiplication of complex numbers back in the 1500s, but other
mathematicians were slow to accept the concept. Bombelli suggested that if we have this quantity, though we don't know where to put
it on the number line, that obeys all of our other rules for addition and multiplication then we can use that as a placeholder.
Whenever an equations requires √(-1) then we will use i instead. We know that i has the characteristic that
i2 = -1. (Mazur, 2003) Imaginary numbers were more accepted once Leonhard Euler and Carl Gauss worked with them. (Nahin, 1998)
Podcast introducing imaginary numbers
BBC Podcast
Imaginary numbers have been accepted since the 1800s. It was in northern Italy that Cardano came up with a form to solve cubic
functions or equations with degree three. The problem was that this formula led to taking the square root of negative numbers
(Bragg, Yusotoi, Stewart, & Sirius, 2010). But what number when squared is negative. Not a negative number since the poet W.H.
Auden wrote: "Minus times minus is plus. The reason for this we need not discuss." (Nahin, 1998) About 20 years later Bombelli,
another Italian mathematician, proposed that we don't really need to know exactly what that number is. If we use a placeholder then
we get consistent results. Trust the algebra that if the equation asks to square √(-3) then you just get -3. So its just a calculating
device. Though we still can't put i on the number line. It was Argand that proposed that instead of just have a line that we plot
imaginary numbers on a plane with two axes. There will be a real axis and an imaginary axis. (Bragg, Yusotoi, Stewart, & Sirius, 2010)