Lesson 1 · from §116, 117 of the 1910 text
Imaginary and Complex Numbers
Imaginary numbers
When the square root of a negative number arose in our previous work, it was called an imaginary, and no attempt was then made to use it or to explain its meaning. The treatment of imaginaries was deferred because there were so many topics of more importance to the beginner. It must not be supposed, however, that imaginaries are not of great value in mathematics. They are also of much use in certain branches of applied science; and it is unfortunate that symbols which can be used in numerical computations to obtain practical results should ever have been called imaginary. By such a name something unreal and fanciful is suggested. To obviate this it has been proposed to call imaginary numbers orthotomic numbers, but this name has been little used.
The equation x^2 + 1 = 0, or x^2 = -1, asks the question, "What is the number whose square is -1?" By defining a new number, \sqrt{-1}, as a number whose square is -1, we obtain one root for the equation x^2 + 1 = 0. Similarly \sqrt{-5} is a number whose square is -5. And, in general, \sqrt{-n} is a number whose square is -n. Obviously \sqrt{-5} means something very different from \sqrt5.
The positive numbers are all multiples of the unit +1, and the negative numbers are all multiples of the unit -1. Similarly pure imaginary numbers are real multiples of the imaginary unit \sqrt{-1}.
Thus \sqrt{-1} + \sqrt{-1} = 2\sqrt{-1} and \sqrt{-1} + 2\sqrt{-1} = 3\sqrt{-1}, etc. Further, \sqrt{-4} = 2\sqrt{-1}; \sqrt{-a^2} = a\sqrt{-1}; \sqrt{-5} = \sqrt5\sqrt{-1}.
The imaginary unit \sqrt{-1} is often denoted by the letter i; that is, 3\sqrt{-1} = 3i.
If a real number be united to a pure imaginary by a plus sign or a minus sign, the expression is called a complex number.
Thus -2 + \sqrt{-1} and 3 - 2\sqrt{-4} are complex numbers. The general form of a complex number is a + bi, in which a and b may be any real numbers.
Addition and subtraction of imaginaries
The fundamental operations of addition and subtraction are performed on imaginary and complex numbers as they are performed on real numbers and ordinary radicals of the same form.
Thus 2\sqrt{-1} + 4\sqrt{-1} = 6\sqrt{-1}, and 5\sqrt{-1} - 3\sqrt{-1} = 2\sqrt{-1}. Also 3 + 5\sqrt{-1} + 4 - 2\sqrt{-1} = 7 + 3\sqrt{-1}. Similarly a + bi + c + di = a + c + (b + d)i.
Before adding, write each imaginary as a multiple of \sqrt{-1}: \sqrt{-18} = \sqrt{9 \cdot 2}\,\sqrt{-1} = 3\sqrt2\,i. Imaginary terms with different radicals, like 7i and 6\sqrt2\,i, are not similar, and are left as two terms.
How to type imaginaries
- Type the imaginary unit as
i:3i,2 - 5i,sqrt(2) i,2x i. You may also type it assqrt(-1), the way the book writes it:5sqrt(-1). - Give each answer in the form a + bi: work out every power of i, collect the real terms and the imaginary terms, and leave no i in a denominator. Write the square root of a negative number as a multiple of i:
sqrt(-9)becomes3i,sqrt(-2)becomessqrt(2) i. - A fraction may stand over its common denominator:
(1 + 3i)/5and1/5 + 3/5 iare both fine. - Letters stand for positive numbers, so \sqrt{-a^2} = a\sqrt{-1} = ai.
From the history of algebra
Karl Friedrich Gauss. Standing in the very front rank of mathematicians, with Archimedes and Newton, is Karl Friedrich Gauss (1777–1855). He was the son of a bricklayer and was afforded an education, much against the will of his parents, by a nobleman who had noticed his remarkable talents.
Up to his time the imaginary numbers were not clearly understood, and were usually thought of as absurd and inadmissible. The situation reminds one of the time when negative numbers were similarly regarded, and the veil was removed from both in about the same way. It was found that negative numbers really had a significance; that they could be used in problems that involve debt, opposite directions, and many other everyday relations. The interpretation of imaginary numbers is not quite so obvious, but none the less actual and simple. As soon as it was seen that they could be represented with real numbers as points on a plane, the ice was broken, and it needed only the insight and authority of a man like Gauss to give them their proper place in mathematics. He also used the letter i for \sqrt{-1}.
Gauss was the first man to prove that every rational, integral equation in one unknown, whatever its degree, possesses a root. In fact, during his life he gave three distinct demonstrations of this theorem which had baffled all the attempts of mathematicians before his time.
In collaboration with another professor at the University of Göttingen he invented the telegraph independently of the American, S. F. B. Morse, and probably earlier.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Simplify:
3\sqrt{-1} + 4\sqrt{-1} - 2\sqrt{-1}
\sqrt{-4}
\sqrt{-25}
5\sqrt{-1} + \sqrt{-9}
\sqrt{-4} + \sqrt{-16}
5\sqrt{-36x^2} - 2\sqrt{-49x^2}
\sqrt{-18} + \sqrt{-8}
(-12)^{\frac12} + (-27)^{\frac12}
3 + 2\sqrt{-1} + 5 - 6\sqrt{-1}
5\sqrt{-x^2} - 7a - 3\sqrt{-x^2}
4 - 8\sqrt{-1} + 16 - 3\sqrt{-4}
6 - 2\sqrt{-64x^2} - 3\sqrt{-25x^2} + 8
18 - 3(-1)^{\frac12} + 6(-2)^{\frac12} + (-100)^{\frac12} + 4
5\sqrt{-3} + 3\sqrt{-2} - \sqrt{-27} + 2\sqrt{-8}
6\sqrt{-4a^4} - 7a^2\sqrt{-9} + 3\sqrt{-6} - 5\sqrt{-24}
\left(12 - 6\sqrt{-9}\right) - \left(15 + 2\sqrt{-36}\right)
3a - 2x - \left(2a\sqrt{-a^2} - 5ra^2\sqrt{-1}\right)
(x - iy) - (n - iv)
Write as a multiple of \sqrt{-1} (that is, of i):
\sqrt{-10}
\sqrt{-6}
2\sqrt{-3}
\sqrt{-a}
a\sqrt{-b}
\sqrt{-a - b}