Zero - Charles Seife [51]
A linear equation like 4x - 12 = 0 is extremely simple to solve, and such problems didn’t entertain algebraists for very long. So they soon turned to more difficult problems: quadratic equations—equations that begin with an x2 term, like x2 - 1 = 0. Quadratic equations are more complicated than regular equations; for one thing, they can have two different roots. For instance, x2 - 1 = 0 has two solutions: 1 and -1. (Substitute -1 or 1 for x in the equation and you’ll see what happens.) Either one of those solutions works; as it turns out, the expression x2 - 1 splits into (x - 1)(x + 1), making it easy to see that if x is 1 or -1, the expression goes to zero.
Though quadratic equations are more complicated than linear equations, there is a simple way to figure out what the roots of a quadratic equation are. It’s the famous quadratic formula, which is the crowning achievement of high-school algebra class. The formula for finding the roots of a quadratic equation ax2 + bx + c = 0 is: . The + sign gives us one root, while the - sign gives us the other. The quadratic formula has been known for centuries; the ninth-century mathematician al-Khowarizmi knew how to solve almost every quadratic equation, though he didn’t seem to consider negative numbers as roots. Not long after that, algebraists learned to accept negative numbers as valid solutions to equations. Imaginary numbers, though, were a little different.
Imaginary numbers never appeared in linear equations, but they began to crop up in quadratic ones. Consider the equation x2 + 1 = 0. No number seems to solve the equation; plugging in –1, 3, –750, 235.23, or any other positive or negative number you could think of doesn’t yield the correct answer. The expression simply will not split. Worse yet, when you try to apply the quadratic equation, you get two silly-sounding answers:
These expressions don’t seem to make any sense. The Indian mathematician Bhaskara wrote in the twelfth century that “there is no square root of a negative number, for a negative number is not a square.” What Bhaskara and others realized was that when you square a positive number, you get a positive number back; 2 times 2 equals 4, for instance. When you square a negative number, you still get a positive number: –2 times –2 also equals 4. When you square zero, you get zero. Positive numbers, negative numbers, and zero all give you nonnegative squares, and those three possibilities cover the whole number line. This means that there is no number on the number line that gives you a negative number when you square it. The square root of a negative number seemed like a ridiculous concept.
Descartes thought that these numbers were even worse than negative numbers; he came up with a scornful name for the square roots of negatives: imaginary numbers. The name stuck, and eventually, the symbol for the square root of –1 became i.
Algebraists loved i. Almost everyone else hated it. It was wonderful for solving polynomials—expressions like x3 + 3x + 1 that have x raised to various powers. In fact, once you allow i into the realm of numbers, every polynomial becomes solvable: x2 + 1 suddenly splits into (x - i)(x + i)—the roots of the equation are +i and -i. Cubic expressions like x3 - x2 + x - 1 split three ways, such as (x - 1)(x + i)(x - i). Quartic expressions—ones with a leading x4 term—always split into four terms, and quintics—ones with a leading x5 term—split five ways. All polynomials of degree n—those that have a leading term of xn—split into n distinct terms. This is the fundamental theorem of algebra.
As early as the sixteenth century, mathematicians were using numbers with i included—the so-called complex numbers—to solve cubic and quartic polynomials. And while many mathematicians saw the complex numbers as a convenient fiction, others saw God.
Leibniz thought that i was a bizarre mix between existence and nonexistence, something like a cross between 1 (God) and 0 (Void) in his binary scheme. Leibniz likened i to the Holy Spirit: both have an ethereal and barely