If the product of the two numbers is positive, they must have the same sign, and since their sum is positive, this means they must both be positive. Finding two numbers that add to 3 and multiply to 10 seems an impossible challenge. This meant $latex x^2-8x+12$ could be factored as $latex (x-2)(x-6)$, and expressing this polynomial as a product of two factors made solving the equation $latex x^2-8x+12=0$ easy.īut it wasn’t so easy to do this for equations like $latex x^2-3x+10=0$. He had no trouble solving equations like $latex x^2-8x+12=0 $, because it was easy to find two numbers whose sum was 8 and whose product was 12: namely, 2 and 6. In the 1500s, the master equation solver Girolamo Cardano was trying to solve polynomial equations. But the real numbers aren’t enough to solve all our math problems. We can add, subtract, multiply and divide real numbers, and we use them to answer questions both in classrooms and in our everyday lives. The “real numbers” are some of our most familiar mathematical objects: They are all the numbers that can be represented in decimal notation, like 5, 8.2, -13.712, 0, 10.33333… and $latex \pi \approx$ 3.141592…. Let’s take a look at how these unfamiliar numbers are rooted in the numbers we know, but at the same time, are unlike anything we have imagined. And they are the first step into a world of strange number systems, some of which are being proposed as models of the mysterious relationships underlying our physical world. They have a far-reaching impact in physics, engineering, number theory and geometry. Have you ever sat in a math classroom and wondered, “When will I ever use this?” You might have asked yourself this question when you first encountered “imaginary” numbers, and with good reason: What could be less practical than a number described as imaginary?īut imaginary numbers, and the complex numbers they help define, turn out to be incredibly useful.
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