What Is i?
Question
There is no such ________________ number where you can take the square of and get a negative number out of it.
Hi, my name is Bassem Saad. I'm a Math Ph.D. candidate at U.C. Davis, and I'm here today for About.com to answer the question: “What is i?”i is defined to be the square root of negative one. And by now you've seen lots of different numbers: whole numbers like one or two; rational numbers, like one-fifth; or even irrational numbers, like the square root of two. And all of these numbers are real numbers -- so is i a real number? After all, it doesn't look so different from the square root of two.To answer this question, we can redefine i as being the number, such that if you take the square of it -- equals negative one. But there is no such real number where you can take the square of and get a negative number out of it. So it's not a real number; it's actually what we call an imaginary number.Now if we treat i as a variable, with the added property that i squared equals negative one, we can do all sorts of familiar operations to add, subtract, multiply, and divide i. For example, if we have five times i, well we just treat it like we would a variable, as coefficient five in front of i. Or we can have 6i minus 3i. Well we just subtract the coefficients like we would with a variable to get 3i.And with this property, we can multiply i times itself – say eight times. That's the same thing as saying i squared, times i squared, times i squared, times i squared. Well, this is minus one, times minus one, times minus one, times minus one, which just equals one.We have just answered the question: “What is i?” Thanks for watching, and to learn more, visit us on the web at About.com.