Topics Numbers and logic

What are imaginary numbers?

Imaginary numbers are numbers that, when multiplied by themselves, give a negative result. The basic one is called i, defined as the square root of negative one. No ordinary number does that, since a positive times a positive is positive and a negative times a negative is also positive. So mathematicians added a new number to the toolbox, and the rules still hold together.

What makes this interesting is the name. Imaginary suggests made up, but i is no less a tool than negative numbers or fractions, both of which were once met with suspicion. Combine a regular number with a multiple of i and you get a complex number, which can be drawn as a point on a flat plane. That picture turns rotation into arithmetic, and it is why imaginary numbers show up in electrical engineering, signal processing and quantum physics.

An episode would start with the puzzle of a number that squares to negative one, walk through how the idea grew out of solving equations, and show the plane picture slowly. bre's hosts are AI, so they can slip, and you can press Talk to ask them to slow down or try a different example.

What a bre episode would cover

An outline of the episode bre would make for this question. Every episode is written fresh when you ask, so yours will differ.

  1. The number that should not existWhy no ordinary number squares to a negative, and what it means to define i as the square root of negative one.
  2. A bad name that stuckHow imaginary began as a mild insult, and why historians note that the idea crept in while people solved equations.
  3. Complex numbers and the flat planeAdding a real part to an imaginary part, and plotting the result as a point you can see.
  4. Multiplying by i is a quarter turnThe picture that makes the strange rules feel natural: multiply by i and a point rotates ninety degrees.
  5. Where imaginary numbers get usedAlternating current, waves and signals, and quantum physics, where complex numbers are a standard working tool.
  6. Are they real?A fair look at what real means in math, and why the label is a matter of history rather than truth.

How the episode might open

A sample exchange between two of bre’s AI hosts, bre and Tess. Both are AI; this is written by AI, as every bre episode is.

  1. breAI host

    Okay, a question I bet you have already asked in a math class: what number times itself gives negative one? Because I tried a few, and nothing works.

  2. TessAI host

    Nothing works. Positive times positive is positive. Negative times negative is also positive. So the honest answer is that there is no such number. Done. Next topic?

  3. breAI host

    Ha. Except mathematicians looked at that wall and said, fine, what if we just invent one and see whether anything breaks?

  4. TessAI host

    And did it break? Because when I hear invent a number, I picture my bank doing creative accounting.

  5. breAI host

    Not at all. They called it i, defined it as the square root of negative one, and all the usual rules of arithmetic kept working. That is the surprising part.

  6. TessAI host

    Okay, but real talk. Why is it called imaginary? That makes it sound like it is not allowed in the club.

  7. breAI host

    It started as a bit of a put-down. The name stuck anyway, even though the numbers turned out to be very useful.

  8. TessAI host

    So the name is bad branding. Fine. Show me what they actually do.

Questions people also ask

Are imaginary numbers real?
They are real in the sense that matters in math: they are well defined, follow consistent rules and can be used to get correct results. The word imaginary is a historical label, not a claim that they are fake. Negative numbers and zero also met resistance once.
What is i in math?
The symbol i stands for the square root of negative one, so i times i equals negative one. It is the basic unit of imaginary numbers. Any imaginary number is some ordinary number multiplied by i, such as 3i or 0.5i.
What is a complex number?
A complex number combines an ordinary number with an imaginary one, written like 2 + 3i. It can be plotted as a point on a plane, with the ordinary part going across and the imaginary part going up. This makes many calculations easier to picture.
What are imaginary numbers used for?
They are a standard tool in electrical engineering, signal processing and physics. Alternating current circuits, waves and quantum mechanics are all commonly described using complex numbers, because they handle rotation and oscillation neatly.

Related topics

More: all 300 topics, numbers and logic, or the longer reads on /learn.

bre’s hosts are AI, and every episode is generated, so they can be wrong: check anything that matters. This page outlines what an episode would cover. It is for interest and learning, not medical, financial or legal advice.