# What is the Square root of minus 1? **URL:** \ **Category:** Factual Questions\ **Created:** [November 25, 2005, 10:25pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736 "2005-11-25T22:25:24Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Wattschri](https://avatars.discourse-cdn.com/v4/letter/w/b782af/32.png) [@Wattschri](https://boards.straightdope.com/u/Wattschri)\ **Post date:** [November 25, 2005, 10:25pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/1 "2005-11-25T22:25:24Z")
oh and 0 thinking about it? ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [November 25, 2005, 10:30pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/2 "2005-11-25T22:30:06Z")
The sqaure root of -1 is defined as “i” (imaginary number). ---
**Author:** ![hibernicus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hibernicus/32/2802_2.png) [@hibernicus](https://boards.straightdope.com/u/hibernicus)\ **Post date:** [November 25, 2005, 10:30pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/3 "2005-11-25T22:30:35Z")
> [@Wattschri](#): > > oh and 0 thinking about it? There’s no real number which, when multiplied by itself, gives minus 1. The imaginary number _i_ has been defined as the square root of minus 1. The other square root of minus one is -_i_. ---
**Author:** ![Bricker](https://avatars.discourse-cdn.com/v4/letter/b/977dab/32.png) [@Bricker](https://boards.straightdope.com/u/Bricker)\ **Post date:** [November 25, 2005, 10:31pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/4 "2005-11-25T22:31:57Z")
> [@Wattschri](#): > > oh and 0 thinking about it? There is no real number that multiplied by itself will yield -1. So… the answer is an imaginary number. In fact, the definition of the imaginary number _i_ is that _i_[sup]2[/sup]= -1. With that, you can easily see that the square root of -4 = 2_i_, the square root of -9 = 3_i_, and so on. ---
**Author:** ![Harmonious\_Discord](https://avatars.discourse-cdn.com/v4/letter/h/74df32/32.png) [@Harmonious\_Discord](https://boards.straightdope.com/u/Harmonious_Discord)\ **Post date:** [November 25, 2005, 10:33pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/5 "2005-11-25T22:33:20Z")
Imaginary Numbers [http://www.math.toronto.edu/mathnet/answers/imaginary.html](http://www.math.toronto.edu/mathnet/answers/imaginary.html) ---
**Author:** ![Exploding\_Kitchen](https://avatars.discourse-cdn.com/v4/letter/e/c89c15/32.png) [@Exploding\_Kitchen](https://boards.straightdope.com/u/Exploding_Kitchen)\ **Post date:** [November 25, 2005, 10:37pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/6 "2005-11-25T22:37:00Z")
The other part of the question is easy. The square root of 0 is 0. ---
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [November 26, 2005, 12:27am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/7 "2005-11-26T00:27:41Z")
> [@John Mace](#): > > The sqaure root of -1 is defined as “i” (imaginary number). No! No! It’s **j** I’m an EE. 😉 ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [November 26, 2005, 1:45am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/8 "2005-11-26T01:45:46Z")
And just to forestall the comment that everybody knows is coming: Although mathematicians talk about the “real” numbers and the “imaginary” numbers, the “imaginary” numbers are exactly as real as the “real” numbers. In fact, both are components of the complex numbers (a + bi). Don’t let the terminology confuse anyone into a false belief. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [November 26, 2005, 2:26am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/9 "2005-11-26T02:26:46Z")
> [@Exapno Mapcase](#): > > And just to forestall the comment that everybody knows is coming: > > Although mathematicians talk about the “real” numbers and the “imaginary” numbers, the “imaginary” numbers are exactly as real as the “real” numbers. In fact, both are components of the complex numbers (a + bi). > > Don’t let the terminology confuse anyone into a false belief. To take this a little further. The use of the term “imaginary” goes way back to ancient mathematicians. Occasionally they ran into equations like x[sup]2[/sup] + 1 = 0. They knew of no number in their set of numbers that could replace x and make this a true statement. So they speculated that such equations must have “imaginary” solutions. Once a system of numbers was devised that could be used to solve these equations the term “imaginary” was retained for such numbers but that is “imaginary” in a different sense than that of “not existing.” The solution for this equation is the notorious square root of -1. Other equations might be x[sup]2[/sup] + 4 = 0. This becomes x[sup]2[/sup] = -1_4 = square root of -2 \* square root of 4 = i_2 or 2i. And so on. ---
**Author:** ![TJdude825](https://avatars.discourse-cdn.com/v4/letter/t/dec6dc/32.png) [@TJdude825](https://boards.straightdope.com/u/TJdude825)\ **Post date:** [November 26, 2005, 3:07am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/10 "2005-11-26T03:07:18Z")
> [@David Simmons](#): > > Other equations might be x[sup]2[/sup] + 4 = 0. This becomes x[sup]2[/sup] = -1_4 = square root of -2 \* square root of 4 = i_2 or 2i. > > And so on. Typo: square root of -1 \* square root of 4 Anyway, “j” and “i” mean the same thing, it’s just that j is used by electrical engineers because “i” already has a different meaning for them. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [November 26, 2005, 3:14am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/11 "2005-11-26T03:14:57Z")
> [@David Simmons](#): > > To take this a little further. The use of the term “imaginary” goes way back to ancient mathematicians. Occasionally they ran into equations like x[sup]2[/sup] + 1 = 0. They knew of no number in their set of numbers that could replace x and make this a true statement. So they speculated that such equations must have “imaginary” solutions. Once a system of numbers was devised that could be used to solve these equations the term “imaginary” was retained for such numbers but that is “imaginary” in a different sense than that of “not existing.” > > The solution for this equation is the notorious square root of -1. > > Other equations might be x[sup]2[/sup] + 4 = 0. This becomes x[sup]2[/sup] = -1_4 = square root of -2 \* square root of 4 = i_2 or 2i. > > And so on. Actually, historically it wasn’t quadratic equations that led to the discovery of imaginary numbers, it was cubics. Quadratic equations with no real solutions weren’t really considered unusual, or to be “hiding” anything (such as imaginary numbers)–these equations simply didn’t have a solution, and nobody was really troubled by it. It wasn’t until the cubic formula was discovered that people began to discover imaginary numbers. The cubic formula was discovered in the 16th century; the strange thing was that while occasionally the steps required you to take the square root of a negative number, the final result would give you an actual (real number) solution. How could this method ultimately work, even when the intermediate steps led you through “fictitious” (imaginary) numbers? (I believe it was Cardano who first began to question what was going on with what he termed as the “fictitious numbers” involved in the steps of the cubic formula). He didn’t manage to figure it out, but this was the genesis of the discovery of imaginary numbers. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [November 26, 2005, 4:16am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/12 "2005-11-26T04:16:04Z")
> [@Cabbage](#): > > Actually, historically it wasn’t quadratic equations that led to the discovery of imaginary numbers, it was cubics. > > Quadratic equations with no real solutions weren’t really considered unusual, or to be “hiding” anything (such as imaginary numbers)–these equations simply didn’t have a solution, and nobody was really troubled by it. > > It wasn’t until the cubic formula was discovered that people began to discover imaginary numbers. The cubic formula was discovered in the 16th century; the strange thing was that while occasionally the steps required you to take the square root of a negative number, the final result would give you an actual (real number) solution. How could this method ultimately work, even when the intermediate steps led you through “fictitious” (imaginary) numbers? (I believe it was Cardano who first began to question what was going on with what he termed as the “fictitious numbers” involved in the steps of the cubic formula). He didn’t manage to figure it out, but this was the genesis of the discovery of imaginary numbers. You might very well be right. In that case I’m going to demand some of my money back of the University of Iowa where my algebra professor recounted the history of the quadratic in introducing imaginary and complex numbers. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [November 26, 2005, 4:39am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/13 "2005-11-26T04:39:59Z")
Here’s a cite I found on the history of complex numbers: > [@](#): > > The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. > **[Complex number | History](https://en.wikipedia.org/wiki/Complex_number#History)** > > The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary. > Work on the problem of general pol... Though I gotta admit I’m not really familiar with the reference to Heron of Alexandria in the first sentence. ---
**Author:** ![Plynck](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Plynck](https://boards.straightdope.com/u/Plynck)\ **Post date:** [November 26, 2005, 6:09am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/14 "2005-11-26T06:09:26Z")
> [@Cabbage](#): > > Though I gotta admit I’m not really familiar with the reference to Heron of Alexandria in the first sentence. There was a “Hero of Alexander” who invented the steam engine, so that might answer the inventor part. ---
**Author:** ![Bricker](https://avatars.discourse-cdn.com/v4/letter/b/977dab/32.png) [@Bricker](https://boards.straightdope.com/u/Bricker)\ **Post date:** [November 26, 2005, 1:27pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/15 "2005-11-26T13:27:02Z")
> [@Exapno Mapcase](#): > > Although mathematicians talk about the “real” numbers and the “imaginary” numbers, the “imaginary” numbers are exactly as real as the “real” numbers. In fact, both are components of the complex numbers (a + bi). Hmmm. Simple mind that I have, I’d argue that the real numbers are more “real” than the imaginary numbers, because some of the set of real numbers are rational, and rational numbers are more “real” than the irrational numbers. And Lord knows the integers are more “real” still… right? ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [November 26, 2005, 3:03pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/16 "2005-11-26T15:03:19Z")
> [@Bricker](#): > > Hmmm. > > Simple mind that I have, I’d argue that the real numbers are more “real” than the imaginary numbers, because some of the set of real numbers are rational, and rational numbers are more “real” than the irrational numbers. > > And Lord knows the integers are more “real” still… right? Isn’t 2i/3 merely 2/3 measured along the i axis? Why is that less rational than 2/3 measured along the real axis? ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [November 26, 2005, 3:06pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/17 "2005-11-26T15:06:50Z")
> [@Bricker](#): > > Hmmm. > > Simple mind that I have, I’d argue that the real numbers are more “real” than the imaginary numbers, because some of the set of real numbers are rational, and rational numbers are more “real” than the irrational numbers. > > And Lord knows the integers are more “real” still… right? Numbers are just things we make up to understand the world. It’s not so much that imaginary numbers are just as “real” as real numbers are, but that “real” numbers are just as imaginary as imaginary numbers are. IOW, both sets are made up (by us). ---
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [November 26, 2005, 3:21pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/18 "2005-11-26T15:21:02Z")
> [@John Mace](#): > > Numbers are just things we make up to understand the world. It’s not so much that imaginary numbers are just as “real” as real numbers are, but that “real” numbers are just as imaginary as imaginary numbers are. IOW, both sets are made up (by us). I understand. But why stop at 2-dimesions? Could we invent another axis that is orthogonal to the real axis and imaginary axis, thereby creating a 3-D number? ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [November 26, 2005, 3:22pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/19 "2005-11-26T15:22:55Z")
> [@Crafter\_Man](#): > > I understand. But why stop at 2-dimesions? Could we invent another axis that is orthogonal to the real axis and imaginary axis, thereby creating a 3-D number? Yep. It’s called a vector. ---
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [November 26, 2005, 3:38pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/20 "2005-11-26T15:38:25Z")
> [@Crafter\_Man](#): > > I understand. But why stop at 2-dimesions? Could we invent another axis that is orthogonal to the real axis and imaginary axis, thereby creating a 3-D number? If you want to have multiplication as well as addition, you have to go to four dimensions, where you have the [quaternions](http://en.wikipedia.org/wiki/Quaternions). But, you have to give up commutativity. The next step is to eight dimensions, where you find the [octonions](http://en.wikipedia.org/wiki/Octonion) (or Cayley numbers), but you have to give up associativity. I like the quote at the end of that article: > [@John Baez](#): > > The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are **nonassociative**. In an appropriately defined sense, the reals, complex numbers, quaternions, and octonions are the only cases with nicely behaved multiplications. [Next page](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736.md?page=2)