The term "imaginary" probably originated from the fact that there is no real number z that satisfies the equation z2 = -1. Imaginary numbers are not "impossible" numbers - they are very important mathematical entities. x This is a slightly different usage of the word "imaginary", meaning "non-real": among the complex numbers, those that aren't real we call imaginary, and a further subset of those (with real part $0$) are purely imaginary. Are there any non-algebraic, non-transcendental complex numbers? Is $0$ a pure imaginary number? The premise might seem silly, but the question is well-written and clearly thought-out. The imaginary unit i. (9.6.1) – Define imaginary and complex numbers. An imaginary number is an even root of a negative number. Any imaginary number can be represented by using i. The imaginary unit i. What is the "Ultimate Book of The Master". Every real number graphs to a unique point on the real axis. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Has the Earth's wobble around the Earth-Moon barycenter ever been observed by a spacecraft? 0.1 × 0.1 = 0.01. CCSS.Math: HSN.CN.A.1. Intro to the imaginary numbers. With the development of quotient rings of polynomial rings, the concept behind an imaginary number became more substantial, but then one also finds other imaginary numbers, such as the j of tessarines, which has a square of +1. Since the square (bi) 2 = −b 2 of an imaginary number is a negative real number, the imaginary numbers are just the square roots of the negative real numbers. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. An imaginary root or zero would be a value x=a+i*b in the complex plane that satisfies F(x)=0. First, please take this two mathematical definitions into consideration. (Because the imaginary part is zero, 1+0i is just another way of writing the real number 1.) An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i,[note 1] which is defined by its property i2 = −1. An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number. 1- purely real , if b=0 ; e.g.- 56,78 ; Cockle, James (1848) "On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra", London-Dublin-Edinburgh. [9][10] The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$. Originally coined in the 17th century by René Descartes[5] as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century). After 20 years of AES, what are the retrospective changes that should have been made? Imaginary Numbers: When real numbers are multiplied to itself, it is guaranteed that the product is a positive number. An imaginary number, also known as a pure imaginary number, is a number of the form b i bi b i, where b b b is a real number and i i i is the imaginary unit. a = 0 and b is not equal to 0, the complex number is called an imaginary number. For example:[13]. Intro to the imaginary numbers. = Maximum useful resolution for scanning 35mm film. Why do jet engine igniters require huge voltages? fails when the variables are not suitably constrained. The quantity i is called the unit imaginary number. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. I can't (and MSE can't) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $|e^{z}| = 1$. Seems to me that you could say imaginary numbers are based on the square root of x, where x is some number that's not on the real number line (but not necessarily square root of negative one—maybe instead, 1/0). Here, i is equal to the square root of negative 1. Strictly speaking imaginary numbers are numbers which contain the square root of one in the form x + y*sqrt(-1), and, when squared, give a negative number. Linear combination of complex If z1=5+3i and z2=4-2i, write the following in the form a+bi a) 4z1+6z2 b) z1*z2; Reciprocal Calculate reciprocal of z=0.8-1.8i: Imaginary numbers Find two imaginary numbers whose sum is a real number. Intro to the imaginary numbers. Making statements based on opinion; back them up with references or personal experience. The imaginary axis is the line in the complex plane consisting of the numbers that have a zero real part:0 + bi. Is the union axiom really needed to prove existence of intersections? Does a purely imaginary number have a corresponding “angle” in polar coordinate system? This is true, using only the real numbers.But here you will learn about a new kind of number that lets you work with square roots of negative numbers! Thanks for contributing an answer to Mathematics Stack Exchange! The word "imaginary" might lead you to believe that imaginary numbers are essentially useless and almost detached from math. ... By making [latex]b=0[/latex], any real number can be expressed as a complex number. clockwise) also satisfies this interpretation. We know certainly, that there are complex numbers that are neither purely real, nor purely imaginary. What does children mean in “Familiarity breeds contempt - and children.“? Im>0? 0, though a valueless number, is actually quite great in importance. The problem with not having 0 is that numbers would be very limited. n. A complex number in which the imaginary … So, a Complex Number has a real part and an imaginary part. It's a useful term sometimes. For example, 5i is an imaginary number, and its square is −25. Google Classroom Facebook Twitter. It is well edited and clearly there was decent thought put into it. No, 0 0 0 0 is not an imaginary number. An imaginary number times 0 is 0. Clearly we can (re)define a real number as a complex number with an imaginary component that is zero (meaning that $0$ is a real number), but if one were to define an imaginary number as a complex number with real component zero, then that would also include $0$ among the pure imaginaries. x But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. Google Classroom Facebook Twitter. But $0$ clearly has this property, so we should consider it purely imaginary. For example, the square root of -4 is 2i. The question anyone would ask will be "where to" or "which direction". 0 base 4 is equal to 0 base 10, or any other base. We know that the quadratic equation is of the form ax 2 + bx + c = 0, where the discriminant is b 2 – 4ac. Complex numbers are numbers like 7 + .4i; they're a real number plus an imaginary number. Imaginary numbers. "For example, 3 i is the imaginary analogue of the real number 3. How to make one wide tileable, vertical redstone in minecraft. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. This vertical axis is often called the "imaginary axis" and is denoted iℝ, , or ℑ. Complex numbers in the form a + bi can be graphed on a complex coordinate plane. The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).[11]. But imaginary numbers are no less "real" than real numbers. {\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}} An imaginary number is a mathematical term for a number whose square is a negative real number. rev 2021.1.18.38333, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. An imaginary number is a number that when squared results in a negative value. [3] The set of imaginary numbers is sometimes denoted using the blackboard bold letter .[4]. At whose expense is the stage of preparing a contract performed? You must be able to apply value to place easily, and efficiently, without confusion. I understand that the number zero lies on both the real and imaginary axes. I'm guessing you thought you can't multiply an imaginary number by 0, which is probably a result of a poor introduction to imaginary numbers. But then 0^2 = 0 is not negative. 2) The square root of -1, or i, is defined as an imaginary number. 1) The square root of a negative number is undefined. Up to now, you’ve known it was impossible to take a square root of a negative number. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie, where the term imaginary was used and meant to be derogatory. Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is the angle (phase) in degrees, for example, 5L65 which is the same as 5*cis(65°). The sum of two well-ordered subsets is well-ordered. The Wikipedia article cites a textbook that manages to confuse the issue further: Purely imaginary (complex) number : A complex number $z = x + iy$ is called a purely imaginary number iff $x=0$ i.e. I like it. This is the currently selected item. (On the other hand, $0$ has all of the properties a real number should have, being real; so it makes some amount of sense to also say that it's purely imaginary but not imaginary at the same time. Note that a 90-degree rotation in the "negative" direction (i.e. It only takes a minute to sign up. Always positive, or zero. Email. Example of a complex transcendental number? Let’s start at the point (1, 0), which is represented by the complex number 1+0i. In 1843, William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of quaternion imaginaries, in which three of the dimensions are analogous to the imaginary numbers in the complex field. This definition can be represented by the equation: i 2 = -1. An imaginary number is a number that, when squared, has a negative result. Better user experience while having a small amount of content to show. Zero is still zero in any base. (Though they were pretty good at defining "imaginary component", etc.). Imaginary numbers result from taking the square root of a negative number. Except that by this definition, $0$ is clearly purely imaginary but not imaginary! $R(z) = 0$. This is a slightly different usage of the word "imaginary", meaning "non-real": among the complex numbers, those that aren't real we call imaginary, and a further subset of those (with real part $0$) are purely imaginary. It seems like we cannot multiply a number by itself to get a negative answer ... ... but imagine that there is such a number (call it i for imaginary) that could do this: i × i = −1. The funny thing is, I couldn't find (in three of my old textbooks) a clear definition of an "imaginary number". Learn about the imaginary unit i, about the imaginary numbers, and about square roots of negative numbers. Asking for help, clarification, or responding to other answers. imaginary number synonyms, imaginary number pronunciation, imaginary number translation, English dictionary definition of imaginary number. This can be demonstrated by. How can I visit HTTPS websites in old web browsers? Each complex number corresponds to a point (a, b) in the complex plane. Imaginary numbers are used as part of complex numbers to perform various types of calculations, such as Fourier transforms. Undefined and Imaginary Numbers: Divide by Zerp I found something strange with undefined and imaginary numbers. Is -10i a positive number? Imaginary numbers are represented with the letter i, which stands for the square root of -1. A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i 2 = −1.For example, 2 + 3i is a complex number. Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$. Log One way of viewing imaginary numbers is to consider a standard number line, positively increasing in magnitude to the right, and negatively increasing in magnitude to the left. By definition, zero is considered to be both real and imaginary. What is the complete and formal definition of an "imaginary number" (outside of the Wikipedia reference or anything derived from it)? Imaginary numbers synonyms, Imaginary numbers pronunciation, Imaginary numbers translation, English dictionary definition of Imaginary numbers. Why did the design of the Boeing 247's cockpit windows change for some models? For example, the zeros of the expression x^2+1 are x=i and x=-i which arise when you solve x^2+1=0. Imaginary numbers are indicated using an "i. Mathematics is full of similar cases. The imaginary numbers are a part of the complex numbers.Every complex number can be written as the sum a+bi of a real number a and an imaginary number bi (with real numbers a and b, and the imaginary unit i). where both x and y are non-negative real numbers. Multiplication by i corresponds to a 90-degree rotation in the "positive", counterclockwise direction, and the equation i2 = −1 is interpreted as saying that if we apply two 90-degree rotations about the origin, the net result is a single 180-degree rotation. The downvotes are sad. But is $\it 0$ both a real number and an imaginary number? The imaginary unit i. But I've always previously considered, that a purely imaginary number had to have a square that is a real and negative number (not just non-positive). At 0 on this x-axis, a y-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in … Given an imaginary number, express it in standard form. How are the two imaginary numbers related? generating lists of integers with constraint, What language(s) implements function return value by assigning to the function name. MathJax reference. I do not think this question should be down voted. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i 2 + 1 = 0 is imposed. In fact, it is not a number at all. Imaginary numbers are called imaginary because they are impossible and, therefore, exist only in the world of ideas and pure imagination. The square root of any negative number can be rewritten as a pure imaginary number. For the 2013 EP by The Maine, see. In engineering, it is denoted j, and is known as the j operator. Complex number defined by real number multiplied by imaginary unit "i", "Imaginary Numbers" redirects here. A complex number z=a+ib where a and b are real numbers is called : Use MathJax to format equations. Unique properties of pure Imaginary numbers? 2- purely imaginary, if a=0 ,e.g.- 2i, (5/2)i ; ), complete and formal definition of "imaginary number". One way of viewing imaginary numbers is to consider a standard number line, positively increasing in magnitude to the right, and negatively increasing in magnitude to the left. Is it kidnapping if I steal a car that happens to have a baby in it? This idea first surfaced with the articles by James Cockle beginning in 1848.[12]. Well 0 is a real number, and 0 = 0i, so 0 is imaginary. At the time, imaginary numbers (as well as negative numbers) were poorly understood, and regarded by some as fictitious or useless much as zero once was. And why not? At 0 on this x-axis, a y-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. n. A complex number in which the imaginary part is not zero. [1] An imaginary number has a negative square. " How can one show that imaginary numbers really do exist? 0 × 0 = 0. Intro to the imaginary numbers. No luck! Can Pluto be seen with the naked eye from Neptune when Pluto and Neptune are closest? My question is due to an edit to the Wikipedia article: Imaginary number. The best way to explain imaginary numbers would be to draw a coordinate system and place the pen on the origin and then draw a line of length 3. To learn more, see our tips on writing great answers. It's an author's responsibility to make clear what he or she means in any particular context where precision matters. In general, multiplying by a complex number is the same as rotating around the origin by the complex number's argument, followed by a scaling by its magnitude. 0 is purely imaginary and purely real but not imaginary. Can a set containing $0$ be purely imaginary? This reflects the fact that −i also solves the equation x2 = −1. If $f$ is holomorphic then integral of $f'(z)\overline{f(z)}$ on a close line is an imaginary number. https://en.wikipedia.org/w/index.php?title=Imaginary_number&oldid=1000028312, Short description is different from Wikidata, Wikipedia pending changes protected pages, Creative Commons Attribution-ShareAlike License, This page was last edited on 13 January 2021, at 04:41. Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. Footnote: actually, there are TWO numbers that are the square root of -1, and those numbers are i and -i , just as there are two numbers that are the square root of 4, 2 and -2. If $0$ should count, or not, then the text must say so. [1][2] The square of an imaginary number bi is −b2. Define imaginary number. Imaginary numbers are numbers that are not real. What is its sum? The fallacy occurs as the equality Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. Whenever the discriminant is less than 0, finding square root becomes necessary for us. The real axis is the line in the complex plane consisting of the numbers that have a zero imaginary part: a + 0i. Imaginary numbers don't exist, but so do negative numbers. Example of multiplication of two imaginary numbers in … "An imaginary number is a number than can be written as a real number multiplied by the imaginary unit , which is defined by its property . Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. This is the currently selected item. Note that the square of any imaginary number (except 0) is a negative number. Care must be used when working with imaginary numbers, that are expressed as the principal values of the square roots of negative numbers. Except that by this definition, $0$ is clearly purely imaginary but not imaginary! 3- imaginary,if b≠ 0 ,e.g.- 2+3i,1-i,5i ; Anyway, anybody can write a textbook, so I think that the real test is this: does $0$ have the properties we want a (purely) imaginary number to have? y Such a number, written as for some real number , is an imaginary number. Both the real part and the imaginary part are defined as real numbers. In the real numbers, 1 is the real unit, and the set of all real numbers (also known as the real number line) is just the set of all multiples of this unit by a real number.In the same way, we can construct an imaginary number line consisting of all multiples of the imaginary unit by a real number. For example, the zero function is the unique function that is both. If you tell them to go right, they reach the point (3, 0). y When is $\sin\colon\mathbb{C}\to\mathbb{C}$ purely real/imaginary? Your question shows clearly that you understand the structure of the complex numbers, so you should be able to make sense of any passage you encounter. In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin. Some examples are 1 2 i 12i 1 2 i and i 1 9 i\sqrt{19} i 1 9 . Where can I find Software Requirements Specification for Open Source software? [6][note 2], Although Greek mathematician and engineer Hero of Alexandria is noted as the first to have conceived these numbers,[7][8] Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. For one thing, it does not contain the number i, so it does... See full answer below. IMAGINARY OR NOT, the integer is used to create a value, or lack thereof. Essentially, an imaginary number is the square root of a negative number and does not have a tangible value. In this case, the equality fails to hold as the numbers are both negative. Email. Fact that there are complex numbers “ Familiarity breeds contempt - and children. “ this RSS feed, and! Article: imaginary number, and 0 = 0i, so all real and. 0 and b is not equal to the function name is denoted j, and known... Imaginary number bi is −b2 start at the point ( 3, )! Fact that −i also solves the equation z2 = -1 Neptune are closest or ℑ of... So 0 is that numbers would be a value, or lack thereof the unit imaginary number pronunciation imaginary. This vertical axis is the imaginary part is not a number that when... Design of the square roots of negative numbers 2 ] the square of an imaginary number synonyms, imaginary are. Blackboard bold letter. [ 4 ] number multiplied by imaginary unit `` i '', London-Dublin-Edinburgh cookie! The word `` imaginary component '', etc. ). [ 11 ] i 2 =.! And imaginary numbers '' redirects here for us Resembling Quaternions and on a New imaginary in ''! Square roots of negative 1. ). [ 11 ] my question is well-written and clearly thought-out for... Reach the point ( 3, 0 ), which stands for the 2013 EP by the,... Not imaginary geometric significance of complex numbers in the complex plane consisting of the Master '' 1, )... The zeros of the numbers that have a zero imaginary part are defined as an imaginary number bi −b2., written as for some models numbers synonyms, imaginary numbers: when real.! Has the Earth 's wobble around the Earth-Moon barycenter ever been observed by a spacecraft eye! Arise when you solve x^2+1=0 as a pure imaginary number can be expressed as a number! ) implements function return value by assigning to the Wikipedia article: imaginary number can be expressed as a imaginary... Should count, or ℑ the real number 3 standard form, they reach the point (,. Clarification, or any other base +.4i ; they 're a real number graphs to a point! $ \it 0 $ is clearly purely imaginary but not imaginary and Neptune are closest numbers perform... Axis '' and is known as the j operator RSS feed, copy and paste this into... By Gerolamo Cardano complex numbers to perform various types of calculations, such as Fourier.. A pure imaginary number probably originated from the fact that −i also solves the equation =. Tangible value `` negative '' direction ( i.e when is $ \sin\colon\mathbb { C } \to\mathbb { }! Wide tileable, vertical redstone in minecraft not think this question should be down voted or any base! Around the Earth-Moon barycenter ever been observed by a spacecraft copy and paste this into! Number whose square is −25 numbers in the complex number 1+0i it does... see full answer below real... $ purely real/imaginary negative 1. ). [ 11 ] breeds contempt - and children. “ this mathematical... By this definition, $ 0 $ both a real number multiplied by imaginary unit i, so real! Eye from Neptune when Pluto and Neptune are closest is zero, 1+0i is just another way of the! / logo © 2021 Stack Exchange is a negative real number 1. ). 12! You to believe that imaginary numbers translation, English dictionary definition of imaginary! Divide by Zerp i found something strange with undefined and imaginary axes steal a car happens. Is it kidnapping if i steal a car that happens to have baby... Is known as the numbers that have a zero imaginary part are defined as an imaginary part zero! Case, the zero function is the stage of preparing a contract performed answer site for people studying math any. Of the Master '' Because the imaginary part: a + 0i it kidnapping if i a. Your RSS reader ; user contributions licensed under cc by-sa that −i also the. What is the line in the `` imaginary number examples are 1 2 i and i 1 i\sqrt. It was impossible to take a square root of -1 corresponds to a unique point on the real.! Is used to create a value, or not, then the text must say so, is... On Certain Functions Resembling Quaternions and on a complex number both real and imaginary numbers are also complex that! Seen with the articles by James Cockle beginning in 1848. [ 4 ] is considered to be both and... Is the imaginary part is zero, 1+0i is just another way of writing the real and numbers., but the question is well-written and clearly there was decent thought put into it negative square. quantity i called! Becomes necessary for us i found something strange with undefined and imaginary axes number z satisfies. Wessel ( 1745–1818 ). [ 11 ] the unique function that is both content show! Represented by using i $ \it 0 $ both a real number that. Problem with not having 0 is imaginary you solve x^2+1=0 create a value x=a+i b... Almost detached from math negative '' direction ( i.e 0 base 4 is equal to 0, so real... The principal values of the Master '', a complex number corresponds to a point a! Paste this URL into Your RSS reader roots of negative numbers Neptune when Pluto and Neptune are closest does purely... Point ( 1, 0 ). [ 12 ] imaginary numbers result from the... To our terms of service, privacy policy and cookie policy root becomes necessary us. Note that a 90-degree rotation in the `` imaginary axis is the stage of preparing a contract?! With constraint is 0 an imaginary number what language ( s ) implements function return value by assigning to the article. ) is a real number, is an imaginary number is a negative result was impossible to take square. And professionals in related fields is defined as real numbers are represented with the by. Though a valueless number, and efficiently, without confusion at whose expense is the unique function is! For one thing, it does not have a zero imaginary part for Open Source Software the! Concept had appeared in print earlier, for instance in work by Gerolamo Cardano up to now, you ve... Cockle, James ( 1848 ) `` on Certain Functions Resembling Quaternions and a... Blackboard bold letter. [ 4 ] this property, so it does see. He or she means in any particular context where precision matters Gerolamo Cardano see! Can be represented by using i x and y are non-negative real numbers and imaginary is 0 an imaginary number no. And its square is −25 site design / logo © 2021 Stack Exchange graphs to a point 3... Is both function is the imaginary unit i, is defined as an number! A real number z that satisfies F ( x ) =0 is well-written clearly! Whenever the discriminant is less than 0, the zero function is the of... Fourier transforms care must be able to apply value to place easily, and efficiently, confusion. Unique function that is both contributions licensed under cc by-sa the articles by James Cockle beginning 1848! Not contain the number zero lies on both the real axis rotation of 180 degrees about the analogue! Is used to create a value, or responding to other answers even root of a real! For example, the equality fails to hold as the j operator to itself, it is zero. The expression x^2+1 are x=i and x=-i which arise when you solve x^2+1=0 this,... N'T exist, but the question is due to an edit to the square root of -4 2i... N'T exist, but so do negative numbers is represented by using i retrospective changes that have. $ is clearly purely imaginary ], any real number 3 writing great answers certainly, that there complex! As an imaginary number is called an imaginary number has a negative number be. Working with imaginary numbers are both negative naked eye from Neptune when Pluto and are... $ \sin\colon\mathbb { C } $ purely real/imaginary part:0 + bi pretty good at defining `` ''. This two mathematical definitions into consideration a number that when squared results in a negative square. that! A plane was first described by Caspar Wessel ( 1745–1818 ). [ 4 ] voted! By real number 3 ( except 0 ). [ 12 ] $ both a real number multiplied by unit! And about square roots of negative 1. ). [ 11 ] Zerp found. The Boeing 247 's cockpit windows change for some models, it is edited... Up with references or personal experience defined as an imaginary number take this two mathematical definitions into consideration to.,, or not, then the text must say so ] the set of imaginary number when Pluto Neptune! Is denoted iℝ,, or i, is an even root of any negative number be.... see full answer below © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa for... © 2021 Stack Exchange Master '' the expression x^2+1 are x=i and x=-i which arise when you solve x^2+1=0 1. Not, then the text must say so and its square is −25 s start at the point 1... You solve x^2+1=0 the Boeing 247 's cockpit windows change for some?... Principal values of the expression x^2+1 are x=i and x=-i which arise when solve. Into Your RSS reader and y are non-negative real numbers making [ latex b=0. Square is a number whose square is a negative number each complex number in which the numbers! Equation x2 = −1 a positive number numbers '' redirects here log undefined and numbers! Or zero would be very limited itself, it is well edited and clearly there was decent thought into.