# is 0 an imaginary number

My question is due to an edit to the Wikipedia article: Imaginary number. This vertical axis is often called the "imaginary axis" and is denoted iℝ, , or ℑ. where both x and y are non-negative real numbers. 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. No luck! 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. Zero is still zero in any base. Cockle, James (1848) "On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra", London-Dublin-Edinburgh. Mathematics is full of similar cases. 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. Log Imaginary numbers synonyms, Imaginary numbers pronunciation, Imaginary numbers translation, English dictionary definition of Imaginary numbers. [1] An imaginary number has a negative square. " 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 can be demonstrated by. The imaginary axis is the line in the complex plane consisting of the numbers that have a zero real part:0 + bi. Some examples are 1 2 i 12i 1 2 i and i 1 9 i\sqrt{19} i 1 9 . fails when the variables are not suitably constrained. Better user experience while having a small amount of content to show. The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).[11]. Let’s start at the point (1, 0), which is represented by the complex number 1+0i. CCSS.Math: HSN.CN.A.1. Use MathJax to format equations. 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! The premise might seem silly, but the question is well-written and clearly thought-out. Complex number defined by real number multiplied by imaginary unit "i", "Imaginary Numbers" redirects here. Note that a 90-degree rotation in the "negative" direction (i.e. Imaginary numbers are called imaginary because they are impossible and, therefore, exist only in the world of ideas and pure imagination. Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$. 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. An imaginary number is a number that when squared results in a negative value. Imaginary numbers are used as part of complex numbers to perform various types of calculations, such as Fourier transforms. 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. "For example, 3 i is the imaginary analogue of the real number 3. y How can one show that imaginary numbers really do exist? [3] The set of imaginary numbers is sometimes denoted using the blackboard bold letter .[4]. 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). Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. 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). [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). Each complex number corresponds to a point (a, b) in the complex plane. 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. Such a number, written as for some real number , is an imaginary number. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Google Classroom Facebook Twitter. Well 0 is a real number, and 0 = 0i, so 0 is imaginary. Thanks for contributing an answer to Mathematics Stack Exchange! I can't (and MSE can't) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $|e^{z}| = 1$. 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. Are there any non-algebraic, non-transcendental complex numbers? But imaginary numbers are no less "real" than real numbers. It's a useful term sometimes. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. 3- imaginary,if b≠ 0 ,e.g.- 2+3i,1-i,5i ; The imaginary unit i. MathJax reference. Is -10i a positive number? What is its sum? It only takes a minute to sign up. 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. An imaginary number is a number that, when squared, has a negative result. Imaginary Numbers: When real numbers are multiplied to itself, it is guaranteed that the product is a positive number. The square root of any negative number can be rewritten as a pure imaginary number. Learn about the imaginary unit i, about the imaginary numbers, and about square roots of negative numbers. Every real number graphs to a unique point on the real axis. 2- purely imaginary, if a=0 ,e.g.- 2i, (5/2)i ; If you tell them to go right, they reach the point (3, 0). Care must be used when working with imaginary numbers, that are expressed as the principal values of the square roots of negative numbers. (Though they were pretty good at defining "imaginary component", etc.). How are the two imaginary numbers related? The real axis is the line in the complex plane consisting of the numbers that have a zero imaginary part: a + 0i. So, a Complex Number has a real part and an imaginary part. It is well edited and clearly there was decent thought put into it. In this case, the equality fails to hold as the numbers are both negative. Intro to the imaginary numbers. Is it kidnapping if I steal a car that happens to have a baby in it? Imaginary numbers result from taking the square root of a negative number. The sum of two well-ordered subsets is well-ordered. Email. The term "imaginary" probably originated from the fact that there is no real number z that satisfies the equation z2 = -1. 1- purely real , if b=0 ; e.g.- 56,78 ; 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. 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 … [1][2] The square of an imaginary number bi is −b2. Except that by this definition, $0$ is clearly purely imaginary but not imaginary! Is the union axiom really needed to prove existence of intersections? Example of multiplication of two imaginary numbers in … I understand that the number zero lies on both the real and imaginary axes. 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). But is $\it 0$ both a real number and an imaginary number? 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. Making statements based on opinion; back them up with references or personal experience. 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. (Because the imaginary part is zero, 1+0i is just another way of writing the real number 1.) 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. Undefined and Imaginary Numbers: Divide by Zerp I found something strange with undefined and imaginary numbers. No, 0 0 0 0 is not an imaginary number. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. a = 0 and b is not equal to 0, the complex number is called an imaginary number. But $0$ clearly has this property, so we should consider it purely imaginary. For one thing, it does not contain the number i, so it does... See full answer below. Asking for help, clarification, or responding to other answers. This reflects the fact that −i also solves the equation x2 = −1. Imaginary numbers. Does a purely imaginary number have a corresponding “angle” in polar coordinate system? Essentially, an imaginary number is the square root of a negative number and does not have a tangible value. It's an author's responsibility to make clear what he or she means in any particular context where precision matters. The question anyone would ask will be "where to" or "which direction". Example of a complex transcendental number? 1) The square root of a negative number is undefined. Except that by this definition, $0$ is clearly purely imaginary but not imaginary! 0 × 0 = 0. 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°). 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. What is the complete and formal definition of an "imaginary number" (outside of the Wikipedia reference or anything derived from it)? Complex numbers are numbers like 7 + .4i; they're a real number plus an imaginary number. The downvotes are sad. The problem with not having 0 is that numbers would be very limited. In fact, it is not a number at all. Imaginary numbers are not "impossible" numbers - they are very important mathematical entities. 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. "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 . ... By making $b=0$, any real number can be expressed as a complex number. This is the currently selected item. Any imaginary number can be represented by using i. 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? For example, the zero function is the unique function that is both. The imaginary unit i. 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. I like it. Always positive, or zero. Imaginary numbers don't exist, but so do negative numbers. First, please take this two mathematical definitions into consideration. If $f$ is holomorphic then integral of $f'(z)\overline{f(z)}$ on a close line is an imaginary number. Intro to the imaginary numbers. 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. (9.6.1) – Define imaginary and complex numbers. imaginary number synonyms, imaginary number pronunciation, imaginary number translation, English dictionary definition of imaginary number. 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. = 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. 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. 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. If $0$ should count, or not, then the text must say so. Has the Earth's wobble around the Earth-Moon barycenter ever been observed by a spacecraft? An imaginary number is an even root of a negative 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. The imaginary unit i. After 20 years of AES, what are the retrospective changes that should have been made? generating lists of integers with constraint, What language(s) implements function return value by assigning to the function name. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. 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. Intro to the imaginary numbers. Imaginary numbers are numbers that are not real. Why did the design of the Boeing 247's cockpit windows change for some models? y An imaginary number is a mathematical term for a number whose square is a negative real number. We know that the quadratic equation is of the form ax 2 + bx + c = 0, where the discriminant is b 2 – 4ac. 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. Given an imaginary number, express it in standard form. To learn more, see our tips on writing great answers. Intro to the imaginary numbers. 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. This idea first surfaced with the articles by James Cockle beginning in 1848.[12]. 0 base 4 is equal to 0 base 10, or any other base. $R(z) = 0$. The quantity i is called the unit imaginary number. A complex number z=a+ib where a and b are real numbers is called : x ), complete and formal definition of "imaginary number". This definition can be represented by the equation: i 2 = -1. Imaginary numbers are represented with the letter i, which stands for the square root of -1. What does children mean in “Familiarity breeds contempt - and children.“? Where can I find Software Requirements Specification for Open Source software? The fallacy occurs as the equality (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. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary. IMAGINARY OR NOT, the integer is used to create a value, or lack thereof. I do not think this question should be down voted. But then 0^2 = 0 is not negative. Google Classroom Facebook Twitter. When is $\sin\colon\mathbb{C}\to\mathbb{C}$ purely real/imaginary? Im>0? x 0.1 × 0.1 = 0.01. 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 engineering, it is denoted j, and is known as the j operator. Why do jet engine igniters require huge voltages? Can Pluto be seen with the naked eye from Neptune when Pluto and Neptune are closest? Here, i is equal to the square root of negative 1. Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$. You must be able to apply value to place easily, and efficiently, without confusion. What is the "Ultimate Book of The Master". Can a set containing $0$ be purely imaginary? We know certainly, that there are complex numbers that are neither purely real, nor purely imaginary. 0, though a valueless number, is actually quite great in importance. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. [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. Maximum useful resolution for scanning 35mm film. An imaginary number times 0 is 0. 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. Note that the square of any imaginary number (except 0) is a negative number. n. A complex number in which the imaginary … At whose expense is the stage of preparing a contract performed? 2) The square root of -1, or i, is defined as an imaginary number. clockwise) also satisfies this interpretation. Email. How can I visit HTTPS websites in old web browsers? For the 2013 EP by The Maine, see. The word "imaginary" might lead you to believe that imaginary numbers are essentially useless and almost detached from math. 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. Complex numbers in the form a + bi can be graphed on a complex coordinate plane. Up to now, you’ve known it was impossible to take a square root of a negative number. How to make one wide tileable, vertical redstone in minecraft. And why not? n. A complex number in which the imaginary part is not zero. Whenever the discriminant is less than 0, finding square root becomes necessary for us. {\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}} Imaginary numbers are indicated using an "i. This is the currently selected item. Is $0$ a pure imaginary number? An imaginary root or zero would be a value x=a+i*b in the complex plane that satisfies F(x)=0. For example, the square root of -4 is 2i. 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. In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin. Define imaginary number. For example:[13]. Both the real part and the imaginary part are defined as real numbers. 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. Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. 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). 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. The funny thing is, I couldn't find (in three of my old textbooks) a clear definition of an "imaginary number". Unique properties of pure Imaginary numbers?