0 is purely imaginary and purely real but not imaginary. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Imaginary numbers are called imaginary because they are impossible and, therefore, exist only in the world of ideas and pure imagination. Intro to the imaginary numbers. If $f$ is holomorphic then integral of $f'(z)\overline{f(z)}$ on a close line is an imaginary 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., $I(z) \ne 0$. I can't (and MSE can't) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $|e^{z}| = 1$. The quantity i is called the unit imaginary number. 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. 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. [3] The set of imaginary numbers is sometimes denoted using the blackboard bold letter .[4]. $R(z) = 0$. No, 0 0 0 0 is not an imaginary number. Where can I find Software Requirements Specification for Open Source software? It's a useful term sometimes. Use MathJax to format equations. where both x and y are non-negative real numbers. Imaginary numbers are represented with the letter i, which stands for the square root of -1. 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. At whose expense is the stage of preparing a contract performed? By definition, zero is considered to be both real and imaginary. Imaginary numbers. n. A complex number in which the imaginary … Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. In engineering, it is denoted j, and is known as the j operator. Email. To learn more, see our tips on writing great answers. Mathematics is full of similar cases. Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. 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. 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? The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. 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? 0.1 × 0.1 = 0.01. For example, the square root of -4 is 2i. Well 0 is a real number, and 0 = 0i, so 0 is imaginary. fails when the variables are not suitably constrained. Why did the design of the Boeing 247's cockpit windows change for some models? What does children mean in “Familiarity breeds contempt - and children.“? 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. An imaginary number is a number that when squared results in a negative value. How can one show that imaginary numbers really do exist? Care must be used when working with imaginary numbers, that are expressed as the principal values of the square roots of negative numbers. Example of multiplication of two imaginary numbers in … For one thing, it does not contain the number i, so it does... See full answer below. If $0$ should count, or not, then the text must say so. So, a Complex Number has a real part and an imaginary part. Imaginary numbers are indicated using an "i. Complex number defined by real number multiplied by imaginary unit "i", "Imaginary Numbers" redirects here. 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. The downvotes are sad. y 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. x In fact, it is not a number at all. When is $\sin\colon\mathbb{C}\to\mathbb{C}$ purely real/imaginary? If you tell them to go right, they reach the point (3, 0). 2) The square root of -1, or i, is defined as an imaginary number. Except that by this definition, $0$ is clearly purely imaginary but not imaginary! Can a set containing $0$ be purely imaginary? x 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. 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. 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