Understanding the Importance and Definition of Imaginary Numbers

The Letter I in Mathematical Language

In mathematics, i is an imaginary unit. Its square is equal to -1. This unit allows us to solve equations that are unsolvable in real numbers.

Ask any physicist or engineer, and they will tell you that almost nothing in physics works without imaginary numbers. But what exactly are they?

Definition

Imaginary numbers are a useful tool for mathematicians, because they allow us to take a complex number field, like the complex numbers, and make it a real algebra. So, for example, every polynomial of a given degree in this algebra can be represented as a product of its coefficients and the imaginary unit.

But how does a mathematician rigorously define this imaginary unit? There are several different ways, but perhaps the most common is to say that i and -i are quantitatively opposite, in the sense that they both have equal claim to squaring to be -1. This definition is often called the Argand plane.

Origin

The letter i represents the sound of a high front vowel similar to English long e. In Semitic it was a neutral consonant, but in Latin and the Romance languages it has largely assimilated to long e, so that in most words with this sound the i is almost invisible. In some Latin-derived nouns and some borrowings from Arabic, the i is pronounced as a separate consonant (e.g., tiya).

In chemistry the i symbol is used to represent the element iodine. The letter is also sometimes used as a diacritic, especially before vowels in handwritten manuscripts. The dot on the lowercase i first appeared in manuscripts around the 11th century to distinguish it from the stroke of another letter, facilitating reading in the crowded letters of medieval manuscripts.

A suffix forming nouns from adjectives that denote the state, quality, or measure of the adjective; a form of the plural inflection ending -i of Latin second declension and cognate with Greek – (see –). In Turkish, this forms a number of personal names and is found after certain foreign borrowings.

Symbol

In mathematics, the symbol i (or ) denotes an imaginary number—that is, a number that when squared gives -1. It is also the symbol for the iodine ion.

In phonology, I is the vowel sound that is the neutral vowel between e and o in English, but it has palatal affinities and (as in pique) a Continental sound. It forms many combinations with e, as in heid and believe; see also ie, oi, ui.

The i prefix has become a trademark of Apple products, such as the iPod, iPad, and iPhone, as a way to emphasize the devices’ internet connectivity and user-friendliness. It has also been adopted by other technology companies, such as Hewlett-Packard and Motorola, to designate products that are part of a new generation of smartphones. The i-band also represents an industry that is rapidly expanding, as more bands are starting to offer online music services to consumers. See I-generation, Internet. The i-generation refers to those who grew up during the time of the development of the Internet.

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