\n 1. What is an Imaginary Number?<\/a>\n
\n\t 2. Imaginary Number Symbol<\/a>\n\t
\n\t 3. Powers of Imaginary Numbers<\/a>\n
\n 4. Imaginary & Complex Numbers<\/a><\/font>\n <\/div>\n
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Before learning about complex numbers, it’s key to learn specifically about imaginary numbers first, including the symbol for imaginary numbers.\n On this Page: 1. What is an Imaginary Number? 2. Imaginary Number Symbol 3. Powers of Imaginary Numbers 4. Imaginary & Complex Numbers Before learning about complex numbers, it’s key to learn specifically about imaginary numbers first, including the symbol for imaginary numbers. Complex and imaginary numbers aren’t quite exactly the same thing, but they… Read More »Complex and Imaginary Numbers,
\nComplex and imaginary numbers aren’t quite exactly the same thing, but they are best introduced together in Math.<\/p>\n
\n<\/a>\n\n\n\nWhat is an Imaginary Number?<\/span><\/h2>\n\n\n\nImaginary numbers appear when we are to deal with the square root of a negative number.\n
\nNow normally we think that we can’t have a square root of a negative number, because a number multiplied by itself, either negative or positive, results in a positive number.\n
\nFor example, 3<\/b> × 3<\/b> = 9<\/b> , –3<\/b> × –3<\/b> = 9<\/b> etc.\n
\nSo \\bf{\\sqrt{9}}<\/span> = +<\/u> 3<\/b>.\n
\nBut we don’t seem to be able to find a result for \\bf{\\sqrt{{\\text{-}}9}}<\/span>.\n
\nThis is where imaginary numbers come in.\n
\n<\/a>\n\n\n\nSymbol for Imaginary Numbers<\/span><\/h3>\n\n\n\nThere is a special number we can use, represented by the letter i<\/i><\/font><\/font>.\n
\nWhich has the value: i<\/i><\/b><\/font> = \\bf{\\sqrt{{\\text{-}}1}}<\/span>.\n
\nThis i<\/i><\/font><\/font> now enables us to find \\bf{\\sqrt{{\\text{-}}9}}<\/span>.\n
\n\\bf{\\sqrt{{\\text{-}}9}} \\space = \\space {\\sqrt{9 \\times {\\text{-}}1}} \\space = \\space {\\sqrt{9}}{\\sqrt{{\\text{-}}1}}<\/span> = <\/font>\\bf{\\sqrt{9}}<\/span>i<\/i><\/b><\/font>\n
\n\\bf{\\sqrt{9}}<\/span>i<\/i><\/b><\/font> = 3<\/b>i<\/i><\/b><\/font> , –3<\/b>i<\/i><\/b><\/font>\n
\nGeneral Case: {\\sqrt{{\\text{-}}a}}<\/span> = {\\sqrt{a}}<\/span>i<\/i><\/b><\/font>\n
\nThough sometimes we can write as i<\/i><\/b><\/font>{\\sqrt{a}}<\/span>, which is also perfectly fine as the order doesn’t matter.<\/font>\n
\n<\/a>\n\n\n\nPowers of Imaginary Numbers:<\/span><\/h3>\n\n\n\nSomething interesting happens when we raise i<\/i><\/font><\/font> to a power and multiply it with itself.\n
\ni<\/i><\/b><\/font><\/font>1<\/font><\/sup><\/sup> = i<\/i><\/b><\/font><\/font>\n
\ni<\/i><\/b><\/font><\/font>2<\/font><\/sup><\/sup> = i<\/i><\/b><\/font><\/font> × i<\/i><\/b><\/font><\/font> = \\bf{\\sqrt{{\\text{-}}1}}\\times{\\sqrt{{\\text{-}}1}}<\/span> = –1<\/b> \n
\ni<\/i><\/b><\/font><\/font>3<\/font><\/sup><\/sup> = –1<\/b> × i<\/i><\/b><\/font><\/font> = –i<\/i><\/b><\/font><\/font>\n
\ni<\/i><\/b><\/font><\/font>4<\/font><\/sup><\/sup> = –i<\/i><\/b><\/font><\/font> × i<\/i><\/b><\/font><\/font> = –\\bf{\\sqrt{{\\text{-}}1}}\\times{\\sqrt{{\\text{-}}1}}<\/span> = -( (\\bf{\\sqrt{{\\text{-}}1}}\\times{\\sqrt{{\\text{-}}1}}<\/span> ) = -( –1<\/b> ) = 1<\/b>\n
\nSo we have:\n
\ni<\/i><\/b><\/font><\/font>1<\/font><\/sup><\/sup> = i<\/i><\/b><\/font><\/font> , i<\/i><\/b><\/font><\/font>2<\/font><\/sup><\/sup> = –1<\/b> , i<\/i><\/b><\/font><\/font>3<\/font><\/sup><\/sup> = –i<\/i><\/b><\/font><\/font> , i<\/i><\/b><\/font><\/font>4<\/font><\/sup><\/sup> = 1<\/b>.\n
\nWhat’s interesting is that this pattern continues over and over every 4 powers.\n
\nSo i<\/i><\/b><\/font><\/font>5<\/font><\/sup><\/sup> = i<\/i><\/b><\/font><\/font> , i<\/i><\/b><\/font><\/font>6<\/font><\/sup><\/sup> = –1<\/b> ……. and so on<\/font>.\n
\n<\/a>\n\n\n\n
\n\n\n\nComplex and Imaginary Numbers<\/span><\/h2>\n\n\n\n
\nImaginary numbers in Math enable us to create what are known as ‘complex numbers’.\n
\nWhich are numbers made up of a real part and an imaginary part.\n
\n\n\n\n![\"\"](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/complex-and-imaginary-numbers.png\")
<\/figure>\n\n\n\n
\n4<\/b> + 5i<\/i><\/b><\/font><\/b> is an example of a complex number.\n
\nBut technically, any real number can be thought of as a complex number.\n
\nFor example if we have the number 4<\/b>.\n
\nThis can be thought of as 4<\/b> + 0i<\/i><\/b><\/font><\/b>.\n
\nThe imaginary part just happens to be 0<\/font>, which leaves us with just the number 4<\/font>.\n
\n\n\n\n
\n\n\n\nUses and History of Complex Numbers<\/span><\/h2>\n\n\n\n
\nComplex and imaginary numbers have been around for a few hundred years now.\n
\nWhile at first they did primarily have a use just in Math. In more recent times they have become important in other scientific fields, such as studying electrical currents.\n
\nA good article on the history of imaginary numbers can be seen here<\/i><\/a> at the sciencefocus.com website.\n
\nWhile an informative article on some of the applications of complex and imaginary numbers can be viewed here<\/i><\/a> at the issuu website. \n
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