{"id":451,"date":"2015-10-26T19:08:50","date_gmt":"2015-10-26T19:08:50","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=451"},"modified":"2017-03-31T19:45:01","modified_gmt":"2017-03-31T19:45:01","slug":"express-square-roots-of-negative-numbers-as-multiples-of-%e2%80%89i","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/express-square-roots-of-negative-numbers-as-multiples-of-%e2%80%89i\/","title":{"raw":"Express square roots of negative numbers as multiples of \u2009i","rendered":"Express square roots of negative numbers as multiples of \u2009i"},"content":{"raw":"
\r\n

We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number<\/strong>. The imaginary number [latex]i[\/latex] is defined as the square root of negative 1.<\/p>\r\n\r\n

[latex]\\sqrt{-1}=i[\/latex]<\/div>\r\n

So, using properties of radicals,<\/p>\r\n\r\n

[latex]{i}^{2}={\\left(\\sqrt{-1}\\right)}^{2}=-1[\/latex]<\/div>\r\n

We can write the square root of any negative number as a multiple of i<\/em>. Consider the square root of \u201325.<\/p>\r\n\r\n

[latex]\\begin{cases} \\sqrt{-25}=\\sqrt{25\\cdot \\left(-1\\right)}\\hfill \\\\ \\text{ }=\\sqrt{25}\\sqrt{-1}\\hfill \\\\ \\text{ }=5i\\hfill \\end{cases}[\/latex]<\/div>\r\n

We use 5i\u00a0<\/em>and not [latex]-\\text{5}i[\/latex]\u00a0because the principal root of 25 is the positive root.<\/p>\r\n\r\n\r\n[caption id=\"attachment_2527\" align=\"aligncenter\" width=\"487\"]\"Showing Figure 1<\/b>[\/caption]\r\n

A complex number<\/strong> is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a\u00a0<\/em>+ bi<\/em>\u00a0where a<\/em>\u00a0is the real part and bi<\/em>\u00a0is the imaginary part. For example, [latex]5+2i[\/latex] is a complex number. So, too, is [latex]3+4\\sqrt{3}i[\/latex].\r\n<\/span><\/p>\r\n

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative\u00a0real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.<\/p>\r\n\r\n

\r\n

A General Note: Imaginary and Complex Numbers<\/h3>\r\n

A complex number<\/strong> is a number of the form [latex]a+bi[\/latex] where<\/p>\r\n\r\n