Square Root of a Negative Number

We have seen that taking the square root of a negative number is not possible in the Real number system. The set of real numbers is the union of the set of rational numbers and the set of irrational numbers and can be shown on a real number line. In order to take the square root of a negative number, we have to consider a new number system. But, we really need only one new number to start working with the square roots of negative numbers. That number is the square root of negative one, [latex]\sqrt{-1}[/latex] which we define to be the imaginary number [latex]i[/latex].

Another way of saying this is that [latex]{{i}^{2}}=-1[/latex]. This is because [latex]\sqrt{i^2}=i[/latex] and [latex]i=\sqrt{-1}[/latex] imply that [latex]\sqrt{i^2}=\sqrt{-1}[/latex] and, consequently, [latex]i^2=-1[/latex].

The number [latex]−1[/latex] allows us to work with roots of negative Real numbers, not just [latex]\sqrt{-1}[/latex]. There are two important square root rules to remember: [latex]\sqrt{-1}=i[/latex], and [latex]\sqrt{ab}=\sqrt{a}\sqrt{b}[/latex] for. We will use the latter rule to rewrite the square root of a negative number as the square root of a positive number times [latex]\sqrt{-1}[/latex], then rewrite[latex]\sqrt{-1}[/latex] as [latex]i[/latex].

Notice that [latex]\sqrt{-4}=2i[/latex] and not [latex]\pm2i[/latex]. This is because the radical sign [latex]\sqrt{}[/latex] represents the principal square root, which is always ≥ 0.

The only way to end up with a negative answer is if there is a negative sign in front of the radical.

Imaginary Numbers

The set of imaginary numbers is defined as the set of all numbers of the form [latex]ai[/latex], where [latex]a[/latex] is a real number.  [latex]2i,\; -7i,\; 9658i,\; \frac{3}{4}i,\; -\frac{9}{5}i,\; \sqrt{3}\,i[/latex] are all imaginary numbers. When there is a square root in front of the [latex]i[/latex], it is best to write the [latex]i[/latex] in front of the radical: [latex]i\sqrt{3}[/latex] so that it is obvious that the [latex]i[/latex] is not under the radical. In set builder notation, the set of imaginary numbers is defined as [latex]\{ai\; \large | \normalsize \;\;a\;\in\mathbb{R}\}[/latex]. The set of imaginary numbers is NOT a subset of the set of real numbers.

To rewrite the radicals as imaginary numbers, we used the product rule for square roots: [latex]\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}[/latex]. This rule works for all real numbers [latex]a[/latex] and [latex]b[/latex]. However, be aware that this rule only works backwards [latex]\left (\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}\right )[/latex] when [latex]a[/latex] and [latex]b[/latex] are greater than or equal to zero. In other words, [latex]\sqrt{a}\cdot \sqrt{b}\neq\sqrt{ab}[/latex] if [latex]a[/latex] or [latex]b[/latex] are negative.

For example, [latex]\sqrt{-4}\cdot\sqrt{-9}\neq\sqrt{-4\cdot (-9)}[/latex], because [latex]\sqrt{-4}\cdot\sqrt{-9}=2i\cdot 3i=6i^2=6\cdot (-1)=-6[/latex] and [latex]\sqrt{-4\cdot (-9)}=\sqrt{36}=6[/latex].

Complex Numbers

The set of real numbers and the set of imaginary numbers have only one number in common: zero. Zero can be written as [latex]0[/latex] in the set of real numbers and [latex]0i[/latex] in the set of imaginary numbers. [latex]0i=0[/latex]. When we union the set of real numbers with the set of imaginary numbers, we get a new set of numbers called the complex numbers.

A complex number is expressed in standard form when written [latex]a+bi[/latex] where [latex]a[/latex] is the real part and [latex]bi[/latex] is the imaginary part. For example, [latex]5+2i[/latex] is a complex number. So, too, is [latex]3+4i\sqrt{3}[/latex].

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number, while the square of a real number (either positive or negative) is a positive real number. Complex numbers are a combination of real and imaginary numbers.

In a number with a radical as part of b, such as [latex]-\frac{3}{5}+i\sqrt{2}[/latex], [latex]i[/latex] should be written in front of the radical. Although writing this number as [latex]-\frac{3}{5}+\sqrt{2}i[/latex] is technically correct, it makes it much more difficult to tell whether [latex]i[/latex] is inside or outside of the radical. Putting it before the radical, as in [latex]-\frac{3}{5}+i\sqrt{2}[/latex], clears up any confusion.

By making [latex]b=0[/latex], any real number can be expressed as a complex number. The real number [latex]a[/latex] is written as [latex]a+0i[/latex] in complex form. Similarly, any imaginary number can be expressed as a complex number. By making [latex]a=0[/latex], any imaginary number [latex]bi[/latex] can be written as [latex]0+bi[/latex] in complex form.

These examples show that the set of real numbers is a subset of the complex numbers and the set of imaginary numbers is a subset of the complex numbers.

[latex]\mathbb{R}\subset\mathbb{C}\text{ and }\mathbb{I}\subset\mathbb{C}[/latex]

The video, shows more examples of how to write complex numbers.

Standard Form

The standard form of a complex number is [latex]a+bi[/latex]. So, [latex]\frac{5}{2}+\frac{3}{2}i[/latex] is a complex number written in standard form. However, if we used the common denominator to add the fractions, [latex]\frac{5+3i}{2}[/latex] is NOT in standard form. If a complex number is the numerator of a fraction, we can simplify the fraction to write the answer in standard form by splitting up the fractions.

The same technique can be used when there are radicals involved.