NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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Operations with Negative Surds
Lesson

Writing numbers in complex form

To be able to write some numbers in complex form, a little algebraic manipulation may be necessary, mostly involving the fact that $\sqrt{-1}=i$1=i or that $-1=i^2$1=i2.  Let's have a look at some examples.

Example 1

Rewrite  $\sqrt{-50}$50 in the form $z=x+yi$z=x+yi

$\sqrt{-50}$50 $=$= $\sqrt{-1\times50}$1×50

using properties of surds

$\sqrt{ab}=\sqrt{a\times b}=\sqrt{a}\times\sqrt{b}$ab=a×b=a×b

  $=$= $\sqrt{-1}\times\sqrt{50}$1×50  
  $=$= $\sqrt{50}i$50i

and now simplify the $\sqrt{50}$50

$\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}$50=25×2=25×2=52

  $=$= $5\sqrt{2}i$52i  

So in the form $x+yi$x+yi, the real component is $0$0, ie. that $x=0$x=0 and the imaginary component is $y=5\sqrt{2}$y=52, so $x+yi=0+5\sqrt{2}i$x+yi=0+52i

 

Properties of Radical Numbers (surds)

Consider first the fact that a square root sign is actually an index.  It is the power half. 

$\sqrt{a}=a^{\frac{1}{2}}$a=a12

Now, also remember all the index laws.

$\left(ab\right)^n$(ab)n $=$= $a^n\times b^n$an×bn (1)
$\left(\frac{a}{b}\right)^n$(ab)n $=$= $\frac{a^n}{b^n}$anbn (2)

We can now look at what these rules look like with regards to square roots.

Let's start here with (1) $\left(ab\right)^n=a^n\times b^n$(ab)n=an×bn with $n=\frac{1}{2}$n=12

$\left(ab\right)^{\frac{1}{2}}$(ab)12 $=$= $a^{\frac{1}{2}}\times b^{\frac{1}{2}}$a12×b12 which is equivalent to
$\sqrt{ab}$ab $=$= $\sqrt{a}\times\sqrt{b}$a×b  

 

The other equation (2)  $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$(ab)n=anbn with $n=\frac{1}{2}$n=12 becomes 

$\left(\frac{a}{b}\right)^{\frac{1}{2}}$(ab)12 $=$= $\frac{a^{\frac{1}{2}}}{b^{\frac{1}{2}}}$a12b12 which becomes
$\sqrt{\frac{a}{b}}$ab $=$= $\frac{\sqrt{a}}{\sqrt{b}}$ab  

$$ which becomes $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ab=ab

 

This covers multiplication and division with square roots.  

Multiplication and Division with Square Roots

Multiplication of Square Roots $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ab=a×b

Division of Square Roots $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ab=ab

Now, what about addition and subtraction I hear you ask.  Well, there are no simplification laws for general addition and subtraction of square roots, unless the value in the radicand is the same.

Addition and Subtraction of Square Roots

Addition and subtraction only works if the value in the square root is identical. 

Thus

$a\sqrt{c}+b\sqrt{c}=\left(a+b\right)\sqrt{c}$ac+bc=(a+b)c

and

$a\sqrt{c}-b\sqrt{c}=\left(a-b\right)\sqrt{c}$acbc=(ab)c

 

But  $\sqrt{a}+\sqrt{b}$a+b is definitely NOT equal to $\sqrt{a+b}$a+b

 

When the square root value is negative

Combining together now our ability to convert $i^2=\sqrt{-1}$i2=1, and our knowledge of the properties of surds we can simplify a whole myriad of questions involving negative surds. 

Things to remember when doing this

  • Separate a value like $-49$49 in under the square root sign into two components, $-1\times49$1×49 and then take the square root of each component
  • Always simplify the surd as far as you can, look for square numbers that could be factors
  • Be careful when simplifying the product of 2 negative surds. 

Here is an example of what I mean by that last point

Example 2

Simplify $\sqrt{-2}\times\sqrt{-30}$2×30

$\sqrt{-2}\times\sqrt{-30}$2×30 $=$= $\sqrt{-1\times2}\times\sqrt{-1\times30}$1×2×1×30
$=$= $i\sqrt{2}\times i\sqrt{30}$i2×i30
$=$= $i^2\sqrt{2\times30}$i22×30
$=$= $-\sqrt{4\times15}$4×15
$=$= $-2\sqrt{15}$215
Watch out!

A very common error for students to make here is to assume that the negative 2 multiplied by the negative 30 would give positive 60 and hence they would end up with the answer of $2\sqrt{15}$215.  So deal with each $\sqrt{-1}$1 component separately.  

More Worked Examples

QUESTION 1

Express $\sqrt{-100}$100 in terms of $i$i.

QUESTION 2

Express $-\sqrt{-29}$29 in terms of $i$i.

QUESTION 3

Find the value of $\frac{\sqrt{-33}\times\sqrt{-3}}{\sqrt{11}}$33×311.

Outcomes

M8-9

Manipulate complex numbers and present them graphically

91577

Apply the algebra of complex numbers in solving problems

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