Cartesian Form
What is an imaginary number?
- Up until now, when we have encountered an equation such as
we would have stated that there are “no real solutions”
- The solutions are
which are not real numbers
- The solutions are
- To solve this issue, mathematicians have defined one of the square roots of negative one as
; an imaginary number
- The square roots of other negative numbers can be found by rewriting them as a multiple of
- using
- using
What is a complex number?
- Complex numbers have both a real part and an imaginary part
- For example:
- The real part is 3 and the imaginary part is 4i
- For example:
- Complex numbers are often denoted by
- We refer to the real and imaginary parts respectively using
and
- We refer to the real and imaginary parts respectively using
- Two complex numbers are equal if, and only if, both the real and imaginary parts are identical.
- For example,
and
are not equal
- For example,
- The set of all complex numbers is given the symbol
What is Cartesian Form?
- There are a number of different forms that complex numbers can be written in
- The form z = a + bi is known as Cartesian Form
- a, b ∈
- This is the first form given in the formula booklet
- In general, for z = a + bi
- Re(z) = a
- Im(z) = b
- A complex number can be easily represented geometrically when it is in Cartesian Form
- Your GDC may call this rectangular form
- When your GDC is set in rectangular settings it will give answers in Cartesian Form
- If your GDC is not set in a complex mode it will not give any output in complex number form
- Make sure you can find the settings for using complex numbers in Cartesian Form and practice inputting problems
- Cartesian form is the easiest form for adding and subtracting complex numbers
Exam Tip
- Remember that complex numbers have both a real part and an imaginary part
- 1 is purely real (its imaginary part is zero)
- i is purely imaginary (its real part is zero)
- 1 + i is a complex number (both the real and imaginary parts are equal to 1)
Worked Example
a)
Solve the equation ![x squared equals negative 9](data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2223%22%20width%3D%2261%22%20wrs%3Abaseline%3D%2217%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsup%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsup%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E9%3C%2Fmn%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A'math143f4d31b04031e49f5eb18baba'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUNBwAAAVgAAAA6Z2x5ZoPi2VsAAAGUAAAA%2FGhlYWQQC2qxAAACkAAAADZoaGVhCGsXSAAAAsgAAAAkaG10eE2rRkcAAALsAAAADGxvY2EAHTwYAAAC%2BAAAABBtYXhwBT0FPgAAAwgAAAAgbmFtZaBxlY4AAAMoAAABn3Bvc3QB9wD6AAAEyAAAACBwcmVwa1uragAABOgAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACAD0iEv%2F%2FAAAAPSIS%2F%2F%2F%2FxN3wAAEAAAAAAAAAAAFUAywAgAEAAFYAKgJYAh4BDgEsAiwAWgGAAoAAoADUAIAAAAAAAAAAKwBVAIAAqwDVAQABKwAHAAAAAgBVAAADAAOrAAMABwAAMxEhESUhESFVAqv9qwIA%2FgADq%2FxVVQMAAAIAgADrAtUCFQADAAcAZRgBsAgQsAbUsAYQsAXUsAgQsAHUsAEQsADUsAYQsAc8sAUQsAQ8sAEQsAI8sAAQsAM8ALAIELAG1LAGELAH1LAHELAB1LABELAC1LAGELAFPLAHELAEPLABELAAPLACELADPDEwEyE1IR0BITWAAlX9qwJVAcBV1VVVAAEAgAFVAtUBqwADADAYAbAEELEAA%2FawAzyxAgf1sAE8sQUD5gCxAAATELEABuWxAAETELABPLEDBfWwAjwTIRUhgAJV%2FasBq1YAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%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%3D%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2217%22%3Ex%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2213.5%22%20y%3D%2212%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math143f4d31b04031e49f5eb18baba%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2225.5%22%20y%3D%2217%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22math143f4d31b04031e49f5eb18baba%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2242.5%22%20y%3D%2217%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2255.5%22%20y%3D%2217%22%3E9%3C%2Ftext%3E%3C%2Fsvg%3E)
b)
Solve the equation
, giving your answers in Cartesian form.
Complex Addition, Subtraction & Multiplication
How do I add and subtract complex numbers in Cartesian Form?
- Adding and subtracting complex numbers should be done when they are in Cartesian form
- When adding and subtracting complex numbers, simplify the real and imaginary parts separately
- Just like you would when collecting like terms in algebra and surds, or dealing with different components in vectors
How do I multiply complex numbers in Cartesian Form?
- Complex numbers can be multiplied by a constant in the same way as algebraic expressions:
- Multiplying two complex numbers in Cartesian form is done in the same way as multiplying two linear expressions:
- This is a complex number with real part
and imaginary part
- The most important thing when multiplying complex numbers is that
- Your GDC will be able to multiply complex numbers in Cartesian form
- Practise doing this and use it to check your answers
- It is easy to see that multiplying more than two complex numbers together in Cartesian form becomes a lengthy process prone to errors
- It is easier to multiply complex numbers when they are in different forms and usually it makes sense to convert them from Cartesian form to either Polar form or Euler’s form first
- Sometimes when a question describes multiple complex numbers, the notation
is used to represent each complex number
How do I deal with higher powers of i?
- Because
this can lead to some interesting results for higher powers of i
- We can use this same approach of using i2 to deal with much higher powers
- Just remember that -1 raised to an even power is 1 and raised to an odd power is -1
Exam Tip
- When revising for your exams, practice using your GDC to check any calculations you do with complex numbers by hand
- This will speed up using your GDC in rectangular form whilst also giving you lots of practice of carrying out calculations by hand
Worked Example
a)
Simplify the expression
.
b)
Given two complex numbers
and
, find
.
Complex Conjugation & Division
When dividing complex numbers, the complex conjugate is used to change the denominator to a real number.
What is a complex conjugate?
- For a given complex number
, the complex conjugate of
is denoted as
, where
- If
then
- You will find that:
is always real because
- For example:
- For example:
is always imaginary because
- For example:
- For example:
is always real because
(as
)
- For example:
- For example:
How do I divide complex numbers?
- To divide two complex numbers:
- STEP 1: Express the calculation in the form of a fraction
- STEP 2: Multiply the top and bottom by the conjugate of the denominator:
- This ensures we are multiplying by 1; so not affecting the overall value
- STEP 3: Multiply out and simplify your answer
- This should have a real number as the denominator
- STEP 4: Write your answer in Cartesian form as two terms, simplifying each term if needed
- OR convert into the required form if needed
- Your GDC will be able to divide two complex numbers in Cartesian form
- Practise doing this and use it to check your answers if you can
Exam Tip
- We can speed up the process for finding
by using the basic pattern of
- We can apply this to complex numbers:
(using the fact that)
- So
multiplied by its conjugate would be
- So
Worked Example
Find the value of .