A complex number is a number that contains two parts: a real part and an 'imaginary' part.^{[1]} The real part is any real number. The imaginary part is a real number multiplied by the imaginary unit written as the lowercase letter 'i'. The imaginary unit i represents .
Complex numbers are typically written in the form . where a is the real part and bi is the imaginary part. can also be written or
An imaginary number is a complex number with no real part, such as or . This is sometimes called a pure imaginary number.
Classifying NumbersFor each number, decide which class or classes to which each number belongs. Click 'yes' or 'no' under each class of numbers. A number may belong to more than one class. 

Number  Complex  Real  Pure Imaginary 

66i  yes noYes is correct. 66i has a real part (6) and an imaginary part (6i).No is incorrect. 66i has a real part (6) and an imaginary part (6i).  yes noYes is incorrect. 6i is the imaginary part. A real number has no imaginary part so 66i is not a real number.No is correct. 6i is the imaginary part. A real number has no imaginary part so 66i is not a real number.  yes noYes is incorrect. 66i has a real part (6). An imaginary number has no real part so 66i is not a imaginary number.No is correct. 66i has a real part (6). An imaginary number has no real part so 66i is not a imaginary number. 
5  yes noYes is correct. 5 has a real part (5). The imaginary part is (0i). All real numbers are also complex numbers.No is incorrect. 5 has a real part (5). The imaginary part is (0i). All real numbers are also complex numbers.  yes noYes is correct. 5 is the real part. While the imaginary part can be considered to be 0i, a complex number with 0i is also a real number.No is incorrect. 5 is the real part. While the imaginary part can be considered to be 0i, a complex number with 0i is also a real number.  yes noYes is incorrect. 5 has a real part (5). An imaginary number has no real part so 5 is not a imaginary number.No is correct. 5 has a real part (5). An imaginary number has no real part so 5 is not a imaginary number. 
2i  yes noYes is correct. 2i has an imaginary part (2i). The real part is (0). All pure imaginary numbers are also complex numbers.No is incorrect. 2i has an imaginary part (2i). The real part is (0). All imaginary numbers are also complex numbers.  yes noYes is incorrect. 2i has an imaginary part. A real number has no imaginary part, so 2i can not be imaginary.No is correct. 2i has an imaginary part. A real number has no imaginary part, so 2i can not be real.  yes noYes is correct. 2i has an imaginary part (2i). Since it has no real part, 2i is an imaginary number.No is incorrect. 2i has an imaginary part (2i). Since it has no real part, 2i is an imaginary number. 
3+4x  yes noYes is incorrect. 3+4x is an expression, not a number.No is correct. 3+4x is an expression, not a number.  yes noYes is incorrect. 3+4x is an expression, not a number.No is correct. 3+4x is an expression, not a number.  yes noYes is incorrect. 3+4x is an expression, not a number.No is correct. 3+4x is an expression, not a number. 
Table 1: Classifying numbers 

Complex numbers can be represented as points on the complex plane.^{[2]} Like the Cartesian plane, the complex plane has two axes: the real axis and the imaginary axis. The real axis is the horizontal axis. The real axis is also the real number line, put into the complex plane. The vertical axis is the imaginary axis. The complex plane is also called an Argand diagram. In figure 1, find the point 4+6i, which is red. Note that it corresponds to 4 on the real axis and 6 on the imaginary axis. Each point is plotted the same way. If a number has no imaginary part, it is a real number, and is found on the real axis. In figure 1, 6 is a real number on the real axis. If a number is an imaginary number, it is found on the imaginary axis. The number 4i in figure 1 is an example of an imaginary number on the imaginary axis. 
 When simplifying expressions containing the imaginary constant i, it is important to know the values of the integral powers of i. To understand this, remember the definition of i: . Since the square root function is the inverse of the square function, . So, . Table 1 gives the values of the first five integral powers of i. Can you figure out the next three? Come up with your answer, then click on 'Click for answer' to check your answer. 

When adding complex numbers, add
corresponding
parts. Add the real parts together and the imaginary parts together:
a+bi + c+di = (a+c)+(b+d)i
Here are some examples:
3+2i + 13i = (3+1)+(23)i = 41i = 4i
4+1i + 2+2i = (42)+(1+2)i = 2+3i
To multiply complex numbers, cross multiply the parts:
×(a+bi)
×(c+di)
×(a·c + bi·c
×(a·c + a·di + bi·di
+ac + (ad + bc)i + bdi^{2}
It is important to remember that i^{2} = 1.
Substitute 1 for i^{2}
in the equation:
= ac + (ad + bc)i + bd·(1)
= ac + (ad + bc)i  bd·
= ac  bd + (ad + bc)i
Here is an example:
(13i)·(6+2i)
= 1·(6)+1·2i+(3)·(6)i+(3)·2i^{2}
= 6+2i+18i6i^{2}
= 6+20i6i^{2}
Substitute 1 for i^{2} in the
equation:
6+20i6i^{2}
= 6+20i6(1)
= 6+20i+6
= 6+6+20i
= 0+20i
= 20i
Property  Value  Example  Description 

Additive Identity  The additive identity is the number that, when added to any complex number, gives the original number unchanged. The additive identity for complex numbers is .  
Multiplicative Identity  The multiplicative identity is the number that, when multiplied by any complex number, gives the original number unchanged. The multiplicative identity for complex numbers is .  
Additive Inverse  The additive inverse of a complex number is the number that when added to the original number gives the additive identity. The additive identity of an arbitrary complex number is .  
Multiplicative Inverse  The multiplicative inverse of a complex number is the number that when
multiplied by the original number gives the multiplicative identity. The
multiplicative inverse of an arbitrary complex number
is
.  
Associate Property of Addition  The associative property of addition states that the order in which complex numbers are added does not matter.  
Commutative Property of Addition  The commutative property of addition states that addends of complex numbers can be switched without changing the result.  
Associate Property of Multiplication  The associative property of multiplication states that the order in which complex numbers are added does not matter.  
Commutative Property of Multiplication  The commutative property of multiplication states that multiplicands of complex numbers can be switched without changing the result.  
Table 1: Properties of complex numbers. 
Manipulative 2: Addition of complex numbers. Created with GeoGebra. 
Manipulative 3: Multiplication of complex numbers. Created with GeoGebra. 
Manipulative 4: Additive inverse of a complex number. Created with GeoGebra. 
Manipulative 5: Multiplicative inverse of a complex number. Created with GeoGebra. 
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