Commutative Property of Multiplication of Complex Numbers

Here we will discuss about the commutative property of multiplication of complex numbers.

Commutative property of multiplication of two complex numbers:

For any two complex number z\(_{1}\) and z\(_{2}\), we have z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\).

Proof:

Let z\(_{1}\) = p + iq and z\(_{2}\) = r + is, where p, q, r and s are real numbers. Them

z\(_{1}\)z\(_{2}\) = (p + iq)(r + is) = (pr - qs) + i(ps - rq)

and z\(_{2}\)z\(_{1}\) = (r + is) (p + iq) = (rp - sq) + i(sp - qr)

             = (pr - qs) + i(ps - rq), [Using the commutative of multiplication of real numbers]

Therefore, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\)

Thus, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\) for all z\(_{1}\), z\(_{2}\) Ο΅ C.

Hence, the multiplication of complex numbers is commutative on C.

Examples on commutative property of multiplication of two complex numbers:

1. Show that multiplication of two complex numbers (2 + 3i) and (3 + 4i) is commutative.

Solution:

Let, z\(_{1}\) = (2 + 3i) and z\(_{2}\) = (3 + 4i)

Now, z\(_{1}\)z\(_{2}\) = (2 + 3i)(3 + 4i)

= (2 βˆ™ 3 - 3 βˆ™ 4) + (2 βˆ™ 4 + 3 βˆ™ 3)i

= (6 - 12) + (8 + 9)i

= - 6 + 17i

Again, z\(_{2}\)z\(_{1}\) = (3 + 4i)(2 + 3i)

= (3 βˆ™ 2 - 4 βˆ™ 3) + (3 βˆ™ 3 + 2 βˆ™ 4)i

= (6 - 12) + (9 + 8)i

= -6 + 17i

Therefore, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\)

Thus, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\) for all z\(_{1}\), z2 Ο΅ C.

Hence, the multiplication of two complex numbers (2 + 3i) and (3 + 4i) is commutative.

 

2. Show that multiplication of two complex numbers (3 - 2i) and (-5 + 4i) is commutative.

Solution:

Let, z\(_{1}\) = (3 - 2i) and z\(_{2}\) = (-5 + 4i)

Now, z\(_{1}\)z\(_{2}\) = (3 - 2i)(-5 + 4i)

= (3 βˆ™ (-5) - (-2) βˆ™ 4) + ((-2) βˆ™ 4 + (-5) βˆ™ (-2))i

= (-15 - (-8)) + ((-8) + 10)i

= (-15 + 8) + (-8 + 10)i

= - 7 + 2i

Again, z\(_{2}\)z\(_{1}\) = (-5 + 4i)(3 - 2i)

= ((-5) βˆ™ 3 - 4 βˆ™ (-2)) + (4 βˆ™ 3 + (-2) βˆ™ 4)i

= (-15 + 8) + (12 - 8)i

= -7 + 2i

Therefore, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\)

Thus, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\) for all z\(_{1}\), z\(_{2}\) Ο΅ C.

Hence, the multiplication of two complex numbers (3 - 2i) and (-5 + 4i) is commutative.




11 and 12 Grade Math 

From Commutative Property of Multiplication of Complex Numbers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 16, 25 02:45 AM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More

  2. Perimeter of a Figure | Perimeter of a Simple Closed Figure | Examples

    Jul 16, 25 02:33 AM

    Perimeter of a Figure
    Perimeter of a figure is explained here. Perimeter is the total length of the boundary of a closed figure. The perimeter of a simple closed figure is the sum of the measures of line-segments which hav…

    Read More

  3. Formation of Numbers | Smallest and Greatest Number| Number Formation

    Jul 15, 25 11:46 AM

    In formation of numbers we will learn the numbers having different numbers of digits. We know that: (i) Greatest number of one digit = 9,

    Read More

  4. 5th Grade Quadrilaterals | Square | Rectangle | Parallelogram |Rhombus

    Jul 15, 25 02:01 AM

    Square
    Quadrilaterals are known as four sided polygon.What is a quadrilateral? A closed figure made of our line segments is called a quadrilateral. For example:

    Read More

  5. 5th Grade Geometry Practice Test | Angle | Triangle | Circle |Free Ans

    Jul 14, 25 01:53 AM

    Name the Angles
    In 5th grade geometry practice test you will get different types of practice questions on lines, types of angle, triangles, properties of triangles, classification of triangles, construction of triang…

    Read More