What Are Complex Numbers?

Click For Summary
SUMMARY

A complex number is defined as an ordered pair (a, b) that adheres to specific multiplication rules. The multiplication of two complex pairs (a, b) and (c, d) is calculated using the formula (a, b) x (c, d) = (a x c - b x d, a x d + b x c). Complex numbers are visualized on the complex plane, where the first axis represents real numbers and the second axis represents imaginary numbers. While complex numbers exhibit commutative and associative properties, they are not ordered, meaning one cannot determine if (a, b) is greater or lesser than (c, d), although their magnitudes can be compared.

PREREQUISITES
  • Understanding of ordered pairs and basic algebra
  • Familiarity with the concept of imaginary numbers
  • Knowledge of the complex plane visualization
  • Basic grasp of commutative and associative properties in mathematics
NEXT STEPS
  • Study the properties of complex numbers in detail
  • Learn about the geometric interpretation of complex numbers on the complex plane
  • Explore applications of complex numbers in engineering and physics
  • Investigate advanced topics such as complex number functions and transformations
USEFUL FOR

Students in mathematics, engineers working with signal processing, and anyone interested in the theoretical foundations of complex numbers will benefit from this discussion.

Ijjapwar
Messages
1
Reaction score
0
what is the defination of complex no?
 
Physics news on Phys.org
A complex number is an ordered pair which obeys a special set of rules for multiplication.

The usual visualization is the complex plane: the first axis is a real number, the second axis is an imaginary number.

Then given two complex pairs (a,b) and (c,d) the multiplication rule is:

(a,b) x (c,d) = (a x c - b x d, a x d + b x c), which is what you would get if you were to write it out as
(a + bj) x (c + dj) and treat jxj=-1, and regroup the resulting set of terms as real and imaginary as a pair.

The resulting algebra is commutative and associative, but the complex numbers are not "ordered" ... you cannot say that (a,b) is greater or lesser than (c,d), though you can determine the magnitudes (distance from the origin of the plane) ... then all complex numbers lying on the same circle have the same magnitude. Two complex numbers are equal if corresponding elements of each pair are equal.
 
Is that really so difficult to check wikipedia or google for the definition?
 

Similar threads

Undergrad Formal definition of multiplication for real and complex numbers
  • · Replies 2 ·
Replies
2
Views
2K
Insights Views On Complex Numbers
  • · Replies 108 ·
4
Replies
108
Views
11K
High School Is this a complex number at the second quadrant?
  • · Replies 7 ·
Replies
7
Views
2K
High School Getting from complex domain to real domain
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Is it valid to express a complex number as a vector?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Division of a complex number by zero
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Trouble with infinity and complex numbers
  • · Replies 12 ·
Replies
12
Views
2K
MHB Finding Real Part of $z$ for Complex Numbers
  • · Replies 1 ·
Replies
1
Views
1K
High School Principal square root of a complex number, why is it unique?
  • · Replies 13 ·
Replies
13
Views
6K
High School How can complex numbers be elevated to complex powers?
  • · Replies 12 ·
Replies
12
Views
3K