What are complex numbers and how do they differ from real numbers?

In summary, complex numbers are defined as pairs of real numbers with specific rules for addition and multiplication. They cannot be ordered like real numbers, and the concept of "positive" and "negative" does not apply. The imaginary number, i, is defined as (0,1) and is the square root of -1 when multiplied by itself.
  • #1
newton1
152
0
actually what is complex number...
i know it's root of the -1
but how can we imagine the kind of number exist??
 
Last edited by a moderator:
Mathematics news on Phys.org
  • #2
Please read this article:
http://physicspost.com/articles.php?articleId=118
 
Last edited:
  • #3
Of course, you can't DEFINE i as "square root of -1", not because -1 doesn't have a square root, but because, like any number, it has TWO.

Silly "paradoxes" like: i= [sqrt](-1) so i*i= i2= [sqrt](-1)*[sqrt](-1)= [sqrt](-1*-1)= [sqrt](1)= 1 depend on that ambiguity.

When we are working in the real numbers, we can specify sqrt[x] as meaning the POSITVE root. In complex numbers, we don't have any way of distinguishing "positive" or "negative" (the complex numbers cannot be an ordered field).

The way complex numbers are properly defined is as PAIRS of real numbers (a,b) with addition defined as (a,b)+ (c,d)= (a+b, c+d) and multiplication defined as (a,b)*(c,d)= (ac-bd,ad+bc). It then follows that numbers of the form (a,0) act like real numbers while (0,1)*(0,1)= (0*0-1*1,0*1+1*0)= (-1,0). If we identify (0,1) with i (having dodged the question of how to distinguish between roots), we can write any complex number as (a, b)= a+ bi and have i*i= (-1,0)= -1.
 

FAQ: What are complex numbers and how do they differ from real numbers?

What is a complex number?

A complex number is a number that consists of a real and an imaginary part. It can be written in the form a + bi, where a is the real part and bi is the imaginary part, with i representing the square root of -1.

How is a complex number represented?

A complex number can be represented in various forms, such as rectangular form (a + bi), polar form (r(cosθ + isinθ)), or exponential form (re^iθ). Each form has its own advantages and is used in different applications.

What are the properties of complex numbers?

Complex numbers have many properties, including addition, subtraction, multiplication, and division. They also follow the associative, commutative, and distributive laws, making them useful in solving equations and representing mathematical relationships.

What is the purpose of using complex numbers?

Complex numbers are used in many fields of science, engineering, and mathematics. They are particularly useful in electrical engineering, quantum mechanics, and signal processing. They are also used to solve equations that involve imaginary numbers and to represent real-world phenomena such as alternating currents and harmonic oscillations.

How are complex numbers related to the real world?

Although complex numbers may seem abstract, they have many real-world applications. They are used to model and analyze systems and phenomena that involve both real and imaginary quantities, such as electrical circuits, sound waves, and quantum mechanics. They also help us understand the behavior of natural phenomena, such as the motion of planets and the growth of populations.

Similar threads

Replies
7
Views
2K
Replies
3
Views
1K
Replies
7
Views
3K
Replies
13
Views
2K
Replies
1
Views
960
Replies
8
Views
2K
Back
Top