What Are Complex Numbers?

In summary, a complex number is an ordered pair that follows specific rules for multiplication and can be visualized on a complex plane. The multiplication rule involves treating the imaginary number as j and results in a pair of real and imaginary terms. The algebra is commutative and associative, but complex numbers cannot be ordered. Two complex numbers are considered equal if their corresponding elements are equal. It is recommended to check Wikipedia or Google for a more detailed definition.
  • #1
Ijjapwar
1
0
what is the defination of complex no?
 
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  • #2
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.
 
  • #3
Is that really so difficult to check wikipedia or google for the definition?
 

FAQ: What Are Complex Numbers?

What are complex numbers?

Complex numbers are numbers that consist of a real part and an imaginary part. They are written in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (equal to the square root of -1).

How are complex numbers different from real numbers?

Complex numbers have both a real and an imaginary part, while real numbers only have a single real part. Additionally, complex numbers follow different rules for addition, subtraction, multiplication, and division compared to real numbers.

What is the geometric interpretation of complex numbers?

Complex numbers can be represented graphically on a two-dimensional plane called the complex plane. The real part is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis. This allows for visualizing operations on complex numbers, such as addition and multiplication.

What are some applications of complex numbers in science?

Complex numbers have many applications in science, including in electrical engineering, quantum mechanics, and fluid dynamics. They are also used in signal processing, control systems, and image processing.

How are complex numbers used in everyday life?

Complex numbers are used in many areas of everyday life, such as in computer graphics and animation, GPS technology, and financial modeling. They are also used in music theory to represent sound waves and in sports to calculate trajectory and motion.

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