What is complex number

The various operations of addition, subtraction, multiplication , division of natural numbers can also be performed for complex numbers also. The details of the various arithmetic operations of complex numbers are as follows. Th addition of complete numbers is similar to the addition of natural numbers.

Here in complex numbers, the real part is added to the real part and the imaginary part is added to the imaginary part. The complex numbers follow all the following properties of addition. The subtraction of complex numbers follows a similar process of subtraction of natural numbers. Here for any two complex numbers, the subtraction is separately performed across the real part and then the subtraction is performed across the imaginary part.

The multiplication of complex numbers is slightly different from the multiplication of natural numbers. The multiplication of complex numbers is polar form is slightly different from the above mentioned form of multiplication. Here the absolute values of the two complex numbers are multiplied and their arguments are added to obtain the product of the complex numbers. The division of complex numbers makes use of the formula of reciprocal of a complex number.

All the algebraic identities apply equally for complex numbers. The addition and subtraction of complex numbers and with exponents of 2 or 3 can be easily solved using algebraic identities of complex numbers. This equation is as same as the one we saw in the beginning of this page. Substitute these values in the quadratic formula:. Example 2: Express the sum, difference, product, and quotient of the following complex numbers as a complex number.

A complex number is a combination of real values and imaginary values. The complex number is used to easily find the square root of a negative number. Further to find the negative roots of the quadratic equation , we used complex numbers. If a complex number is considered as a vector representation in the argand plane, then the module of the complex number is the magnitude of that vector.

A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the set of ordinary real numbers can be thought of as a subset of the set of complex numbers. It is beneficial to think of the set of complex numbers as an extension of the set of real numbers.

Complex numbers are used in many scientific fields, including engineering, electromagnetism, quantum physics, and applied mathematics, such as chaos theory. Complex numbers allow for solutions to certain equations that have no real number solutions.

For example, the equation:. Complex numbers can be added and subtracted by adding the real parts and imaginary parts separately. Calculate the sums and differences of complex numbers by adding the real parts and the imaginary parts separately.

Complex numbers can be added and subtracted to produce other complex numbers. This is done by adding the corresponding real parts and the corresponding imaginary parts. Note that this is always possible since the real and imaginary parts are real numbers, and real number addition is defined and understood. Thus we would write:.

In a similar fashion, complex numbers can be subtracted. The key again is to combine the real parts together and the imaginary parts together, this time by subtracting them.

Thus you could write:. Note that it is possible for two non-real complex numbers to add to a real number. Because a complex number is a binomial — a numerical expression with two terms — arithmetic is generally done in the same way as any binomial, by combining the like terms and simplifying. For example:. For multiplication, you employ the FOIL method for polynomial multiplication: multiply the First, multiply the Outer, multiply the Inner, multiply the Last, and then add.

Division, however, becomes more complicated and requires using conjugates. Multiplying complex conjugates causes the middle term to cancel out. When dividing complex numbers, determine the conjugate of the denominator and multiply the numerator and denominator by the conjugate. For example,. Complex numbers behave exactly like two dimensional vectors.

Indeed real numbers are one dimensional vectors on a line and complex numbers are two dimensional vectors in a plane. The natural question to ask is 'could there be three dimensional numbers corresponding to three dimensional vectors or could there even be higher dimensional numbers? The answer is no. The only sets of numbers which satisfy all the usual rules of elementary algebra that is satisfy the field axioms have dimension one or two.

We can define division of complex numbers but we cannot define division of three dimensional vectors. There are no three dimensional or higher dimensional numbers obeying all the rules of elementary algebra.