Complex Number Formula
Complex Number Formula
Complex Number Formula can be used to calculate the square root of a negative number. Hero of Alexandria, a Greek mathematician, first mentioned Complex Number Formula in the first century when attempting to find the square root of a negative number. However, he simply changed the negative into a positive and took the numeric root value. Furthermore, in the 16th century, Italian mathematician Gerolamo Cardano defined the real identity of a Complex Number Formula while searching for the negative roots of cubic and quadratic polynomial expressions.
Many scientific fields use Complex Number Formula, including signal processing, electromagnetism, fluid dynamics, quantum mechanics, and vibration analysis. Here students can learn about the definition, terminology, visualization, properties, and operations of Complex Number Formula.
A Complex Number Formula is the result of adding a real and an imaginary number. A complex number has the formula a + ib and is usually represented by the symbol z. Both a and b are real numbers in this case. The value ‘a’ is known as the real part, which is denoted by Re(z), and the value ‘b’ is known as the imaginary part im (z). ib is also known as an imaginary number.
What are Complex Numbers?
The letter is known as the iota, and it is used to represent the imaginary part of a Complex Number Formula. Furthermore, the iota(i) function is very useful for calculating the square root of negative numbers. Students have the value of i2 = -1, which is used to calculate the value of -4 = i24 = +2i. The fundamental aspect of a Complex Number Formula is the value of i2 = -1. The powers of i are given below
i = √-1
i2 = -1
i3 = i.i2 = i(-1) = -i
i4 = (i2)2 = (-1)2 = 1
i4n = 1
i4n + 1 = i
i4n + 2 = -1
i4n + 3 = -i
Graphing of Complex Numbers
The Complex Number Formula is made up of a real and an imaginary part, which can be represented as coordinate points in the Euclidean plane as an ordered pair (Re(z), Im(z)). The complex plane, named after Jean-Robert Argand, is the Euclidean plane with reference to Complex Number Formula. The complex number z = a + ib is represented by the real part – a on the x-axis and the imaginary part – ib on the y-axis. Let students try to understand two key terms related to Complex Number Formula representation in the Argand plane.
Modulus of the Complex Number
The modulus of the Complex Number Formula is the distance of the complex number represented as a point in the argand plane (a, ib). This distance is measured as a linear distance from the origin (0, 0) to the point (a, ib).
This is also derived from Pythagoras’ theorem, where the modulus represents the hypotenuse, the real part is the base, and the imaginary part is the altitude of the right-angled triangle.
Argument of the Complex Number
The complex number argument is the angle formed by the line connecting the geometric representation of the Complex Number Formula and the origin with the positive x-axis in the anticlockwise direction. The complex number’s argument is the inverse of the tan of the imaginary part divided by the Complex Number Formula real part.
Polar Representation of a Complex Number
With the modulus and argument of a complex number, as well as its representation in the Argand plane, students have a new type of complex number representation known as the polar form of a complex number. In polar form, z = r ( Cos θ + iSin θ ) represents the complex number z = a + ib.
Properties of a Complex Number
The following Complex Number Formula qualities will help students better comprehend complex numbers and execute various arithmetic operations on them.
Conjugate of a Complex Number
The complex number’s conjugate is generated by taking the identical real component of the complex number and altering the imaginary part to its additive inverse. When the sum and product of two Complex Number Formula are both real numbers, they are referred to as conjugate complex numbers.
The Complex Number Formula and its conjugate are added, and the product of both complex numbers is
Reciprocal of a Complex Number
The reciprocal of a complex number is useful for dividing one complex number by another complex number. Complex number division is equal to the product of one complex number and the reciprocal of another complex number.
Equality of Complex Numbers
It is not feasible to rank complicated numbers. Complex Number Formula cannot be sorted, while real numbers and other related number systems may. Complex numbers lack the structure of an ordered field, and there is no ordering of complex numbers that are addition and multiplication compatible. In addition, the non-trivial sum of squares in an ordered field is a number not equal to 0, although, in a complex number, it is equivalent to i2 + 12 = 0. The magnitude of Complex Number Formula, which is their distance from the origin, may be measured and represented in a two-dimensional Argand plane.
Ordering of Complex Numbers
According to Euler’s formula, for any real value, eiθ = Cosθ + iSinθ represents the complex number in the coordinate plane, where Cos is the real part and is represented with respect to the x-axis, Sin is the imaginary part and is represented with respect to the y-axis, θ is the angle made with respect to the x-axis, and is the imaginary line that connects the origin and the Complex Number Formula.
Operations on Complex Numbers
Natural number operations such as addition, subtraction, multiplication, and division may also be done on complex numbers. The following are the specifics of complex number arithmetic procedures.
Addition of Complex Numbers
Complex Number Formula addition is comparable to natural number addition. In Complex Number Formula, the real component is multiplied by the real part, while the imaginary part is multiplied by the imaginary part. For two complex numbers of the type z 1 = a +id and z 2 = c +id, the sum of complex numbers z 1 + z 2 = ( a + c ) +i( b + d ). All of the following additional characteristics apply to complex numbers.
- The sum of two complex numbers is also a complex number, according to the Closure Law.
- The product of two complex numbers, z 1 and z 2, is itself a complex number.
- Commutative Law states that for two complex integers z 1 and z 2, z 1 + z 2 = z 2 + z 1.
- Associative Law: For the given three complex numbers z 1, z 2, and z 3, students obtain z 1 + ( z 2 + z 3 ) = ( z 1 + z 2 ) + z 3.
- Additive Inverse: There exists a complex number -z = -a -ib for the complex number z = a + ib such that z + (-z) = (-z) + z = 0. In this case, -z is the additive inverse.
Subtraction of Complex Numbers
The procedure of subtracting complex numbers is identical to subtracting natural numbers. In this case, for any two complex numbers, the subtraction is done individually over the real component and then across the imaginary part. For the complex numbers z 1 = a + ib and z 2 = c +id, students obtain z 1 z 2 = (a – c) +i(b – d)
Multiplication of Complex Numbers
The multiplication of complex numbers differs somewhat from the multiplication of natural numbers. In this case, students must use the formula i2 = 1. The product of the two complex numbers
z 1 = a + ib and z 2 = c + id
z 1. z 2 = (ca – bd) + i(ad + bc)
Complex number multiplication in polar form differs somewhat from the previously discussed method of multiplication. To obtain the product of the complex numbers, multiply the absolute values of the two complex numbers by their arguments.
Division of Complex Numbers
Algebraic Identities of Complex Numbers
For complex numbers, all algebraic identities are true. The addition and subtraction of complex numbers with exponents of 2 or 3 may be readily solved using complex number algebraic identities.
Complex Numbers Tips and Tricks:
All real numbers are complex numbers, but not all complex numbers are real numbers.
All imaginary numbers are complex numbers, however not all complex numbers must be imaginary.
The conjugate of a complex number z = a + ib is z = a -ib.
Complex Numbers Examples
One of the most crucial concepts that students must master is how to complete the Square Formula. The chapter has significant ramifications and is one of the most often utilized strategies.
Mathematics is one of the most essential topics that kids study in school. Students frequently understand the subject, which is generally due to considerable study for the subject. Students cannot grasp the material immediately, and they must regularly practise these ideas and chapters. Students can only master the subject and the Complex Number Formula by regularly practising the problems.
Practice Questions on Complex Numbers
If pupils create a habit of returning to the mathematical exercises and solving the problems on a frequent basis. Students generally read the chapter and answered questions to understand more about the Complex Number Formula. After finishing this procedure, students begin working on the exercise connected to the Complex Number Formula. There are several formulae related to Complex Number Formula, and memorizing all of the subtle phases is tough. When students have completed the problems associated with Complex Number Formula, they usually go on to the following chapter.
FAQs (Frequently Asked Questions)
1. Complex Numbers
In general, the problems in the Complex Number Formula exercises are more difficult than in other chapters. And Complex Number Formula has far-reaching effects. Students should be particularly attentive and meticulous when answering these chapters since the principles they acquire in Complex Number Formula repeat more in later chapters.
On the Extramarks website, students may get any information connected to the Complex Number Formula. Extramarks just debuted its mobile application, which has made a significant influence on the academic world. All of the content on the Extramarks website and mobile app was created by highly experienced specialists with years of relevant experience educating students about Complex Number Formula. As a result, students may be certain that whatever information they access on the website or the mobile application, which has been launched across several platforms, is correct and free of faults and inaccuracies.