Complex Roots
Complex roots are the imaginary root of quadratic or polynomial functions. These complex roots are a form of complex numbers and are represented as α = a + ib, and β = c + id. The quadratic equation having a discriminant value lesser than zero (D<0) have imaginary roots, which are represented as complex numbers.
Complex roots have a real part and an imaginary part, and the formula of i2 = -1 is useful for the computation of complex roots. Let us learn more about the properties and operations of complex roots, with the help of examples, FAQs.
| 1. | What Are Complex Roots? |
| 2. | Properties Of Complex Roots |
| 3. | Operations On Complex Roots |
| 4. | Examples On Complex Roots |
| 5. | Practice Questions |
| 6. | FAQs On Complex Roots |
What Are Complex Roots?
Complex roots are the imaginary roots of quadratic equations which have been represented as complex numbers. The square root of a negative number is not possible and hence we transform it into a complex number. The quadratic equations having discriminant values lesser than zero b2 - 4ac < 0, is transformed using i2 = -1, to obtain the complex roots. Here -D is written as i2D.
Complex roots are expressed as complex numbers a + ib. The complex root is made up of a real part and an imaginary party. The complex root is often represented as Z = a + ib. Here 'a' is the real part of the complex number, which is denoted by Re(Z), and 'b' is the imaginary part, which is represented as I'm(Z). Here ib is the imaginary number.
In the imaginary part of the complex number, the alphabet 'i' is referred as iota. The iota - i is very useful to find the square root of any negative number. Here i2 = -1, and the negative number -N is represented as i2N, and it has now transformed into a positive number.
Properties Of Complex Roots
Magnitude of Complex Roots:
The complex root α = a + ib is represented as a point (a, +b) in the argand plane, and the distance of this point from the origin (0, 0) is called the modulus of the complex number. The distance is the simple linear distance, which is measured as r = |\(\sqrt{a^2 + b^2}\)|. This can be easily understood with the use of Pythagoras theorem, and here the modulus of the complex root is represented by the hypotenuse of the right triangle, the base is the real part, and the height is the imaginary part.
Argument of Complex Roots
The complex root can be represented in the argand plane as a point, and the line joining this point with the origin, makes an angle θ with the positive x-axis in the argand plane, which is referred to as the argument of the complex number. The argument of the complex root obtained from the inverse of the trigonometric tan of the imaginary part divided by the real part, which is equal to Argz (θ) = \(Tan^{-1}\frac{b}{a}\).
Polar Representation of Complex Roots
The complex root can be represented in the polar form with the help of the modulus and the argument of the complex number in the argand plane. The complex root α = a + ib can be represented in polar form as α = r(Cosθ + iSinθ). Here r is the modulus of the complex root and θ is the argument of the complex root. The modulus of the complex root is computed as (r = \sqrt{a^2 + n^2}\), and the argument of the complex root is computed using the formula θ = \(Tan^{-1}\frac{b}{a}\).Conjugate of Complex Roots
Reciprocal of Complex Roots
The division of one complex root with another complex root is possible with the help of the reciprocal of a complex root. The division of one complex root with another complex root is equal to the product of one complex root with the reciprocal of another complex root. The reciprocal of a complex root α = a + ib is \(α^{-1} = \dfrac{ 1 }{ a + ib} = \dfrac{a - ib}{a^2 + b^2} = \dfrac{a}{a^2 + b^2} + \dfrac{i(-b)}{a^2 + b^2}\). Further we can also understand that \(α \neq α^{-1}\).
Operations On Complex Roots
The complex roots of simple algebraic expressions can also be added, subtracted, multiplied, or divided similar to the normal normals. Let us check each of the operations in detail.
Addition Of Complex Roots
The complex roots can also be added similar to the addition of natural numbers. For complex roorts the real part is added to the real part, and the imaginary part is added to the imaginary part. The two complex roots α = a + ib, and β = c + id, on addition we have α + β = (a + c) + i(b + d) . The addition of complex roots follows the closure law, commutative law, associative law, additive identity, and additive inverse, similar to the normal numbers.
Subtraction Of complex Roots
The subtraction of complex roots is similar to the subtraction of normal numbers. Since the complex roots are complex numbers, the real part is subtracted from the real part, and the imaginary part is subtracted from the imaginary part, to find the resultant answer. The two complex roots α = a + ib, and β = c + id, on subtraction we have α - β = (a - c) + i(b - d) .
Multiplication Of Complex Roots
The multiplication of complex roots is slightly different from the multiplication of normal algebraic expressions. Here for multiplying the complex roots we use this important formula \(i^2 = -1\). The two complex numbers α = a + ib, and β = c + id, on multiplication we obtain α × β = (ca - bd) + i(ad + bc). The multiplication of complex roots is also possible in polar form. The multiplication of complex numbers follows all the properties of closure property, commutative property, associative property, and distributive property.
Division Of Complex Roots
The division of complex roots is equal to the multiplication of one complex root with the reciprocal of another complex root. The division of two complex roots α = a + ib, and β = c + id is \(\dfrac{α}{β } = (a + ib) × \dfrac{1}{(c + id) }= (a + ib) × \dfrac{(c - id)}{(c^2 + d^2)}\).
☛Related Topics
FAQs On Complex Roots
What Are Complex Roots In Equations?
Complex roots are the imaginary roots of equations, which are represented as complex numbers. The quadratic equations having discriminant value lesser than zero have imaginary or complex roots. The complex roots are of the form α = a + ib, and β = c + id and it has the real part and the imaginary part.
How Do You Find Complex Roots?
The complex roots of equations can be computed similar to the other roots of the equations through factorization or by using the formula. Further, we use the formula \(i^2 = -1\) in the process of computation of complex roots.
How Do You Know If An Equation Has Complex Roots?
The equation having complex roots are the equations having imaginary roots. The quadratic equations ax2 + bx + c = 0, having discriminant values lesser than zero (D=b2 - 4ac < 0) have complex roots.
What Is The Difference Between Real Roots And Complex Roots?
The real roots are made of real numbers, and the complex roots have a real part and an imaginary part. The imaginary part of the complex number is represented using i and we have the formula \(i^2 = -1\). A quadratic equation having a discriminant value positive or zero have real roots, and for discriminant value lesser than zero it has complex roots.