COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS.

Let us solve these equations.. There is no real number whose square is a negative number..

A COMPLEX NUMBER. The square root of -1 is denoted by i (IOTA)..

z = a + ib. The real part of z = a Re z = a. If a and b are real numbers, the number z = (a + ib ) is defined as a complex number ..

Solution:. (2 + 3i) + (3 + 4i). = (2 + 3) + i(3 + 4).

Solution:. (5 + 3i) – (2 + i). = (5 – 2) + i(3 – 1).

Solution:. (1 + 2i)(1 + 2i). = 1 + 2i + 2i + 4i2.

In this way, we find the repetition of the four values, i.e., i , -1, - i , 1..

Negative powers of i. In this way, we find the repetition of the same four values, i.e ., i , -1, - i , 1..

We can say that.

Solution:. (5 + 3i)2 = 52 + 2.5.(3i) + (3i)2. Express (5 + 3 i ) 2 in the form a + ib ..

Let z = a + ib be a complex number.. The Modulus and the Conjugate of a Complex Number.

MODULUS. The Modulus and the Conjugate of a Complex Number.

The multiplicative inverse of a non-zero complex number is given by.

Important Results. Here, z 1 and z 2 are two complex numbers..

Solution: Let z = 3 + 2 i. Example. Find the multiplicative inverse of 3 + 2 i ..

X. X /. Y. Y /. P( x,y ). The complex number x + iy can be represented geometrically as a unique point P( x,y ) in the XY-plane..

The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane ..

The x-axis and the y-axis in the Argand plane are called the real axis and the imaginary axis respectively..

X. X /. Y. Y /. O. P( x,y ). Q(x,-y). Geometrically, the complex number and its conjugate are the mirror image on the real axis..

X. X /. Y. Y /. |z|. The modulus of the complex number, x+iy is the distance between point P( x,y ) to the origin O(0,0)..

X. X /. Y. Y /. |z|. O. P( x,y ). M.

X. X /. Y. Y /. |z|. O. P( x,y ). M. NOTE: The modulus of the complex number is always positive..

X X / Y Y / r = |z| O P( x,y ) M . Polar Representation of a Complex Number.

X X / Y Y / r=|z| O P( x,y ) M . The polar form of a complex number.

On squaring and adding (1) and (2), we get. The modulus of a complex number.

 is called the argument (or amplitude) of z, which is denoted by arg z ..

X X / Y Y / r=|z| O P( x,y ) M .

X X / Y Y / r=|z| O P( x,y ) M .

Example. Represent the complex number z = 1+ i in the polar form..

Nature of Roots. D > 0. Nature of Roots of Quadratic Equations.

Quadratic Equations with Negative Discriminant. The quadratic equations with a negative discriminant have no real solution. Therefore, the roots are imaginary and are given by.

Solution: D = (-4) 2 – 4(2)(3) = 16 – 24 = -8. Therefore, the equation has no real solution. The roots are.

Following are the steps to find the square root:.

(iv) Find the value of (x 2 + y 2 ) and solve it with x 2 – y 2 ..

Solution:. On squaring both the sides, we get. x 2 – y 2 + 2ixy = 7 + 24i.

Since, (x 2 + y 2 ) 2 =(x 2 – y 2 ) 2 + (2xy) 2 = 7 2 + (24) 2.

Thus, the square root of 7 +24i are 4+ 3i and –4–3i..

Now, on your FINGER TIPS…. Complex numbers: A number of the form x + iy , where x and y are real numbers, is defined as a complex number. Algebra of complex numbers: It deals with mathematical operations such as addition, subtraction, multiplication, division on two complex numbers. Conjugate of a complex number: Let z = a + ib be a complex number. Then the conjugate of z is denoted by and is equal to a - ib ..

Modulus of a complex number: The modulus of a complex number z = a + ib is denoted by | z | and defined as The argand plane and the polar representation: The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane. The polar representation of a complex number will be given as.

If D > 0, the roots are real and distinct. If D = 0, the roots are real and equal. If D < 0, the roots are imaginary and are given by.

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