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A complex number is written as a + bi where a and b are real numbers an i, called the imaginary unit, has the property that i2 = -1. The complex numbers z = a + bi and z = a - bi are called complex conjugate of each other.
-1. The solutions to this equation (x =+ i) cannot be represented by a real number. Complex numbers have many applications in applied mathematics, physics and engineering. A complex number can be thought of as a two dimensional vector (a,b), where a is the real part and b is the imaginary part. The term "imaginary" is
Q Is. v. ?1 a number? A from your Kindergarten teacher Not a REAL number. Daniel Chan (UNSW). Chapter 3: Complex Numbers. Semester 1 2018. 2 / 48 Cool Formula. Let z = a + bi ? C (with a, b ? R of course). We define the conjugate of z to be. ?z = a ? bi. Daniel Chan (UNSW). Chapter 3: Complex Numbers.
usefulness, was proved by Gauss in 1799. From the quadratic formula (1) we know that all quadratic equations can be solved using complex numbers, but what Gauss was the first to prove was the much more general result: Theorem 5 (FUNDAMENTAL THEOREM OF ALGEBRA) The roots of any polynomial equation.
arg (z1) – arg (z2). 5.2.2 Solution of a quadratic equation. The equations ax2 + bx + c = 0, where a, b and c are numbers (real or complex, a ? 0) is called the general quadratic equation in variable x. The values of the variable satisfying the given equation are called roots of the equation. The quadratic equation ax2 + bx + c
Consider now the quadratic equation x2 2 x 10 ? 0. ? b2 ? 4 ac¤ x ?. ¦ 2 зийййййййййййййй. 4 40. 2. 2. 36. 2. 2. 2. 6. 2 j !" 1 # 3 j. This leads us t oa general form for complex numbers z $ a % bj a and b arereal a & real part of z & Re ' z( b ) imaginary part of z ) Im 0 z(. In the previous example; roots 1 z. 12 1 3 3 j, hence a 45 1
The equation x2 + 1 = 0 has no solutions, because for any real number x the square x2 is nonnegative, and so x2 + 1 can never be less than 1. In spite of this it turns out to be very useful to assume that there is a number i for which one has. (1) i2 = ?1. Any complex number is then an expression of the form a + bi, where a
Important Concepts and Formulas of Complex Numbers, Rectangular(Cartesian) Form, Cube Roots of Unity, Polar and Exponential Forms, Convert from Rectangular Form to Polar Form and Exponential Form, Convert from Polar Form to Rectangular(Cartesian) Form, Convert from Exponential Form to
Properties of real numbers. • Solution of linear and quadratic equations. • Representation of a real number on the number line. • Representation of point in a plane. 1.1 COMPLEX NUMBERS. Consider the equation x2 + 1 = 0. (A). This can be written as x2 = –1 or. 1 x ?+=. But there is no real numbers which satisfy x2 = –1.
An important property enjoyed by complex numbers is that every com- plex number has a square root: THEOREM 5.2.1. If w is a non–zero complex number, then the equation z2 = w has a so- lution z ? C. Proof. Let w = a + ib, a, b ? R. Case 1. Suppose b = 0. Then if a > 0, z = va is a solution, while if a < 0, iv?a is a
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