A Most Charming Overview of Complex Numbers

The latest blog, is about nothing else but complex numbers!! Complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies і² = -1. Every complex number can be written in the form a + bi, where a and b are real numbers called the real part and the imaginary part. Complex numbers were first conceived and defined by the Italian mathematician Girolamo Cardano, who called the numbers "fictitious" during attempts to find solutions to cubic equations. Complex numbers are a field and therefore have addition, subtraction, multiplacation and division operations. Complex numbers are used in many different areas of the work force, including applications of engineering, electromagnetism, quantum physics, applied mathematics and chaos theory. The set of all complex numbers is usually denoted by "C".

Examples of complex number equations are:

1. Check that both 3 + 2i and 3 - 2i satisfy the equation x^2 - 6x + 13 = 0

Use the quadratic formula to find the solutions. In this case, the values are a=1, b=-6, and c=13. where Substitute in the values of a=1, b=-6, and c=13. Multiply -1 by the -6 inside the parentheses. Simplify the section inside the radical. Simplify the denominator of the quadratic formula. First, solve the + portion of Simplify the expression to solve for the + portion of the x=3+2i Next, solve the - portion of Simplify the expression to solve for the - portion of the x=3-2i The final answer is the combination of both solutions. x=3+2i,3-2i

2. Solve and Check a) x^2 +4x +5 =0

Use the quadratic formula to find the solutions. In this case, the values are a=1, b=4, and c=5. where Substitute in the values of a=1, b=4, and c=5. Simplify the section inside the radical. Simplify the denominator of the quadratic formula. First, solve the + portion of Simplify the expression to solve for the + portion of the x=-2+i Next, solve the - portion of Simplify the expression to solve for the - portion of the x=-2-i The final answer is the combination of both solutions. x=-2+i,-2-i

b) x^2 -2x +3 =0

Use the quadratic formula to find the solutions. In this case, the values are a=1, b=-2, and c=3. where Substitute in the values of a=1, b=-2, and c=3. Multiply -1 by the -2 inside the parentheses. Simplify the section inside the radical. Simplify the denominator of the quadratic formula. First, solve the + portion of Simplify the expression to solve for the + portion of the Next, solve the - portion of Simplify the expression to solve for the - portion of the The final answer is the combination of both solutions. x = 1 +i√2 -i√2

3. Find two numbers with a sum of 2 and a product of 2.

x + y =2 xy=2 y= x-2 x(x-2) = 2

x^2 -2x +2= 0

x= [-b ± sqrt(b^2 -4ac)]/2a ---> x = -(-2) ± sqrt([-2]^2 - 4(1)(2)])/2(1)

Multiply -1 by the -2 inside the parentheses. Simplify the section inside the radical. Simplify the denominator of the quadratic formula. First, solve the + portion of Simplify the expression to solve for the + portion of the x=1+i Next, solve the - portion of Simplify the expression to solve for the - portion of the x=1-i The final answer is the combination of both solutions x = 1 +i ; 1-i


I used Puplemath.com and Wikipedia to assist me in this blog entry. Also pages 246 and 247 in the Mathematics 11 textbook for British Columbia.