# Section 1.3 - Gordon State College

Section 1.3 Complex Numbers; Quadratic Equations in the Complex Number System IMAGINARY NUMBERS Definition: The number , called the imaginary

unit, is the number such that COMPLEX NUMBERS If and are real numbers and is the imaginary unit, then is called a complex number. The real number is called the real part and the real number is called the imaginary part.

COMPLEX NUMBER ARITHMETIC If and are complex numbers, then Addition: Subtraction:

Multiplication: COMPLEX CONJUGATES The complex numbers and are called complex conjugates or conjugates of each other. The conjugate of a complex number is denoted by .

EXAMPLES: PRODUCT OF CONJUGATES Theorem: The product of a complex number and its conjugate is a nonnegative real number. That is, if , then

DIVIDING COMPLEX NUMBERS To perform the division we multiply the numerator and denominator by the conjugate of the denominator. Then simplify the complex number into standard form.

PROPERTIES OF CONJUGATES The conjugate of the conjugate is the number itself. The conjugate of a sum is the sum of the conjugates. The conjugate of a product is the product of the conjugates.

POWERS OF i And so on. The powers of repeat with every fourth power. PRINCIPAL SQUARE ROOT

OF N Definition: If is a positive real number, we define the principle square root of N, denoted by , as where is the imaginary unit and . THE QUADRATIC FORMULA

In the complex number system, the solutions of the quadratic equation , where , , and are real numbers and , are given by the formula CHARACTER OF THE SOLUTIONS OF A QUADRATIC EQUATION In the complex number system, consider a

quadratic equation with real coefficients. 1. If , there are two unequal real solutions. 2. If , there is a repeated real solution, a double root. 3. If , the equation has two complex solutions that are not real. These solutions are conjugates of each other.

## Recently Viewed Presentations

• Success. 179 pre-primary centres running in project areas. 17 Grade 1 Grade II & III running in Khagrachari. 124 teacher trained (women 83%) 120 Centre Management Committees (CMC) established
• Lessons from Noah & The Flood. Questions about the flood: How could Noah have gotten so many different animals on the ark? Are there any witnesses outside of Scripture? Where is the scientific evidence that such a flood occurred? It...
• Universal Tools. Universal tools are access features of the assessment that are either provided as digitally-delivered components of the test administration system or separate from it.. Available to all students. Embedded (in the test system) and Non-Embedded (not in the...
• Sir William Preece, sjefsingeniør i British Post Office, 1878 : "The Americans have need of the telephone, but we do not. We have plenty of messenger boys."
• Preconscious - Bridge between conscious and unconscious. Thoughts are stored in the unconscious and brought to the conscious through the preconscious mind. Unconscious according to Freud, a reservoir of mostly unacceptable thoughts, wishes, feelings, and memories.
• "The rise of the West transformed the world… the rise of Asia will bring about an equally significant transformation."Professor Kishore MahbubaniThe New Asian Hemisphere: the irresistible shift of global power to the East, 2008. Rapid economic and population growth in...
• crescendo-decrescendo relates directly to and connotes a pattern of highs and lowsin observable behaviors with clearly defined criteria. Relationships. an extension of Aristotle's 'tragic hero' paraphrased in today's language as "the bigger they are, the harder they fall."
• It is more tempting to break the fast if you live in a non Muslim country! Taking part in PE without hydration is hard. The Night of Power. This festival marks the beginning of the revelation to Muhammad. Observing it...