complex numbers

product

(a+bi)(c+di) is defined to be (ac−bd)+(bc+ad)i represents the______.

difference

(a+bi)−(c+di) is defined to be (a+bi)+(−c−di) represents the______.

Complex numbers

A complex number is a number which can be written in the form a+bi where a and b are real numbers and i=−1−−−√ Definition: The set of Complex Numbers is the set of all numbers which can be written in the form a+bi where a and b are real numbers and i=−1−−−√ Standard Symbol: C Symbolic Form: C={a+bi∣∣a∈R,b∈R,and i=−1−−−√}

Reducing

If the numerator and the denominator of a fraction contain common factors, the fraction is equivalent to a fraction without some of those common factors. Removing some of these common factors is called reducing the fraction The process of reducing a fraction depends on the definition of multiplication as illustrated below. abac=aabc=1bc=bc Obsserve that reducing a fraction produces a new fraction which is equivalent to the original fraction.

Polynomial p Divisible by Polynomial d

If the remainder r in the division algorithm is 0, then we say that the _______ __ ___ ___ ___ ___ _______ __

Difference of Complex Numbers

The _____ (a+bi)−(c+di) is defined to be (a+bi)+(−c−di)

Multipicitavie Inverse of a Complex Number

The ______ a+bi is its conjugate divided by its norm a−bi/a2+b2

norm

The ______ of a complex number a+bi is a2+b2.

conjugate

The ______ of a complex number a+bi is a−bi.

Quotient of Complex Numbers

The _______ (a+bi)÷(c+di) is defined to be (a+bi)(c−di/c2+d2)

sum of two complex numbers

The _______ a+bi and c+di is defined by (a+bi)+(c+di)=(a+c)+(b+d)i

multiplicative inverse

The _______ of a complex number a+bi is its conjugate divided by its norm a−bia2+b2

Product of Complex Numbers

The ________ (a+bi)(c+di) is defined to be (ac−bd)+(bc+ad)i

complex unit or the imaginary unit.

The complex number i is sometimes called the

binomials

The product of two complex numbers is best computed as the product of two______.

Reciprocal

The reciprocal of a fraction ab with numerator not zero is constructed by interchanging numerator and denominator. So the reciprocal of ab is ba The reciprocal of a fraction is important because it is the multiplicative inverse of the fraction. Note that abba=1

Norm of a Complex Number

The_____ a+bi is a2+b2.

quotient

The______ (a+bi)÷(c+di) is defined to be (a+bi)(c−di/c2+d2)

equal

Two complex numbers a + bi and c + di are _______ if a = c and b = d.

opposite

a complex number a+bi is −a−bi represents the ______.

complex number

a number that can be written in the form a+bi where a and b are real numbers and i=√−1

sum of two complex numbers

a+bi and c+di is defined by (a+bi)+(c+di)=(a+c)+(b+d)i

Conjugate of a Complex Number

a+bi is a−bi

Opposite of a Complex Number

a+bi is −a−bi.

-1

i2=

real component

of the complex number a+bi is a