Prentice Hall's Online Lesson Quiz Enter Web Code: aga–0506
In the set of real numbers, negative numbers do not have square roots. Recall that an equation like x2 = –4 has no solution. This section introduces us to a new kind of number, called imaginary, so that negative numbers would have square roots and certain equations like the example above would have solutions.
In order to create a new set of numbers, some starting unit or definition was needed. The imaginary unit, named i , with the following understanding or agreement that, by definition:
and
As mathematicians, we assume that i acts like a real number in all other respects. Square roots of all negative numbers can now be expressed as a product of i and a real number.
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The first four powers of i are key elements in finding (simplifying) higher powers of i. It is important to be able to quickly recall these values.
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To multiply imaginary numbers or an imaginary number by a real number, it
is important first to express the imaginary numbers in terms of i.
Then, multiply complex numbers as you would multiply monomials or binomials,
treating the imaginary parts as like terms. Of course, remember
.
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The complex number system defines sums of real and imaginary numbers.
Definition of Complex Numbers
The complex numbers consist of all sums a + bi. where a and b are real numbers and i is the imaginary unit, a is called the real part and bi is called the imaginary part.
Every real number a is a complex number because a = a + 0i. All imaginary numbers bi are also complex because bi = 0 + bi. Therefore, the complex numbers are just an extension of the real number system.
We assume that i acts like a real number, obeying the commutative, associative, and distributive laws. Thus to add or subtract complex numbers we can treat i as we would treat a variable. We combine like terms.
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When you solve equations, answers may involve pure imaginary numbers.
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Equality for complex numbers is based on equality for real numbers. A sentence a + bi = c + di says that a + bi and c + di are two names for the same number. In order for this to be true, a and c must be the same and b and d must be the same. Therefore, the definition follows:
Definition of Equal Complex Numbers
a + bi = c + di if and only if a = c and b = d
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Find the values of m and n that make the equation true:
