Introduction to Complex Numbers |
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The imaginary number i |
In real numbers, we notice that √3, √5, √7 etc. are irrational numbers because 3,5,7 are not perfect square numbers like 4, 9, 16.
What is the square root of -9?
Since the square of every real number must be non-negative there is no real number where its square is negative.
We call the square root of -1, i where ‘i’ is called an imaginary number.
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| then -1 = i2 |
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With this new convention
√-9 = ∓ 3i |
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| We now constitute a new set of numbers called complex numbers. The set is denoted by C and we write |
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We can now solve equations of the form
x2+ 9 = 0
x2 + 7 = 0
x2+ 15 = 0 etc. |
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| The complex numbers are written is the form a + ib |
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Example |
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| In general, we write z = a + ib. |
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Complex numbers in depth |
Definition |
| Any expression of the form a+ ib where a, b represent real numbers is called a complex number. The number ‘a’ is called the real part of the complex number, and the number ‘b’ is called the imaginary part of the complex number. |
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Any complex number can be written as an ordered pair. So
z = a +ib = (a,b)
If in a complex number
a = 0, b ≠ 0 then
z = ib = (0,b) |
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| This is a purely imaginary number. |
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If in a complex number
a ≠ 0, b = 0 then
z = a = (a, 0)
then the number is purely real.
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We can write b = Im (z) in the first case
and a = Re (z) in the second case.
where Imz is imaginary z and Re(z) is real z. |
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Definition |
Two complex numbers are equal provided the real parts are equal and the imaginary parts are equal.
If z1 = a1 + ib1 and
z2= az2 + ibz2 |
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Then z1 = z2 if and only if
a1 = a2
b1 = b2 |
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Definition |
Two complex numbers differing only in the signs of the imaginary parts are called complex conjugates. Each is the conjugate of the other.
That is if z = a + ib is a complex number its complex conjugate
is
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= a - ib. |
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Examples
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1. |
z = 6 + 2i, |
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= 6 - 2i |
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2. |
z = -6 - 2i, |
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= -6 + 2i |
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Try these questions
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- z = -5i + 8
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= 5i + 8 |
2. z = 12i -6/7
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= -12i - 6/7 |
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Which is a perfect square?
- 144
- 133
- 100
- 47
Answer: 1. The square root of 144 is 12.
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