01 · Explore
Introduction to Complex Numbers
In the real number system, the square of any number is non-negative. This means equations like x² + 1 = 0 have no real solutions because x² = -1. To solve such problems, we extend the real numbers to a larger system.
The symbol i is used to denote √-1, such that i² = -1. This allows us to find solutions for equations where the discriminant is negative. A complex number is defined as any number of the form a + ib, where a and b are real numbers.
In the complex number z = a + ib, 'a' is known as the real part, denoted by Re z, and 'b' is known as the imaginary part, denoted by Im z. For instance, in the number 3 + 4i, the real part is 3 and the imaginary part is 4.
Two complex numbers z₁ = a + ib and z₂ = c + id are considered equal if and only if their corresponding parts are equal: a = c and b = d.
| Complex Number (z) | Real Part (Re z) | Imaginary Part (Im z) |
|---|---|---|
| 2 + i3 | 2 | 3 |
| -1 + i√3 | -1 | √3 |
| 4 + i(-1/11) | 4 | -1/11 |
| 5i | 0 | 5 |
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If z = -7 + 2i, what are Re z and Im z?
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Find x and y if x + iy = 3 - 4i.
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02 · Explore
Addition and Subtraction of Complex Numbers
The algebra of complex numbers follows specific rules for combining real and imaginary parts separately.
To add two complex numbers z₁ = a + ib and z₂ = c + id, we sum their real parts and their imaginary parts independently: z₁ + z₂ = (a + c) + i(b + d). This operation follows the closure, commutative, and associative laws.
The additive identity is the zero complex number 0 + i0, denoted as 0. For every complex number z, z + 0 = z. The additive inverse of z = a + ib is -z = -a + i(-b), such that z + (-z) = 0.
Subtraction is defined as adding the additive inverse: z₁ - z₂ = z₁ + (-z₂). For example, (6 + 3i) - (2 - i) = (6 - 2) + i(3 - (-1)) = 4 + 4i.
Adding Complex Numbers
(2 + i3) + (-6 + i5)
Group the real parts: (2 - 6) = -4. Group the imaginary parts: (3 + 5) = 8. Combine to get -4 + i8.
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Question
Simplify (1 - i) - (-1 + i6).
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03 · Explore
Multiplication of Complex Numbers
Multiplication involves the distributive property, keeping in mind that i² = -1.
For z₁ = a + ib and z₂ = c + id, the product is z₁z₂ = (ac - bd) + i(ad + bc). This formula is derived by expanding (a + ib)(c + id) as ac + iad + ibc + i²bd, and substituting i² = -1.
Multiplication is commutative (z₁z₂ = z₂z₁) and associative. The multiplicative identity is 1 + i0, denoted as 1. For every non-zero complex number z, there exists a multiplicative inverse z⁻¹ such that z · z⁻¹ = 1.
The distributive law also holds: z₁(z₂ + z₃) = z₁z₂ + z₁z₃. This allows us to use standard algebraic identities like (z₁ + z₂)² = z₁² + 2z₁z₂ + z₂².
Product of Two Complex Numbers
(3 + i5)(2 + i6)
Using (ac - bd) + i(ad + bc): (3*2 - 5*6) + i(3*6 + 5*2) = (6 - 30) + i(18 + 10) = -24 + i28.
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What is the value of (i)(i)?
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04 · Explore
Powers of i and Square Roots
The imaginary unit i follows a cyclic pattern when raised to integer powers.
The powers of i repeat every four steps: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. In general, for any integer k, i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = -1, and i⁴ᵏ⁺³ = -i.
Negative powers also follow this logic. For example, i⁻¹ = 1/i = (1/i) * (i/i) = i/i² = i/(-1) = -i.
When dealing with square roots of negative numbers, we use the property √-a = √a · √-1 = √a i for any positive real number a. Note that √a · √b = √ab only if at least one of a or b is non-negative. If both are negative, √-1 · √-1 = i² = -1, not √(-1)(-1) = 1.
| Power | Simplified Value |
|---|---|
| i³ | -i |
| i⁴ | 1 |
| i⁵ | i |
| i⁶ | -1 |
| i⁻³⁵ | i |
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Simplify i¹⁸.
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What is the square root of -3?
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05 · Explore
Modulus and Conjugate
The modulus represents the magnitude of a complex number, while the conjugate is its reflection.
For a complex number z = a + ib, the modulus |z| is defined as the non-negative real number √(a² + b²). Geometrically, this is the distance of the point (a, b) from the origin in the Argand plane.
The conjugate of z, denoted as z̄, is a - ib. It is obtained by changing the sign of the imaginary part. Geometrically, z̄ is the mirror image of z with respect to the real axis.
An important relationship exists between the two: z · z̄ = |z|². This property is used to find the multiplicative inverse: z⁻¹ = z̄ / |z|².
Finding Multiplicative Inverse
z = 2 - 3i
1. Find conjugate z̄ = 2 + 3i. 2. Find |z|² = 2² + (-3)² = 4 + 9 = 13. 3. z⁻¹ = z̄ / |z|² = (2 + 3i)/13 = 2/13 + i(3/13).
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Find the conjugate of -3i - 5.
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Calculate |3 + i|.
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06 · Explore
Division of Complex Numbers
Division is performed by multiplying the numerator and denominator by the conjugate of the denominator.
The quotient z₁/z₂ is defined as z₁ · (1/z₂), where 1/z₂ is the multiplicative inverse of z₂. In practice, we multiply both the top and bottom by the conjugate of the denominator to make the denominator a real number.
For example, to simplify (5 + √2i) / (1 - √2i), we multiply by (1 + √2i) / (1 + √2i). The denominator becomes 1² + (√2)² = 3. The numerator is expanded using the distributive law.
Common mistakes include forgetting to change the sign of the imaginary part when finding the conjugate or failing to square the 'i' term correctly (remembering i² = -1).
- 1
Identify the denominator z₂.
- 2
Find the conjugate of the denominator, z̄₂.
- 3
Multiply both numerator and denominator by z̄₂.
- 4
Simplify the denominator using |z₂|² = a² + b².
- 5
Expand the numerator and group real and imaginary parts.
- 6
Write the final result in a + ib form.
The quotient is expressed as a single complex number with a real denominator.
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Question
Simplify 1/i.
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07 · Explore
The Argand Plane
Complex numbers can be represented as points in a two-dimensional coordinate system.
The Argand plane (or complex plane) uses the x-axis as the Real Axis and the y-axis as the Imaginary Axis. A complex number z = x + iy is represented by the unique point P(x, y).
Purely real numbers (a + i0) lie on the real axis, while purely imaginary numbers (0 + ib) lie on the imaginary axis.
The distance of point P(x, y) from the origin O(0, 0) is the modulus |z| = √(x² + y²). This geometric interpretation links algebra with coordinate geometry.
Representing z and its Conjugate
- 1
Point P(x, y)
Represents the complex number z = x + iy in the Argand plane.
- 2
Reflection
Reflect the point P across the horizontal real axis (x-axis).
- 3
Point Q(x, -y)
Represents the conjugate z̄ = x - iy.
The conjugate of a complex number is its mirror image across the real axis in the Argand plane.
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Question
In which quadrant does the point representing -2 + 3i lie?
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08 · Explore
Historical Development
The concept of imaginary numbers evolved over centuries, with significant contributions from Indian and European mathematicians.
Early recognition of the difficulty of square roots of negative numbers dates back to the Greeks. Indian mathematician Mahavira (850 AD) explicitly stated that a negative quantity has no square root because it is not a 'square' quantity. Bhaskara (1150 AD) echoed this sentiment.
In the 16th century, Cardan encountered these numbers while solving cubic equations but dismissed them as 'useless'. Later, Albert Girard (1625) accepted them, noting they allow a polynomial of degree n to have n roots.
Euler introduced the symbol i for √-1. Finally, W.R. Hamilton (1830) provided the modern definition of complex numbers as ordered pairs of real numbers (a, b), removing the 'imaginary' mystery.
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Who introduced the symbol i?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 12, 13.