This chapter introduces complex numbers and their relation to quadratic equations, emphasizing their significance in solving equations without real solutions.
Complex Numbers and Quadratic Equations – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Complex Numbers and Quadratic Equations chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
z = a + ib
z is a complex number where a and b are real numbers. a represents the real part (Re z) and b is the imaginary part (Im z).
D = b² - 4ac
D is the discriminant of the quadratic equation ax² + bx + c = 0. It determines the nature of the roots: D > 0 (two distinct real roots), D = 0 (one real root), D < 0 (two complex roots).
z1 + z2 = (a + c) + i(b + d)
This formula shows the addition of two complex numbers z1 = a + ib and z2 = c + id.
z1 - z2 = (a - c) + i(b - d)
This denotes the difference of two complex numbers z1 = a + ib and z2 = c + id.
z1 * z2 = (ac - bd) + i(ad + bc)
This formula expresses the multiplication of two complex numbers z1 = a + ib and z2 = c + id.
z1 / z2 = (z1 * z2̅) / (z2 * z2̅)
To divide by a complex number, multiply by its conjugate. Here, z2̅ is the conjugate of z2.
|z| = √(a² + b²)
The modulus of a complex number z = a + ib is the distance from the origin in the Argand plane.
z̅ = a - ib
The conjugate of the complex number z = a + ib is denoted as z̅ and involves changing the sign of the imaginary part.
i² = -1
This defines the unit imaginary number, where i represents the square root of -1.
i⁴ = 1
This shows the periodicity of the powers of i with a cycle of 4.
Equations
ax² + bx + c = 0
The standard form of a quadratic equation, where a, b, and c are constants.
x = (-b ± √D) / 2a
The quadratic formula to find the roots of the equation ax² + bx + c = 0, where D is the discriminant.
z1 + z2 = z2 + z1
The commutative property of addition for complex numbers.
z1(z2 + z3) = z1z2 + z1z3
The distributive property of multiplication for complex numbers.
(z1 + z2)² = z1² + 2z1z2 + z2²
This is the expansion of the square of a sum of two complex numbers.
Re(z) = (z + z̅) / 2
This represents the real part of a complex number using its conjugate.
Im(z) = (z - z̅) / (2i)
This represents the imaginary part of a complex number using its conjugate.
z₁ z̅₁ = |z₁|²
The product of a complex number and its conjugate gives the square of its modulus.
z = r(cos θ + i sin θ)
The polar form of a complex number, where r is the modulus and θ is the argument.
D = b² - 4ac = 0
A specific case in the quadratic formula indicating that the equation has exactly one real root.
Complex Numbers and Quadratic Equations – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Complex Numbers and Quadratic Equations chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
z = a + ib
z is a complex number where a and b are real numbers. a represents the real part (Re z) and b is the imaginary part (Im z).
D = b² - 4ac
D is the discriminant of the quadratic equation ax² + bx + c = 0. It determines the nature of the roots: D > 0 (two distinct real roots), D = 0 (one real root), D < 0 (two complex roots).
z1 + z2 = (a + c) + i(b + d)
This formula shows the addition of two complex numbers z1 = a + ib and z2 = c + id.
z1 - z2 = (a - c) + i(b - d)
This denotes the difference of two complex numbers z1 = a + ib and z2 = c + id.
z1 * z2 = (ac - bd) + i(ad + bc)
This formula expresses the multiplication of two complex numbers z1 = a + ib and z2 = c + id.
z1 / z2 = (z1 * z2̅) / (z2 * z2̅)
To divide by a complex number, multiply by its conjugate. Here, z2̅ is the conjugate of z2.
|z| = √(a² + b²)
The modulus of a complex number z = a + ib is the distance from the origin in the Argand plane.
z̅ = a - ib
The conjugate of the complex number z = a + ib is denoted as z̅ and involves changing the sign of the imaginary part.
i² = -1
This defines the unit imaginary number, where i represents the square root of -1.
i⁴ = 1
This shows the periodicity of the powers of i with a cycle of 4.
Equations
ax² + bx + c = 0
The standard form of a quadratic equation, where a, b, and c are constants.
x = (-b ± √D) / 2a
The quadratic formula to find the roots of the equation ax² + bx + c = 0, where D is the discriminant.
z1 + z2 = z2 + z1
The commutative property of addition for complex numbers.
z1(z2 + z3) = z1z2 + z1z3
The distributive property of multiplication for complex numbers.
(z1 + z2)² = z1² + 2z1z2 + z2²
This is the expansion of the square of a sum of two complex numbers.
Re(z) = (z + z̅) / 2
This represents the real part of a complex number using its conjugate.
Im(z) = (z - z̅) / (2i)
This represents the imaginary part of a complex number using its conjugate.
z₁ z̅₁ = |z₁|²
The product of a complex number and its conjugate gives the square of its modulus.
z = r(cos θ + i sin θ)
The polar form of a complex number, where r is the modulus and θ is the argument.
D = b² - 4ac = 0
A specific case in the quadratic formula indicating that the equation has exactly one real root.
Complex Numbers and Quadratic Equations – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Complex Numbers and Quadratic Equations chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
z = a + ib
Where z is a complex number, a is the real part (Re z), and b is the imaginary part (Im z).
|z| = √(a² + b²)
The modulus of a complex number z, representing its distance from the origin in the Argand plane.
z̅ = a - ib
The conjugate of a complex number z, where changing the sign of the imaginary part is essential for operations.
z₁ + z₂ = (a + c) + i(b + d)
The sum of two complex numbers z₁ and z₂, where a, b, c, and d are their respective real and imaginary parts.
z₁ - z₂ = (a - c) + i(b - d)
The difference of two complex numbers z₁ and z₂, calculated by subtracting their real and imaginary parts.
z₁ z₂ = (ac - bd) + i(ad + bc)
The product of two complex numbers z₁ and z₂, demonstrating how to multiply complex numbers.
z₁/z₂ = (z₁ z̅₂) / (z₂ z̅₂)
The division of two complex numbers, involving the multiplicative inverse.
D = b² - 4ac
The discriminant used in quadratic equations to determine the nature of roots: D > 0 (two real roots), D = 0 (one real root), D < 0 (complex roots).
x = (-b ± √D) / (2a)
The quadratic formula to find the roots of the equation ax² + bx + c = 0.
i^2 = -1
The fundamental property of the imaginary unit i, used to derive calculations involving complex numbers.
Equations
x² + 1 = 0
This equation has no real solutions; it leads to complex solutions x = i and x = -i.
z₁ = x + iy; z₂ = a + bi
Defining complex numbers in Cartesian coordinates for operations and conversions.
z̅ = a - ib
Expressing the conjugate of a complex number; critical for division and modulus calculations.
z₁ + z₂ = (a + c) + i(b + d)
Adding two complex numbers using their real and imaginary parts.
z₁ - z₂ = (a - c) + i(b - d)
Subtracting one complex number from another, useful in complex algebra.
z₁z₂ = (ac - bd) + i(ad + bc)
Formula for multiplying complex numbers.
z₁/z₂ = (z₁ z̅₂) / (z₂ z̅₂)
Defining the division of complex numbers through the conjugate.
a + bi
The standard form of a complex number, essential for representation and operations.
x = (-b ± √D) / (2a)
The quadratic formula used to determine the roots of a quadratic equation.
D = b² - 4ac
The discriminant for assessing the nature of roots in quadratic equations.
This chapter introduces the concept of sets, their significance, and basic operations in mathematics.
Start chapterThis chapter explores the concepts of relations and functions in mathematics, focusing on how to connect pairs of objects from different sets and the significance of functions in describing these relationships.
Start chapterThis chapter introduces trigonometric functions, explaining their definitions, properties, and applications. Understanding these concepts is essential for solving various mathematical problems and real-world applications.
Start chapterThis chapter explores linear inequalities in one and two variables, explaining their significance in various real-world applications.
Start chapterThis chapter introduces the concepts of permutations and combinations, essential for counting arrangements and selections in mathematics.
Start chapterThis chapter introduces the binomial theorem, which simplifies the expansion of binomials raised to a power. It is essential for efficiently calculating powers without repeated multiplication.
Start chapterThis chapter discusses sequences, which are ordered lists of numbers, and their importance in mathematics. It covers different types of sequences and series, including arithmetic and geometric progressions, and their applications.
Start chapterThis chapter explores the properties and equations of straight lines in coordinate geometry, emphasizing their significance in mathematics and real-life applications.
Start chapterThis chapter explores conic sections including circles, ellipses, parabolas, and hyperbolas, highlighting their definitions and significance in mathematics and real-world applications.
Start chapterThis chapter introduces the essential concepts of three dimensional geometry, focusing on how to represent points in space using coordinate systems.
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