Unit-4
Algebra
Complex Numbers |Class-12
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| Class-12 Mathematics Complex Number Full Notes 2080/2081 |
COMPLEX NUMBER
Complex numbers are numbers that can be expressed in the form of a+ib, where a and b are real numbers, and 'i' represents the imaginary unit. The imaginary unit, denoted by 'i' or 'j', is defined as the square root of -1, meaning i^2 = -1.
In the complex number a+ib, the real part is 'a' (Re), and the imaginary part is 'b' multiplied by the imaginary unit 'i' (Im). So, a represents the real component, and bi represents the imaginary component.
For example, in the complex number 2+3i, 2 is the real part (Re), and 3i is the imaginary part (Im). This means that the number has a real component of 2 and an imaginary component of 3i.
Complex numbers are widely used in various branches of mathematics, engineering, and physics. They provide a powerful tool for solving equations, representing quantities with both magnitude and direction, and describing phenomena that involve waves and oscillations.
Syllabus
De Moivre's theorem and its application in finding the roots of a complex number, properties of cube roots of unity. Euler's formula.
1.12 state and prove De Moivre's theorem.
1.13 find the roots of a complex number by De Moivre's theorem.
1.14 solve the problems using properties of cube roots of unity.
1.15 apply Euler's formula.
*Some frequently asked questions by students More
What is Complex Numbers?
Complex numbers are numbers that can be expressed in the form of a+ib, where a and b are real numbers, and 'i' represents the imaginary unit. The imaginary unit, denoted by 'i' or 'j', is defined as the square root of -1, meaning i^2 = -1.
Write down the multiplicative identity and inverse of the complex number.
The multiplicative identity of complex numbers is defined as (x+yi). (1+0i) = x+yi. Hence, the multiplicative identity is 1+0i. The multiplicative identity of complex numbers is defined as (x+yi). (1/x+yi) = 1+0i. Hence, the multiplicative identity is 1/x+yi.
WWhat is the demoivre's theorem of complex numbers Class 12?.
De Moivre's theorem is a mathematical theorem that relates to the powers of complex numbers. It states that for any complex number \(z = r(\cos \theta + i \sin \theta)\) and any positive integer \(n\), the \(n\)th power of \(z\) can be expressed as: \[z^n = r^n (\cos(n\theta) + i\sin(n\theta))\] Here, \(r\) represents the magnitude or modulus of the complex number \(z\), and \(\theta\) represents its argument or angle in the complex plane. In other words, to find the \(n\)th power of a complex number, you raise its magnitude to the power of \(n\) and multiply the angle by \(n\). This theorem is particularly useful when working with complex numbers in polar form, as it provides a way to raise them to a power without converting to rectangular form. It also has applications in various areas of mathematics, including trigonometry, calculus, and engineering..
What is a Gaussian integer?
A Gaussian integer is a complex number of the form a + bi, where a and b are integers, and i is the imaginary unit (√-1). In other words, it is a complex number with real and imaginary parts that are both integers. The term "Gaussian integer" is named after the mathematician Carl Friedrich Gauss. The set of Gaussian integers is often denoted by the symbol ℤ[i], where ℤ represents the set of integers. Just like the set of ordinary integers, Gaussian integers form a ring under addition and multiplication. The ring of Gaussian integers is a subset of the complex numbers. Gaussian integers have some interesting properties. For example, prime numbers in the set of Gaussian integers are either ordinary prime numbers of the form 4n + 3 or complex numbers of the form a + bi, where a and b are integers and a^2 + b^2 is a prime number. Gaussian integers are also important in number theory and various areas of mathematics, including algebraic number theory, cryptography, and signal processing. They have applications in solving Diophantine equations, factorization problems, and analyzing certain properties of integer lattices.