Mathematics and Statistics · Class 11 Commerce
Ch 3Complex Numbers — Class 11 Mathematics and Statistics, concept-first.
This Maharashtra Std XI (Commerce) Mathematics and Statistics chapter extends the number system beyond the real numbers so that every quadratic equation has a solution. The chapter draws on the same standard, well-established treatment of complex numbers used in mathematics curricula nationally.
Key concepts
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Algebra of Complex Numbers
This concept owns the imaginary unit and the arithmetic of complex numbers in standard (rectangular) form. In scope: the imaginary unit i with i²=−1 and the cyclic powers i¹=i, i²=−1, i³=−i, i⁴=1, including reducing a hi…
Most relevant Q&A
- The value of $\dfrac{1+i}{1-i}$ is: (A) $1$ (B) $-1$ (C) $i$ (D) $-i$Free
- If $a+ib=\dfrac{3+2i}{2-i}$, find the real numbers $a$ and $b$.Preview
- If $z_1=2+3i$ and $z_2=4-5i$, find (i) $z_1+z_2$ and (ii) $z_1-z_2$.Free
- Simplify $(3+2i)(1-4i)$ and write the result in the form $a+bi$.Free
- Express $\dfrac{3+2i}{1-i}$ in the form $a+bi$.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
The Imaginary Unit and the Complex Number System
This Maharashtra Std XI (Commerce) Mathematics and Statistics chapter extends the number system beyond the real numbers so that every quadratic equation has a solution.
Powers of i
Because , the powers of repeat in a cycle of four, which lets us simplify any integer power of quickly.
Algebra of Complex Numbers — Addition, Subtraction, Multiplication and Division
Complex numbers are added, subtracted and multiplied exactly like ordinary algebraic expressions in , using the single extra rule . Division is handled with the conjugate (Section 4).
Conjugate and Modulus
Two derived quantities attached to a complex number are indispensable for division, for measuring size, and for the polar form.
Argand Diagram, Argument and Polar Form
A complex number can be pictured as a point (or an arrow from the origin) in a plane, which gives it a length and a direction — the modulus and the argument.
Square Roots of a Complex Number
Every non-zero complex number has exactly two square roots, which are negatives of each other. They can be found algebraically without polar form.
Cube Roots of Unity
The equation has three solutions in — the cube roots of unity — with elegant properties used throughout algebra.
Exercises
More questions
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- Example 1Evaluate $i^{50}+i^{23}$.Free
- Example 2If $z_1=2+3i$ and $z_2=4-5i$, find (i) $z_1+z_2$ and (ii) $z_1-z_2$.Free
- Example 3Simplify $(3+2i)(1-4i)$ and write the result in the form $a+bi$.Free
- Example 4Express $\dfrac{3+2i}{1-i}$ in the form $a+bi$.Preview
- Example 5For $z=5-12i$, find (i) the conjugate $\bar z$ and (ii) the modulus $|z|$.Preview
- Example 7Find the modulus and argument of $z=1+i\sqrt3$ and write it in polar form.Preview
- Example 8Find the modulus and the principal argument of $z=-1+i$, and express it in polar form.Preview
- Example 9Find the square roots of $3+4i$.Preview
- Example 11If $\omega$ is a complex cube root of unity, evaluate $(1-\omega+\omega^{2})(1+\omega-\omega^{2})$.Preview
- Example 12Solve the quadratic equation $x^{2}-4x+13=0$ in the complex number system.Preview