Class 11 Mathematics — Chapter 4: COMPLEX NUMBERS AND QUADRATIC EQUATIONS

What you'll learn

• ▸Definition of Complex Numbers and the Imaginary Unit (i)
• ▸Algebra of Complex Numbers: Addition and Subtraction
• ▸Algebra of Complex Numbers: Multiplication and Division
• ▸Powers of the Imaginary Unit (i)
• ▸Modulus and Conjugate of a Complex Number
• ▸Finding the Multiplicative Inverse
• ▸Square Roots of Negative Real Numbers
• ▸Algebraic Identities for Complex Numbers

Chapter Summary

--- PAGE 1 --- Chapter 4 COMPLEX NUMBERS AND QUADRATIC EQUATIONS ❖Mathematics is the Queen of Sciences and Arithmetic is the Queen of Mathematics. – GAUSS ❖ 4.1 Introduction In earlier classes, we have studied linear equations in one and two variables and quadratic equations in one variable. We have seen that the equation x² + 1 = 0 has no real solution as x² + 1 = 0 gives x² = – 1 and square of every real number is non-negative. So, we need to extend the real number system to a larger system so that we can find the solution of the equation x² = – 1. In fact, the main objective is to solve the equation ax² + bx + c = 0, where D = b² – 4ac < 0, which is not possible in the system of real numbers. W. R. Hamilton (1805-1865) 4.2 Complex Numbers Let us denote √-1 by the symbol i. Then, we have i² = –1. This means that i is a solution of the equation x² + 1 = 0. A number of the form a + ib, where a and b are real numbers, is defined to be a complex number. For example, 2 + i3, (–1) + i…

Practice Questions from this Chapter

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1. Describe imaginary numbers' practical uses. Get Solution →
2. Visualize complex numbers on a plane. Get Solution →
3. Trace complex numbers' historical development. Get Solution →
4. What is the value of i²? Get Solution →
5. For the complex number z = a + ib, what is 'a' called? Get Solution →
6. What is the additive identity in the system of complex numbers? Get Solution →
7. If z₁ = a + ib and z₂ = c + id, what is the sum z₁ + z₂? Get Solution →
8. What is the additive inverse of the complex number z = a + ib? Get Solution →

Did you know?

• 💡 Early mathematicians struggled with complex numbers, calling them "fictitious" or "impossible".
• 💡 Complex numbers help engineers design circuits and understand electricity flow.
• 💡 Multiplying a number by 'i' on a graph rotates it exactly ninety degrees.
• 💡 Imaginary numbers are essential for describing real-world phenomena like alternating current.
• 💡 Euler's identity, a famous equation, brilliantly links numbers 0, 1, e, " and i.

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