What Are the Properties and Powers of Imaginary Numbers?
The concept of imaginary numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Imaginary numbers are used when dealing with the square roots of negative numbers—something not possible with real numbers. This concept is essential for students in classes 9, 10, 11, and 12, and also appears in advanced algebra, physics, and engineering.
What Is Imaginary Numbers?
An imaginary number is defined as any number that can be written as a real number multiplied by the imaginary unit i, where i = √-1. This means the square of an imaginary number always results in a negative value. You’ll find this concept applied in areas such as complex numbers, quadratic equations with negative discriminants, and electrical engineering.
Key Formula for Imaginary Numbers
Here’s the standard formula: \( i = \sqrt{-1} \)
When squared: \( i^2 = -1 \)
General imaginary numbers: \( bi \), where \( b \) is a real number.
Properties and Powers of i
| Power | Value | Explanation |
|---|---|---|
| i1 | i | Basic imaginary unit |
| i2 | -1 | By definition, square of i |
| i3 | -i | i × i² = i × (-1) |
| i4 | 1 | i² × i² = (-1) × (-1) |
| in | Repeats every 4 powers | Cycle pattern: i, -1, -i, 1 |
Step-by-Step Illustration
Let’s solve \( (3i) \times (4i) \):
1. Multiply the real numbers: 3 × 4 = 122. Multiply the i terms: i × i = i²
3. Substitute i² = -1: So, 12 × (i²) = 12 × (-1)
4. Final Answer: **-12**
Operations with Imaginary Numbers
Adding, subtracting, multiplying, and dividing imaginary numbers is just like working with algebraic variables—but always remember the special rules for powers of i.
- Addition: \( 2i + 3i = (2 + 3)i = 5i \)
- Subtraction: \( 5i - 2i = (5 - 2)i = 3i \)
- Multiplication: \( (a i) \times (b i) = ab i^2 = -ab \)
Speed Trick or Vedic Shortcut
Here’s a quick trick: When calculating large powers of i, just divide the exponent by 4 and check the remainder. The remainder tells you the result:
- Remainder 1: i
- Remainder 2: -1
- Remainder 3: -i
- Remainder 0: 1
Example: \( i^{23} \). 23 ÷ 4 = 5 remainder 3. So, \( i^{23} = -i \).
Vedantu’s live doubt-solving sessions include more such exam tips to help you master imaginary numbers.
Cross-Disciplinary Usage
Imaginary numbers are not only useful in Maths but also play an important role in Physics (like in alternating current calculations), Computer Science (signal processing), and engineering. Students preparing for JEE, NEET, or board exams will come across these concepts frequently.
Try These Yourself
- Find the value of \( i^{11} \).
- Is \( \sqrt{-9} \) an imaginary number? What is it?
- Add \( 6i + 7i \).
- Multiply \( (2i) \times (5i) \).
Frequent Errors and Misunderstandings
- Forgetting that \( i^2 = -1 \), not 1.
- Confusing imaginary numbers with irrational numbers (example: √2 is irrational, not imaginary).
- Trying to plot imaginary numbers on the usual number line instead of the complex/Argand plane.
Relation to Other Concepts
The idea of imaginary numbers connects closely with topics such as complex numbers, quadratic equations, and real numbers. Mastering this helps with understanding polynomial roots, trigonometry using Euler’s formula, and more advanced maths.
Classroom Tip
A quick way to remember imaginary numbers is to picture i as a “turn on the complex plane”—every multiplication by i rotates a number by 90°. Teachers at Vedantu often use the “powers of i cycle” chart for fast recall: i, -1, -i, 1… and repeat.
We explored imaginary numbers—from definition, formulas, solved examples, and connections to other topics. Continue practicing with Vedantu to become confident in solving problems using this important concept.
Related Topics
FAQs on Imaginary Numbers Explained: Definition, Rules & Uses
1. What are imaginary numbers in maths?
Imaginary numbers are numbers that, when squared, result in a negative value. They are defined as the square root of negative numbers and are represented using the imaginary unit, i, where i = √-1. An imaginary number is typically expressed as a real number multiplied by i; for example, 3i, -5i, or √2i.
2. Why are imaginary numbers called 'imaginary'?
The term 'imaginary' is a historical artifact. These numbers were initially considered to be non-existent or unreal because they cannot be represented on the standard number line of real numbers. However, they are now integral to many branches of mathematics and physics. Their name reflects their initially confusing nature to mathematicians.
3. What is the value of 'i'?
The value of i is defined as the square root of negative one: i = √-1. This fundamental definition allows for the extension of mathematical operations to include negative square roots.
4. What are the powers of i?
The powers of i cycle in a predictable pattern:
• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1
This pattern repeats for higher powers. To find the value of a higher power of i (e.g. i⁷), divide the exponent by 4 and use the remainder to determine the value from this pattern.
5. How do you add imaginary numbers?
To add imaginary numbers, combine the real and imaginary components separately. For example: (a + bi) + (c + di) = (a + c) + (b + d)i
6. How do you subtract imaginary numbers?
Subtracting imaginary numbers is similar to addition. Subtract the real parts and the imaginary parts separately: (a + bi) - (c + di) = (a - c) + (b - d)i
7. How do you multiply imaginary numbers?
Multiply imaginary numbers using the distributive property (FOIL method) and remember that i² = -1. For example: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i
8. How do you divide imaginary numbers?
To divide complex numbers, multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of (a + bi) is (a - bi). This eliminates the imaginary part from the denominator.
9. What are complex numbers?
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1). The real part is a, and the imaginary part is b.
10. What are imaginary numbers used for?
Imaginary numbers have many applications in various fields, including:
• Solving quadratic equations
• Electrical engineering (AC circuits)
• Quantum mechanics
• Signal processing
• Fluid dynamics
11. What is the conjugate of a complex number?
The conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number (a² + b²).
12. Can imaginary numbers be negative?
Yes, an imaginary number can be negative. For instance, -3i is a negative imaginary number. The negative sign applies to the real number part of the imaginary number, not the i itself.
