Question
Properties of Complex Numbers Quick Check
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GPT 3.5
1. A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.
2. Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers.
3. The sum of two complex numbers (a + bi) and (c + di) is (a + c) + (b + d)i.
4. The difference of two complex numbers (a + bi) and (c + di) is (a - c) + (b - d)i.
5. The product of two complex numbers (a + bi) and (c + di) is (ac - bd) + (ad + bc)i.
6. To divide two complex numbers, first multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number (a + bi) is (a - bi).
7. The conjugate of a complex number (a + bi) is (a - bi).
8. The modulus (or absolute value) of a complex number (a + bi) is given by |a + bi| = √(a^2 + b^2).
9. The argument of a complex number (a + bi) is the angle θ between the real axis and the vector representing the complex number in the complex plane. It is given by arg(a + bi) = arctan(b/a).
10. The complex conjugate and modulus of a complex number have the following properties:
- The complex conjugate of the complex conjugate is the original complex number: (a + bi)* = a - bi.
- The modulus of the complex conjugate is the modulus of the original complex number: |(a + bi)*| = |a + bi|.
- The product of a complex number and its conjugate is equal to the square of its modulus: (a + bi)(a - bi) = a^2 + b^2.
These properties help in performing calculations and solving equations involving complex numbers.
2. Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers.
3. The sum of two complex numbers (a + bi) and (c + di) is (a + c) + (b + d)i.
4. The difference of two complex numbers (a + bi) and (c + di) is (a - c) + (b - d)i.
5. The product of two complex numbers (a + bi) and (c + di) is (ac - bd) + (ad + bc)i.
6. To divide two complex numbers, first multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number (a + bi) is (a - bi).
7. The conjugate of a complex number (a + bi) is (a - bi).
8. The modulus (or absolute value) of a complex number (a + bi) is given by |a + bi| = √(a^2 + b^2).
9. The argument of a complex number (a + bi) is the angle θ between the real axis and the vector representing the complex number in the complex plane. It is given by arg(a + bi) = arctan(b/a).
10. The complex conjugate and modulus of a complex number have the following properties:
- The complex conjugate of the complex conjugate is the original complex number: (a + bi)* = a - bi.
- The modulus of the complex conjugate is the modulus of the original complex number: |(a + bi)*| = |a + bi|.
- The product of a complex number and its conjugate is equal to the square of its modulus: (a + bi)(a - bi) = a^2 + b^2.
These properties help in performing calculations and solving equations involving complex numbers.