What is the imaginary unit i defined as? — √(−1).By definition, i = √(−1), so i² = −1. This extends the real numbers to allow square roots of negative numbers.
What is the real part of the complex number 7 − 4i? — 7.For a complex number a + bi, the real part is a. Here a = 7 and b = −4, so the real part is 7.
What is the imaginary part of 5 + 2i? — 2.For a + bi, the imaginary part is the coefficient b (not bi). So the imaginary part of 5 + 2i is 2.
Compute (3 + i) + (5 − 2i). — 8 − i.Add real and imaginary parts separately: (3+5) + (1−2)i = 8 + (−1)i = 8 − i.
What is i²? — −1.By definition, i = √(−1), so i² = −1. This is the fundamental property of the imaginary unit.
What is i³? — −i.i³ = i² · i = (−1) · i = −i. Powers of i cycle: i¹=i, i²=−1, i³=−i, i⁴=1.
What is i⁴? — 1.i⁴ = (i²)² = (−1)² = 1. The cycle of powers of i has period 4.
What is the complex conjugate of 4 − 7i? — 4 + 7i.The complex conjugate of a + bi is a − bi. So the conjugate of 4 − 7i is 4 + 7i.
Compute (0 + 3i)² . — −9.(3i)² = 9i² = 9(−1) = −9. This shows why squaring a purely imaginary number gives a negative real number.
Is 7 a complex number? — Yes, it equals 7 + 0i.Every real number is a complex number with imaginary part 0. So 7 = 7 + 0i is indeed a complex number.
When does a quadratic equation have complex (non-real) roots? — When the discriminant is negative.Complex roots occur when the discriminant b² − 4ac < 0, because you must take the square root of a negative number.
What is the discriminant of x² + 4x + 13 = 0? — −36.Discriminant = b² − 4ac = 16 − 4(1)(13) = 16 − 52 = −36.
What is √(−25) in simplified form? — 5i.√(−25) = √25 · √(−1) = 5 · i = 5i.
If one root of a polynomial with real coefficients is 3 + 5i, what is the other root? — 3 − 5i.Complex roots of polynomials with real coefficients always come in conjugate pairs. So if 3 + 5i is a root, then 3 − 5i must also be a root.
How many real solutions does x² + x + 1 = 0 have? — 0.Discriminant = 1 − 4 = −3 < 0. A negative discriminant means no real solutions — only complex conjugate solutions.
Solve x² + 25 = 0. — x = ±5i.x² = −25 → x = ±√(−25) = ±5i.
Express √(−49) in standard form. — 7i.√(−49) = √49 · √(−1) = 7 · i = 7i.
What does the horizontal axis in the complex plane represent? — Real axis.In the complex plane (Argand plane), the horizontal axis represents the real part of complex numbers, called the real axis.
What does the vertical axis in the complex plane represent? — Imaginary axis.In the complex plane, the vertical axis represents the imaginary part (coefficient of i), called the imaginary axis.
What point in the complex plane represents the number 3 − 4i? — (3, −4).The complex number a + bi is plotted at point (a, b). So 3 − 4i corresponds to (3, −4).
What complex number is represented by the point (0, 5) in the complex plane? — 5i.The point (a, b) represents a + bi. The point (0, 5) represents 0 + 5i = 5i, a purely imaginary number.
What is the modulus of 3 + 4i? — 5.|3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5. This is the famous 3-4-5 Pythagorean triple.
What is the modulus of 0 + 6i? — 6.|0 + 6i| = √(0² + 6²) = √36 = 6. The modulus of a purely imaginary number bi is |b|.
What is the modulus of 5 + 0i (a real number)? — 5.|5 + 0i| = √(5² + 0²) = √25 = 5. For real numbers, the modulus equals the absolute value.
What is the argument of the complex number 0 + i? — π/2.The point (0, 1) lies on the positive imaginary axis, making an angle of π/2 (90°) with the positive real axis.
In which quadrant is the complex number −3 + 4i located? — Q2.−3 + 4i has negative real part (−3) and positive imaginary part (4), placing it in Quadrant II (top-left).
What does the argument of a complex number represent geometrically? — Angle from positive real axis.The argument θ = arg(z) is the angle the line from the origin to z makes with the positive real axis, measured counterclockwise.