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When is it true that is |a|=|b|, then either a=b or a=b*, where a and b are complex numbers and b* is the complex conjugate of b?
The absolute value of a complex number is a measure of its distance from the origin on the complex plane. It is also known as the modulus or magnitude of the complex number and is always a non-negative real number.
The absolute value of a complex number is calculated by taking the square root of the sum of the squares of its real and imaginary parts. In other words, if a complex number is expressed as z = a + bi, where a and b are real numbers, then the absolute value of z is given by |z| = √(a² + b²).
The absolute value of a real number is simply the distance of that number from zero on the number line, while the absolute value of a complex number is the distance of that number from the origin on the complex plane. This means that while the absolute value of a real number is always positive, the absolute value of a complex number can be positive, zero, or even negative.
No, the absolute value of a complex number is always a real number. This is because it represents a distance, which is a real quantity, and cannot have an imaginary component.
The absolute value of a complex number is used in various mathematical operations and concepts such as finding the magnitude of a vector, solving equations involving complex numbers, and calculating complex roots of polynomials. It is also a key concept in understanding the geometry of the complex plane.