Which expression is equivalent to 4+7i?
In the realm of complex numbers, the expression 4+7i represents a complex number with a real part of 4 and an imaginary part of 7. Complex numbers are a fundamental concept in mathematics, particularly in fields such as engineering, physics, and computer science. The expression 4+7i can be expressed in various forms, each with its unique properties and applications. In this article, we will explore some of these equivalent expressions and their significance in the world of complex numbers.
Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary part is a real number multiplied by the imaginary unit, denoted by the letter ‘i’. The imaginary unit is defined as the square root of -1, which is not a real number. In other words, i^2 = -1.
The expression 4+7i can be rewritten in several equivalent forms. One of the most common ways to express this complex number is by using the polar form. In polar form, a complex number is represented as a magnitude (or modulus) and an angle (or argument). The magnitude of a complex number is the distance from the origin to the point representing the number in the complex plane, while the argument is the angle formed by the line connecting the origin and the point with the positive real axis.
To convert the expression 4+7i into polar form, we can use the following formulas:
Magnitude (r) = sqrt(real part^2 + imaginary part^2)
Argument (θ) = arctan(imaginary part / real part)
Applying these formulas to 4+7i, we get:
Magnitude (r) = sqrt(4^2 + 7^2) = sqrt(16 + 49) = sqrt(65)
Argument (θ) = arctan(7 / 4)
Thus, the polar form of 4+7i is (sqrt(65), arctan(7 / 4)).
Another equivalent expression for 4+7i is its conjugate, denoted as 4-7i. The conjugate of a complex number is obtained by changing the sign of its imaginary part. In this case, the real part remains the same, while the imaginary part changes from positive to negative. The conjugate of 4+7i has the same magnitude as 4+7i but an argument that is the negative of the original argument.
In addition to the polar form and the conjugate, 4+7i can also be expressed in exponential form. In exponential form, a complex number is represented as a magnitude raised to the power of the imaginary unit. The exponential form of 4+7i is:
e^(iθ) = cos(θ) + i sin(θ)
where θ is the argument of the complex number. By substituting the argument of 4+7i into this formula, we can obtain its exponential form.
In conclusion, the expression 4+7i can be represented in various forms, each with its unique applications and significance. The polar form, conjugate, and exponential form are just a few examples of the many ways to express this complex number. Understanding these different representations can help us better grasp the nature of complex numbers and their role in various scientific and mathematical fields.
