Question

Show that \((a+b i)+(c+d i)=(c+d i)+(a+b i)\)

Short answer

To prove that \((a+bi)+(c+di)=(c+di)+(a+bi)\), we add the complex numbers individually: \((a+c)+(b+d)i\) and \((c+a)+(d+b)i\). Since addition is commutative, the real parts \((a+c)\) and \((c+a)\) are equal, and the imaginary parts \((b+d)i\) and \((d+b)i\) are equal. Thus, the equality holds.

Step-by-step solution

Add the complex numbers on the left and right sides separately

First, add the left and right side complex numbers individually. For the left side, we have: \[ (a+bi)+(c+di) = (a+c) + (b+d)i \] Similarly for the right side: \[ (c+di)+(a+bi) = (c+a) + (d+b)i \]

Compare the real parts of both sides

Compare the real parts of the two sums above. On the left side, the real part is \((a+c)\) and on the right side, it is \((c+a)\). Since addition is commutative, we can rewrite \((c+a)\) as \((a+c)\). Thus, the real parts are equal.

Compare the imaginary parts of both sides

Now compare the imaginary parts of the two sums. The imaginary part of the left side is \((b+d)i\) while the imaginary part on the right side is \((d+b)i\). Since addition is commutative, we can rewrite \((d+b)i\) as \((b+d)i\). Thus, the imaginary parts are also equal.

Combine the results

Both the real and imaginary parts of the left side are equal to the real and imaginary parts of the right side, so we can now conclude that the equality is true: \((a+bi)+(c+di)=(c+di)+(a+bi)\).

Key concepts

Commutative Property

The commutative property is a fundamental attribute of many arithmetic operations. In the context of addition, it states that changing the order of the numbers being added does not change the sum. In other words, if you have two numbers, let's say x and y, the commutative property assures us that \( x + y = y + x \).

This principle also extends to complex numbers—the mathematical entities that include a real part and an imaginary part. When you add complex numbers, just as with real numbers, the order in which you add them does not affect the result. This property is particularly useful when solving equations or simplifying expressions where complex numbers are present, giving you the flexibility to rearrange the terms to make the calculation easier to understand or compute.

Complex Numbers

Complex numbers are numbers that can be expressed in the form \( a + bi \), where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, which satisfies the equation \( i^2 = -1 \). The number 'a' is called the real part, while 'b' is called the imaginary part of the complex number.

Complex numbers are useful in various areas of mathematics and science because they provide a comprehensive way to express all numbers, including those that involve the square roots of negative numbers, which are not possible within the set of real numbers alone. They allow for the solution of polynomial equations that would otherwise have no solution, and are also essential in the fields of electrical engineering, quantum physics, applied mathematics, and many others.

Real Part of Complex Number

The real part of a complex number is the 'a' in the expression \( a + bi \). It represents the component that lies along the x-axis on the complex plane—a two-dimensional plane used to visualize complex numbers with the real part on the horizontal axis and the imaginary part on the vertical axis.

Understanding the real part of a complex number is crucial as it allows us to perform real-number arithmetic within the context of complex numbers. When we add or compare complex numbers, their real parts combine with other real parts just as standard real numbers do. This simplifies the process of working with complex arithmetic by allowing us to treat real and imaginary components separately.

Imaginary Part of Complex Number

In contrast to the real part, the imaginary part of a complex number is the 'b' in the expression \( a + bi \), where 'i' is the imaginary unit. The imaginary part, 'b', represents the component that lies along the y-axis of the complex plane.

The careful distinction between the real and imaginary parts is essential for operations with complex numbers. When adding or subtracting complex numbers, imaginary parts are combined with other imaginary parts. This is similar to how you would sum like terms in algebra. The imaginary part can often be the source of novel properties and behaviors in higher math and physics, such as in wave forms or in signal processing, where 'i' represents a phase shift of 90 degrees.