How Can You Prove That Any Complex Multiplied with Another Complex Number Gives a Real Number?

The question of whether multiplying complex numbers always results in a real number is multifaceted and depends on specific conditions. A common approach involves the multiplication of a complex number with its complex conjugate. This article will delve into this topic and provide a thorough explanation.

1. Multiplication of Complex Numbers with Their Conjugates

For any complex number (z a bi) (where (a) and (b) are real numbers and (i) is the imaginary unit), multiplying (z) with its complex conjugate (overline{z} a - bi) results in a non-negative real number. Specifically, the product is given by:

[z cdot overline{z} (a bi)(a - bi) a^2 b^2 |z|^2]

This product is always a real number and is the square of the absolute value of (z).

2. Conditions for a Product to Be a Real Number

If (z_0) is a non-zero complex number and (z_0 cdot z) is a real number, say (r), then it can be expressed as:

[z frac{r}{z_0} frac{r cdot overline{z_0}}{z_0^2}]

This implies that (z) is a real multiple by (frac{r}{z_0^2}) of the complex conjugate of (z_0). Conversely, multiplying (z_0) by (k cdot overline{z_0}) (where (k) is a real number) also results in a real number. Thus, for (z_0 eq 0), the product (z_0 cdot z) is a real number if and only if (frac{z}{overline{z_0}}) is a real number.

3. Specific Cases: Products Resulting in Real Numbers

We can also explore scenarios where the product of two complex numbers is real without necessarily being the product of a number and its conjugate. Consider the expression:

[(a bi)(c di) (ac - bd) i(ad bc)]

This product is real if and only if the imaginary part is zero. Mathematically, this means:

[ad bc 0 iff ad -bc]

Case 1: When (a eq 0), let's choose (c) and (d) such that:

[d -frac{bc}{a}]

This implies:

[c di c - frac{bc}{a} cdot i frac{c}{a}(a - bi)]

This results in the two complex numbers not being conjugates but their product being real whenever (c eq a).

Case 2: When (b eq 0), let's choose (c) and (d) such that:

[c -frac{ad}{b}]

This implies:

[c di -frac{d}{b}a - frac{ad}{b}i frac{-d}{b}(a - bi)]

This results in the two complex numbers not being conjugates but their product being real whenever (d eq -b).

4. Conclusion

In summary, the product of a non-zero complex number and another complex number is a real number if and only if the second number is a real multiple of the conjugate of the first number. This aligns with the fact that the product of a complex number and its conjugate is always a positive real number, which is the square of the absolute value of the complex number.

Key Takeaways:

The product (z_0 cdot z) is a real number if and only if (frac{z}{overline{z_0}}) is a real number. Specific conditions such as (ad bc 0) can ensure the product of two complex numbers is real without them being conjugates. The product of a nonzero complex number and its conjugate is always a positive real number.

Further Reading: For a deeper dive into the properties of complex numbers and their operations, consider exploring topics such as the algebra of complex numbers, the geometry of the complex plane, and the application of complex numbers in various fields including physics and engineering.