(4-5i).jpeg "(4+5i)(4-5i)")
Understanding Complex Numbers and Multiplication
In mathematics, complex numbers are numbers that extend the real number system by including the imaginary unit, denoted by 'i', where i² = -1. Complex numbers are written in the form a + bi, where 'a' and 'b' are real numbers.
One key aspect of complex numbers is their multiplication. To multiply two complex numbers, we can use the distributive property, similar to multiplying binomials in algebra.
Exploring the Multiplication (4+5i)(4-5i)
Let's consider the multiplication of (4+5i)(4-5i). We can expand this using the distributive property:
(4+5i)(4-5i) = 4(4-5i) + 5i(4-5i)
Expanding further:
= 16 - 20i + 20i - 25i²
Since i² = -1, we can substitute:
= 16 - 25(-1)
= 16 + 25
= 41
Significance of the Result
The result of multiplying (4+5i)(4-5i) is a real number, 41. This is because (4+5i) and (4-5i) are complex conjugates of each other. Complex conjugates differ only in the sign of the imaginary part. When multiplying complex conjugates, the imaginary terms cancel out, leaving only a real number.
Key Takeaways
- Complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers and i² = -1.
- Multiplying complex numbers involves using the distributive property.
- Complex conjugates, which differ only in the sign of the imaginary part, result in a real number when multiplied.
This understanding of complex numbers and their multiplication is fundamental in various mathematical fields, including algebra, calculus, and physics.