Question

Which expressions represent real numbers and which represent complex numbers?

a) (0 + 4i)

b) (7i^2 + 6i^3)

c) (√((-3)^2))

d) (-1 + 3i^2)

e) (i^4)

f) (√(-3))

g) (409)

h) (-2 + 5i)

Answer 1

Some expressions represent real numbers, while others represent complex numbers. Real numbers include decimals and fractions, while complex numbers have both real and imaginary parts. Examples of real numbers include √((-3)^2) and 409, while examples of complex numbers include (0 + 4i) and (-2 + 5i).

Step-by-step explanation:

To determine whether the expressions represent real numbers or complex numbers, we need to understand the properties of each. A real number is any number that can be written as a decimal or fraction, while a complex number includes both a real part and an imaginary part. Let's analyze each expression:

1. a) (0 + 4i) - This is a complex number with a real part of 0 and an imaginary part of 4.
2. b) (7i^2 + 6i^3) - This expression includes imaginary exponents, which means it represents a complex number. However, without specific values for i^2 and i^3, we cannot determine the real and imaginary parts.
3. c) (√((-3)^2)) - The square root of a real number will always be a real number. So, this expression represents a real number.
4. d) (-1 + 3i^2) - Similar to b, without knowing the values of i^2, we cannot determine the real and imaginary parts. Thus, it represents a complex number.
5. e) (i^4) - Since i^2 is equal to -1, i^4 is equal to 1. Therefore, this expression represents a real number.
6. f) (√(-3)) - The square root of a negative number is not real, but it is considered a complex number. So, this expression represents a complex number.
7. g) (409) - A number without a complex part is a real number, so this expression represents a real number.
8. h) (-2 + 5i) - This expression includes both a real part (-2) and an imaginary part (5i), so it represents a complex number.