Final answer:
Real numbers are numbers without the imaginary unit 'i'. Expressions involving 'i' are complex numbers. The provided expressions are classified accordingly, with the added note that an expression involving 'i' squared becomes real due to 'i' squared being -1.
Step-by-step explanation:
Expressions that represent purely real numbers are those without the imaginary unit 'i', while expressions that include 'i' represent non-real complex numbers. The imaginary unit 'i' is defined as the square root of -1. When we have an expression such as i9, we know it’s a complex number because it involves 'i'. Likewise, the square root of a negative number, such as −(-18), also implies a non-real complex number. However, a negative number without the 'i', such as -289, is a real number. The expression 4 - 3i2 2 −(-49) requires careful evaluation because 'i' squared is -1, thus changing the expression into a real number after multiplication with another negative.
Now, to classify each given expression:
- -289 is a purely real number.
- i9 is a non-real complex number as powers of 'i' represent complex numbers.
- −(-18) as − means taking the square root of a negative number, also a non-real complex number.
- 4 - 3i2 2 −(-49) simplifies to a real number because when you calculate i2, you get -1, which makes the expression 3(-1) 2 −(-49), leading to a real number: purely real number.
- 3 + 2i involves 'i', so it is a non-real complex number.