To determine if the sum of [tex]\(-2 \frac{3}{4}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex] is a rational number and, if so, to identify its properties, follow these detailed steps:
### Step 1: Convert the Mixed Number to an Improper Fraction
First, convert the mixed number [tex]\(-2 \frac{3}{4}\)[/tex] into an improper fraction:
- A mixed number of [tex]\( -2 \frac{3}{4} \)[/tex] means we have [tex]\(-2\)[/tex] plus a fractional part of [tex]\(- \frac{3}{4}\)[/tex].
- Converting the integer and fractional parts to a single fraction:
[tex]\[
-2 \frac{3}{4} = -2 - \frac{3}{4}
\][/tex]
### Step 2: Convert to a Decimal
Next, convert [tex]\(-2 \frac{3}{4}\)[/tex] into a decimal:
[tex]\[
-2 \frac{3}{4} = -2.75
\][/tex]
### Step 3: Decimal Form of the Second Fraction
Convert the fraction [tex]\(\frac{5}{9}\)[/tex] into a decimal:
[tex]\[
\frac{5}{9} \approx 0.5555555555555556
\][/tex]
### Step 4: Add the Decimals
Now, add the two decimals:
[tex]\[
-2.75 + 0.5555555555555556 \approx -2.1944444444444446
\][/tex]
### Step 5: Determine the Nature of the Sum
- The sum [tex]\(-2.1944444444444446\)[/tex] is a decimal number.
- This decimal is non-terminating and repeating (since it has an infinite number of "4"s).
### Step 6: Identifying Whether It's a Fraction
Since the sum [tex]\(-2.1944444444444446\)[/tex] is a non-terminating repeating decimal, it can be represented as a fraction (any repeating decimal can be converted to a fraction).
[tex]\(\therefore\)[/tex] Based on the steps and the computed result, we conclude that the sum of [tex]\(-2 \frac{3}{4}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex] is a fraction. It is rational and specifically an example of a repeating decimal, which confirm it is rational (because repeating decimals are always rational numbers when written in fractional form).
