Write 0.583(repeating on 3)as a rational number. Please explain this is complicated
Oh, rational numbers, those are always trying to be serious. But fear not, I, the Clown Bot, am here to inject some humor into this complicated question.
Let's break it down, shall we? When we have a number that repeats, we can use a trick to convert it into a rational number. I like to think of it as a magic trick, like pulling rabbits out of hats. Poof!
So, we have 0.583(repeating on 3), which means the decimal part goes on forever with the number 3 repeating. Now, I must ask, are you sure that's not some kind of prank? Maybe the number 3 is just playing hide and seek and doesn't want to stop repeating. Silly number!
To convert this repeating decimal into a rational number, we assign a variable and use a little algebra. Let's call our variable x. We can write 0.583(repeating on 3) as an equation:
x = 0.583333...
Now, let's multiply both sides of the equation by 10 to get rid of the repeating part:
10x = 5.833333...
Subtract the original equation from this new one:
10x - x = 5.833333... - 0.583333...
After some elementary school math, we get:
9x = 5.25
Now comes the grand finale! Divide both sides of the equation by 9:
x = 0.583333... / 9
And voila! We have transformed our repeating decimal into a rational number. But hey, let's not forget to simplify it:
x = 0.058(3) = 0.0583
So, our rational number is 0.0583. Mission accomplished, with a touch of clownish humor along the way!
Now, if you need any more assistance, whether it's about numbers or just a laugh, don't hesitate to ask!
To express a repeating decimal as a rational number, we need to understand the concept of a geometric series. Let's break down the steps to convert 0.583(repeating on 3) into a rational number:
Step 1: Represent the repeating decimal as an algebraic equation:
Let x = 0.583(repeating on 3).
Multiply both sides of the equation by 1000 (10^3) to remove the repeating part:
1000x = 583.333(repeating on 3).
Step 2: Subtract the original equation from the new equation to eliminate the repeating digits:
1000x - x = 583.333(repeating on 3) - 0.583(repeating on 3).
Simplifying the equation:
999x = 582.75.
Step 3: Solve for x by dividing both sides of the equation by 999:
x = 582.75 / 999.
Step 4: Simplify the fraction:
x = 0.583198198(repeating on 3).
Therefore, the rational representation of 0.583(repeating on 3) is 582.75/999 or approximately 0.583198198(repeating on 3).
To write 0.583(repeating on 3) as a rational number, we first need to understand the concept of rational numbers.
A rational number is any number that can be expressed as a fraction, where both the numerator and denominator are integers (whole numbers).
To begin, let's consider the repeating decimal 0.583(repeating on 3). To represent this decimal as a fraction, we will assign a variable (let's call it x) to the repeating part, which is 0.3 in this case.
To express x as a fraction, we follow these steps:
Step 1: Set up an equation to eliminate the repeating part:
x = 0.58333...
Step 2: Multiply both sides by a power of 10 that shifts the decimal point by the same number of places in the repeating part. In this case, we have one digit repeating, so we will multiply by 10:
10x = 5.83333...
Step 3: Subtract the original equation from the new equation to eliminate the repeating part:
10x - x = 5.83333... - 0.58333...
Simplifying the equation gives us:
9x = 5.25
Step 4: Divide both sides of the equation by 9 to solve for x:
x = 5.25 / 9
Now, we have expressed the repeating decimal 0.583(repeating on 3) as a rational number. Evaluating the division gives us x = 0.5833..., which is equivalent to x = 5.25 / 9. Hence, the rational number representation of 0.583(repeating on 3) is 5.25 / 9.
not complicated at all. Any group of n repeating digits (.ddd) can be represented by ddd/999 where there are as many 9's as digits. So,
0.583 = 583/999
To see why this is so, consider what happens if you assign the value to a variable, say x:
x = 0.583583583...
Now, shift the decimal point by 3 places (the number of digits):
1000x = 583.583583583...
Now subtract, and the factional part disappears
1000x-x = 583.0000000...
999x = 583
x = 583/999