i'm really confused about if 0.123333... is
A) rational number
B) irrational number
C)integer,rational number
D)whole number, integer,rational number
can somebody help me please
thank you
Well, well, well! Looks like we have a puzzler here! Let's take a look at the number in question: 0.123333... Hmmm, those repeating 3's are throwing a curveball, aren't they?
Now, my friend, if a number can be expressed as the ratio of two integers, we call it a rational number. So, is 0.123333... rational?
You might think, hey, it's a decimal, it's gotta be irrational, right? Well, hold your clown hat! Surprisingly, this little trickster is actually a rational number!
Let's see why. We can rewrite 0.123333... as a fraction, using some clever math. Let's call x = 0.123333... and multiply it by 1000 to get 123.333.... Now, here comes the trick! If we subtract x from 1000x, what happens? The repeating decimals conveniently cancel out, leaving us with a nice equation to solve.
So, 1000x - x = 123.333... - 0.123333... (still with me?)
If we simplify that, we get 999x = 123.
Now, divide both sides by 999, and you'll find that x = 123/999, which means our little friend 0.123333... is a rational number after all!
So, the answer is A) rational number. Mystery solved! Keep those questions coming!
To determine whether 0.123333... is a rational or irrational number, let's analyze the nature of the decimal representation.
The given number 0.123333... has a repeating decimal pattern, with the digit 3 repeating indefinitely.
To express this number in a simpler form, let's assign it a variable:
Let x = 0.123333...
Multiply x by 10 to shift the decimal place:
10x = 1.233333...
Now, subtract x from 10x to eliminate the repeating part:
10x - x = 1.233333... - 0.123333...
= 1.11...
Simplifying the calculation, we have:
9x = 1.11
Divide both sides of the equation by 9 to solve for x:
x = 1.11 / 9
x = 0.123
By simplifying the decimal representation, we find that 0.123333... is equal to 0.123.
Therefore, 0.123333... is a rational number since it can be expressed as a fraction: 123/1000.
The correct answer is: A) rational number.
Of course! I'd be happy to help you understand whether 0.123333... is a rational number, irrational number, integer, rational number, or whole number.
To determine what kind of number 0.123333... is, we can start by understanding what the number represents. When we see the decimal representation "0.123333...", the ellipsis (...) usually means that the digit "3" is repeated infinitely. In other words, the decimal expansion continues as 0.123333333... with an infinite number of "3s" after the decimal point.
Now, let's analyze each option to see which ones apply:
A) Rational number: Rational numbers are numbers that can be expressed as a ratio (fraction) of two integers. To determine if 0.123333... is a rational number, we need to check if it can be expressed as a fraction.
We can represent 0.123333... as a fraction using the following method:
Let x = 0.123333...
Then 1000x = 123.333...
Now, let's subtract the two equations:
1000x - x = 123.333... - 0.123333...
999x = 123.210000...
Simplifying further, we get:
999x = 123.21
So, x = 123.21 ÷ 999
After dividing, we obtain x = 0.123
Therefore, 0.123333... can be expressed as a fraction (0.123). So, it is a rational number.
B) Irrational number: Irrational numbers cannot be expressed as a fraction and have non-repeating, non-terminating decimal expansions. Since we found that 0.123333... can be expressed as a fraction, it cannot be an irrational number.
C) Integer: Integers are whole numbers (positive, negative, or zero) without any fractional or decimal parts. Since 0.123333... has digits after the decimal point, it is not an integer.
D) Whole number: Whole numbers are positive numbers without any fractional or decimal parts. Similarly, 0.123333... has digits after the decimal point, so it is not a whole number.
Therefore, the correct answer is A) rational number.
it is rational, since I can write it as a fraction
Quick way to find that fraction ....
1. write down the digits to the end of the first repeat ---- 123
2. subtract all the digits that don't repeat ----- 123 - 12 = 111
that is your numerator
3. write down a 9 for each repeating digit, followed by a 0 for each non-repeating digit
900 <---- that is your denominator.
so .123333.... = 111/900 = 37/300
check with calculator, it works!
make up your own repeating decimal and try my method