Welcome to divided by, our website explaining the division of two numbers x and y, mainly integers. The division of any two numbers is commonly denoted as x / y; x is the dividend or numerator, and y ≠ 0 is the divisor or denumerator. The division symbol is usually a slash, as in our example, yet the vertical bar (–) and the obelus (÷) can also be used to denote a division.
If both, x and y belong to ℤ, (in other words, either number is an integer also known as whole number), then the result q = x / y is a rational number, which is either terminating (has a finite number of decimal places), or is non-terminating and repeating. If the division of x by y is non-terminating and non-repeating, then it can be concluded that q is an irrational number.
Let’s have a look at these examples:
- Supposed x = 3 and y = 4 we get q = 0.75, a rational number which is terminating.
- With x = 3 and y = 7 we obtain q = 0.42857142857142857142857142…, a rational number which is non-terminating and has the repeating pattern 428571, called repetend or reptend. Better than using an ellipsis is denoting the result with a vinculum, 0.428571, or using parenthesis such as our calculator which you can find further below.
- Assumed x = π, and y = 2, we get 1.57079632679489661923132169…, an irrational number (representing half the ratio of a circle’s circumference to its diameter).
In the next section we elaborate on integers, and there you can also find our calculator which is particularly useful for ratios which have a repeating sequence in the result.
Dividing Whole Numbers
Dividing whole numbers is what this site is mainly about. The basic operation tells us how many times the number x is contained within the number y. Using a sheet of paper in combination with pen or pencil, x by y can be calculated using the short division method if the divisor is simple, else the long division algorithm has to be applied. It gets a bit tricky for those rational number which have a repetend, but our calculator below can handle this nicely for you.
Just enter the denominator and the nominator in their designated fields, our tool then does the calculation for you on the fly. To start over, overwrite your input or hit reset.
In the next paragraph we shed a light on the division of a fraction, and then answer the question what is 0 divided by 0, which is synonym with zero divided by zero.
The mathematical operation of dividing fractions is equivalent to multiplying the first fraction with the inverse of the second fraction (different from zero). We assume u,v,q,r ∈ ℤ:
So, u/v ÷ q/r = u/v * r/q = ur/vq, which equals x/y, for a certain pair (x,y) ∈ ℤ. If only one of the two, either the dividend or the divisor, is a quotient, divide the other number by 1 to have two fractions.
This essentially means that dividing fractions can be seen as the same thing as dividing whole numbers. For converting a fraction to decimal visit this site and check out its unique calculator.
Read on to learn what the division by zero means, followed by the summary of our article.
Divided by 0
The division by 0 is not defined for any x, including 0 divided by 0! The web is full of discussions regarding the division by 0, yet, fact is that divided by zero does not exist.However, it is interesting what happens for q = x / y when y is approaching, but not reaching, 0. For any x ≠ 0 ∈ ℤ the value “explodes”; in mathematical terms is ±∞.
We recommend to you using our search for in the sidebar to find the result of many divisions we have already carried out. Just insert x divided by y with your specific values, and hit the go button.
You have reached the end of our article, in which you have learned about the division of two numbers in a nutshell. This image sums our content up:
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Further information about quotients can be found in the referenced URL below, and web addresses similar to this one are located in the recommended section in the sidebar.