# Quotient Calculator

By Maciej Kowalski, PhD candidate
Last updated: Apr 27, 2021

Welcome to Omni's quotient calculator, where we'll focus on division problems. We start with terminology: define the dividend, divisor, quotient, and remainder. Once we know all the bits and pieces connected with division in math, we move on to examples, e.g., `4` divided by `3`, `12` divided by `4`, or `25` divided by `2`. Our tool specializes as an integer division calculator, although it also returns the answer for decimals.

## Division in math: dividend, divisor, quotient

Division in math is the inverse operation to multiplication. To be precise, if we have some calculations of the form:

`a * b = c`,

then division should take the above product and return one of the factors:

`c / a = b`.

However, division in math uses different terminology than that of multiplication. Instead of factors and product, we have the dividend, divisor, and quotient:

`dividend / divisor = quotient`.

Recall that multiplication (just like addition) is commutative. In other words, the `a` and `b` can exchange places in the first formula from this section, and the result will stay the same.

On the other hand, division (just like subtraction) is not commutative. Therefore, the dividend in math is always the first of the numbers, while the second is the divisor (that's also why they have different names). For instance, `c / a` is something different from `a / c`.

Now that we've come to know the dividend, divisor, and quotient, we move on to the operation itself and the two result variants given by Omni's calculator to find the quotient. In essence, it all boils down to whether we tolerate fractions or not.

## Quotient and remainder

Division problems ask how many copies of something we can have if we distribute a number equally. For instance, say that you bought a large pizza with `8` slices for a family of `4`. To calculate how many slices each person gets, we use division:

`8 / 4 = 2`.

But what if there were `3` people instead? Obviously, everyone could still have two slices, but that would leave two in the box. Now, the question is whether we want to keep them for later, or cut them into smaller pieces. And that's where the quotient and remainder come in handy.

As said in the above section, the result of division is called the quotient. However, not all numbers are divisible by one another (like `4` divided by `3` or `25` divided by `2`). In such cases, we can distribute as much as possible until we leave the last few, the non-divisible parts: that's the remainder.

In the pizza example above, an `8`-slice pizza given to `3` people gives `2` slices each with a remainder of `2`. Other examples would be, say, `4` divided by `3`, giving `1` with a remainder `1`, or `25` divided by `2`, giving `12` with a remainder `1`. Symbolically, we write the result of such division in math by separating the quotient and remainder by a capital `R` (see below).

• `8 / 3 = 2 R 2`

• `4 / 3 = 1 R 1`

• `25 / 2 = 12 R 1`

We can also use this notation when the dividend in math problems is smaller than the divisor:

• `2 / 3 = 0 R 2`

• `17 / 20 = 0 R 17`

Or when the numbers are divisible by one another, like `12` divided by `4`:

• `12 / 4 = 3 R 0`

However, we need to be careful with remainders when dealing with negative numbers. As a rule, we define the remainder in mathematics as a positive number smaller than the (absolute value) of the divisor. Nevertheless, some applications (especially computer sciences) allow negative remainders. To satisfy both parties, Omni's integer division calculator presents both possibilities, and so if you use the calculator to find the quotient of, say, `-4` divided by `3` or `25` divided by `-2`, you'll get two results:

`(-4) / 3 = -1 R (-1) = -2 R 2`
`25 / (-2) = -13 R (-1)= -12 R 1`

Lastly, let's mention that we sometimes want to tackle division problems without mentioning the remainder. If we recall the example with an `8`-slice pizza for `3` people, this would mean cutting the `2` remaining slices into smaller pieces to finish it. Mathematically speaking, this translates to a fractional quotient.

`8 / 3 = 2⅔ ≈ 2.666`

In essence, everyone gets two full pizza slices and another two-thirds of a slice. No remainders here. If needed, make sure to check Omni's tool dedicated to changing between a simple fraction and a decimal one.

As you can see, you can use our calculator to find the quotient, whichever form you need. Quite a tool, wouldn't you say? So let's finish off with some clear instructions on how to operate it.

## Using the quotient calculator

At the top of our tool, you can see the division formula with the names of its consecutive parts, which we use in the quotient calculator below. For instance, if you'd like to see what is `15` divided by `6`, you need to:

1. Input `15` into the "Dividend" variable field.
2. Input `6` into the "Divisor" variable field.
3. Read off the result from underneath in whatever form you need.

As stated in point 3 and in the above section, remember that the quotient calculator gives the answer in different forms: a fractional one (a mixed number or a decimal), or a quotient and remainder. Also, for general integers (i.e., negative numbers), you can expect two variants of the remainder, while for decimals, the quotient calculator simply returns the division result without any additional output (there are no remainders in this case).

Whichever form you choose, we hope our tool gives you what you need. Remember that the quotient calculator is only one of so many other arithmetic tools we have on offer, so make sure to check out the others!

## FAQ

### How do I do division?

To divide two numbers, say, `a` by `b`, you need to:

1. Take the first digit of `a`.
2. Divide that number by `b`.
3. Write the quotient from step 2 as the first digit of the result.
4. Write the remainder from step 2 underneath.
5. Write the next digit of `a` to the right of the number from step 4.
6. Repeat steps 1-5 for subsequent digits of `a`.
7. The quotient consists of the digits from step 3.
8. The remainder is what you got left after running out of digits of `a`.

### How do I find the quotient?

To find the quotient of two numbers, say, `a` and `b`, you need to:

1. Take the first digit of `a`.
2. Divide that number by `b`.
3. Write the quotient from step 2 as the first digit of the result.
4. Write the remainder from step 2 underneath.
5. Write the next digit of `a` to the right of the number from step 4.
6. Repeat steps 1-5 for subsequent digits of `a`.
7. The quotient consists of the digits from step 3.
8. The remainder is what you got left after running out of digits of `a`.

### Is quotient division?

Yes. The result of dividing two numbers (the dividend by the divisor) is called the quotient.

### How do I estimate quotients?

To estimate the quotient of two numbers, say, `a` and `b`, you need to:

1. Take the first digit of `a`.
2. Divide that number by `b`.
3. Write the quotient from step 2 as the first digit of the result.
4. Write the remainder from step 2 underneath.
5. Write the next digit of `a` to the right of the number from step 4.
6. Repeat steps 1-5 for subsequent digits of `a`.
7. Put the decimal dot after the result you got so far.
8. Repeat steps 1-5, but with zeros in step 5.
9. Stop once you reach a satisfactory estimation.

### Is the quotient of two integers always a rational number?

Yes. By definition, a rational number is one that we can represent as one integer divided by another, which is precisely what the question mentioned.

### How do I find the quotient and remainder without actual division?

To find the quotient and remainder of two numbers without actual division, say, of `a` and `b`, you need to:

1. Subtract `b` from `a`.
2. Subtract `b` from what you got in step 1.
3. Repeat until you can no longer subtract `b`.
4. The quotient is how many times you subtracted `b`.
5. The remainder is what you got left after step 3.
6. Enjoy the quotient and remainder of your two numbers.
Maciej Kowalski, PhD candidate  Input two numbers
Dividend
Divisor
Result
Fractional result
Quotient
Remainder
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