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Input in the form of
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Multiplicative inverse

# Multiplicative Inverse Calculator

By Maciej Kowalski, PhD candidate
Last updated: Feb 01, 2021

Welcome to Omni's multiplicative inverse calculator, where we'll learn how to find the multiplicative inverse of an integer, a decimal, a fraction, or a mixed number. In essence, the value we seek is something that gives `1` after multiplying by the original number. As a matter of fact, the inverse of a fraction (a simple one, mind you) is what it all boils down to, and the rest of the process is just getting that form of your input.

So what exactly is the multiplicative inverse of a number? Well, let's find out, shall we?

## What is the multiplicative inverse of a number?

The multiplicative inverse of a number `a` is a value `b` such that `a * b = 1`. The end. Simple definition, so not much to worry about, is there? In fact, we could finish the section here, but we're far too talkative to do that. Nevertheless, we've decided to have the rest of this section's chat in a nice numbered list.

1. Not all numbers have a multiplicative inverse. However, it's not at all tricky to figure out which do, since there's only one that doesn't - zero. After all, multiplying anything by `0` gives `0`, so there is no way to find a value that would return `1`.

2. The sign of the multiplicative inverse must agree with that of the original number. Indeed, we want the two entries to give `1` after multiplication, which is a positive number, and a negative value produces a positive one only when multiplied by another negative value.

3. There are two very special numbers whose multiplicative inverses are the same as the number in question. Those are `1` and `-1` since indeed `1 * 1 = 1` and `(-1) * (-1) = 1` (observe how the signs agree, as mentioned in point 2). All the other numbers have distinct multiplicative inverses. However, note that this is not true if we consider modulo multiplication instead of the regular one.

4. The multiplicative inverse is unique. That means a number `a` can have only one inverse, i.e., if `a * b = a * c = 1`, then we must have `b = c`. Again, this is not the case in the modulo setting.

5. The inverse of the inverse is the number itself. That becomes clear when we look at the equation `a * b = 1`. There, `b` is the multiplicative inverse of `a`, and `a` is the multiplicative inverse of `b` (remember that multiplication is commutative, meaning that `a * b = b * a`).

Alright, that should be enough talk for this introduction. We've seen some properties, some curious facts, so it's time to learn how to find the multiplicative inverse of a number. We begin with the inverse of a fraction.

## The inverse of a fraction

We want it to be perfectly clear that in this section, we look at simple fractions of the form `x / y`. Obviously, we can convert every decimal to a simple fraction, and the same goes for mixed numbers. Nevertheless, for now, let's focus on the case of `x / y`, which is, as a matter of fact, the easiest one.

The name already suggests how to find the multiplicative inverse of a fraction: we simply invert it. In other words, we make the numerator and denominator exchange places. So what is the multiplicative inverse of `x / y`? It's simply `y / x`. No strings attached; it's all there is to it.

Note that this comes from how we multiply fractions and the fact that multiplication is commutative. Indeed, we have:

`(x / y) * (y * x) = (x * y) / (y * x) = (x * y) / (x * y) = 1`.

The last equality is always true, no matter what `x` or `y` are (that is, if neither is zero, which can never appear in the denominator).

Well, this one sure was an easy case. Let's move on to how to find the multiplicative inverse of an integer, a decimal, or a mixed number.

## How to find the multiplicative inverse? Integers, decimals, and mixed numbers

The short answer to the section's title is: convert it to a simple fraction and proceed as in the above section. Therefore, instead of answering the question "What is the multiplicative inverse of anything that isn't a simple fraction?" we'll describe how to change those three types of values to simple fractions.

• Integers
Recall that integers are numbers like `1`, `16`, `2020`, or `-56`. In fact, we can look at them as fractions with a denominator of `1` and numerator equal to the number. In other words, we have `1 = 1/1`, `16 = 16/1`, `2020 = 2020/1`, and `-56 = -56/1`.

• Decimals
Decimals like `0.012` are, in fact, fractions with a denominator equal to `10` to the power of how many digits we have after the decimal dot. In this example, we have three, so the denominator should be `10³ = 1000` (note that the number of zeros is equal to the exponent). Therefore, we get `0.012 = 12/1000`. Remember that sometimes (like in this case), we can write the number as an equivalent fraction with smaller numbers above and below. However, it's not necessary, and the answer will still be valid. Also, if we had, say, `3.012`, the result would be a mixed number of `3` and `12/100`. We deal with such a case in the next bullet point.

• Mixed numbers
These are numbers that have two components: an integer and a fraction (simple or decimal). As an example, consider `2¾` or `3.2`. To convert them into an (improper) fraction, we need to include the whole number (in this case, 2 and 3, respectively) into the fractional part. That amounts to "the number times the denominator of the fractional part," to which we add the numerator of the fraction. If we follow these instructions, our two examples give `2¾ = (2*4 + 3) / 4 = 11/4` and `3.2 = (3*10 + 2) / 10 = 32/10` (for the second, recall how we dealt with decimals in the above bullet point).

Whichever of the above we're facing, once we have the number written as a simple fraction, we simply apply what we've learned in the above section and obtain the result. Note that the answer might not be in its simplest form, so you might wish to reduce the nominator and denominator using tools such as the greatest common factor.

That concludes our elaborate answer to the question "What is the multiplicative inverse of a number?" which means that it's time to leave the theory behind and get on with examples.

## Example: using the multiplicative inverse calculator

Let's put the functionalities of Omni's multiplicative inverse calculator to the test and see how to find the multiplicative inverses of two numbers: `3.25` and `1⅜`.

We begin with `3.25`. The value has a decimal dot, so we start by choosing "an integer/decimal" under "Input in the form of" at the top of our tool. That will show a variable field called "Number" underneath, where we input (surprise, surprise) the number `3.25`. The multiplicative inverse will then appear underneath.

As for `1⅜`, we turn to the option "a mixed number" under "Input in the form of" since it consists of both an integer and a (simple) fraction. That will trigger three variable fields to appear: "Whole number," "Numerator," and "Denominator." Looking at the number at hand, we input `1`, `3`, and `8`, respectively. Just as before, the multiplicative inverse appears underneath the moment you give the last number.

For completion, let's conclude by showing how to find the multiplicative inverses ourselves. We follow instructions given in the above section, which means that in both cases, we first need to convert the numbers into (improper) fractions.

`3.25 = 3¼ = (3*4 + 1) / 4 = 13/4`,

`1⅜ = (1*8 + 3) / 8 = 11/8`.

Note how in the first case, we've reduced `0.25 = 25/100` into `¼` straight away. The multiplicative inverse also does that, but in a later step.

So what are the multiplicative inverses of `3.25` and `1⅜`? We simply flip the two expressions to get the inverses of the fractions: `4/13` and `8/11`, respectively.

Well, that was a piece of cake, wouldn't you say? Arguably, there might be more to arithmetics than just flipping fractions. Fortunately, we have all of Omni's dedicated calculators to help us along the way!

Maciej Kowalski, PhD candidate
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