# Decimal Calculator

Created by Maciej Kowalski, PhD candidate
Reviewed by Steven Wooding
Last updated: Dec 13, 2022

Welcome to Omni's decimal calculator, where we'll learn all about adding, subtracting, multiplying, and dividing decimals, as well as about decimal exponents, square roots of a decimal, and logarithms with decimals. The topic is not too difficult: it all boils down to simple arithmetic, but we'll go through it all nice and slow so that we don't miss a thing. For simplicity, we've split the text below into sections, one for each operation available in our decimal calculator.

To select this option in the decimal calculator, choose Addition in the Operation variable field.

When adding and subtracting decimals, the most important rule is to keep track of the decimal dot. To be precise, the operation boils down to doing the same thing we do when adding integers, as long as the two numbers have the same number of digits after the decimal dot. And if they don't, we force them to.

Decimals are fractions that have very special denominators: a power of 10. As such, we can apply to them the same rules we use when simplifying regular fractions. In this case, it means we can add as many zeros after the last digit (after the decimal dot, mind you) as we like and still have the same value.

\small \begin{aligned} 12.3 &= 12.30 \\ &= 12.30000 \\ &= 12.30000000000 \\ \end{aligned}

For instance, when we want to add two decimals, say, $12.3$ and $1.719$, we begin by fixing them so that they have the same number of digits after the dot: $12.300$ and $1.719$. From there, we can use long addition to find the sum:

\small \begin{aligned} 12&.300 \\ +\quad1&.719 \\ \hline \\[-10pt] 14&.019 \\ \end{aligned}

And that's how the adding decimals calculator works. Let's check the others, shall we?

## Subtracting decimals calculator

To select this option in the decimal calculator, choose Subtraction in the Operation variable field.

In essence, subtraction is very similar to addition. Sure, they are each other's opposites, but we can always change the subtraction into adding the opposite of a number. As a result, the two follow very similar rules.

In particular, when subtracting decimals, we need both expressions to have the same number of digits after the decimal dot. In case they don't, we use the same trick we did in the "adding decimals calculator" section and add a suitable number of zeros to the "shorter" number.

For example, if we want to calculate $5.1 - 2.34$, we write it as $5.10 - 2.34$ and turn to usual long subtraction keeping in mind the position of the decimal dot:

\small \begin{aligned} 5&.10 \\ -\quad2&.34\\ \hline \\[-10pt] 2&.76 \end{aligned}

And that's all there is to the subtracting decimals calculator. We're ready for the next one!

## Multiplying decimals calculator

To select this option in the decimal calculator, choose Multiplication in the Operation variable field.

This one's a bit different. First of all, multiplying decimals doesn't require the two expressions to have the same number of digits after the decimal dot. Secondly, the result will have as many digits after the dot as the two numbers combined (see the addition and subtraction sections later in this article for comparison). Other than that, we follow the same rules we do for long multiplication.

For example, multiplying $1.43$ by $3.5$ will give:

\small \begin{alignat*}{2} 1&.43 \\ \times\quad3&.5\\ \hline \\[-10pt] 3&.5 \qquad && (1.00\times3.5) \\ +\quad1&.4 && (0.4\times3.5)\\ +\quad0&.105 && (0.03\times3.5)\\ \hline \\[-10pt] 5&.005 \\ \end{alignat*}

Voilà! That covers the multiplying decimals calculator. Three down, four to go!

## Dividing decimals calculator

To select this option in the decimal calculator, choose Division in the Operation variable field.

This one's special. In some sense, we don't want to divide decimals. Instead, we find a similar quotient that uses integers.

By the rules of simplifying fractions, we can multiply the dividend and divisor by an arbitrary non-zero number and, as a result, still have the same value. We choose a very specific number: $10$ to the power equal to the number of digits after the decimal dot in the number with more of them. In other words, the multiplier consists of $1$ and as many zeros, as the "longer" number has digits after the dot.

Let's take an example: $6.45 \div 1.5$. The first number (i.e., $6.45$) has one more digit after the decimal dot than the second number ($1.5$). Therefore, we multiply the dividend and divisor by $10$ to how many digits we had:

\small \begin{aligned} & 6.45 \div 1.5 \\ % =& (6.45 \times 10²) \div (1.5 \times 10²) \\ =& (6.45 \times 100) \div (1.5 \times 100) \\ =& 645 \div 150 \\ \end{aligned}

Now that we have integers, we do the usual long division:

\small \begin{aligned} \quad 4.3 \\ 150\ \overline{|\ 645} \\ -600 \\ \hline \\[-10pt] 45 \\ -45 \\ \hline \\[-10pt] 0 \end{aligned}

That concludes the part on the dividing decimals calculator.

With that, we have all four basic arithmetic operations on decimals described. Now we move on to algebra.

## Decimal exponents

To select this option in the decimal calculator, choose Exponent in the Operation variable field.

In case we want to raise a decimal to an integer exponent, we do the usual thing: multiply the number as many times as the exponent says. For example,

\small \begin{aligned} & 3.2^4 \\ =\ &3.2 \times 3.2 \times 3.2 \times 3.2 \\ =\ &104.8576 \\ \end{aligned}

Remember that for the multiplication, you can use the rules from the multiplying decimals calculator section or switch to regular fractions and multiply them in that form.

On the other hand, when the decimal appears in the exponent, the story gets more interesting. Then, we need to treat it as a fraction exponent, i.e.,:

• Change it into a simple fraction (not a mixed number, mind you);
• Raise the number under the initial exponent to the power given by the numerator; and
• Stake the root of the whole thing of order defined by the denominator.

For instance:

\small \begin{aligned} & 1.4^{2.3} \\ =\ & 1.4^{23/10} \\ =\ & \sqrt[10]{1.4^{23}} \\ \approx\ & 2.1682 \\ \end{aligned}

Arguably, decimal exponents proved a tough nut to crack in comparison with the previous operations. Luckily, the idea is very similar to that in the next section.

## (Square) root of a decimal

To select this option in the decimal calculator, choose Root in the Operation variable field.

Fortunately, we can view roots as exponents with the multiplicative inverse of the root order as powers. Symbolically, this means that:

$\small \sqrt[b]{a} = a^{1/b}$

As such, radicals follow similar rules to exponents, e.g., when multiplying or dividing them. In particular, this applies to problems containing decimals.

When the decimal is inside an integer-order root, we can either try to round up the result (the way we do with roots in general) or convert the number into a simple fraction and then split the radical over the numerator and denominator. For instance, if we want to find the cube root $\sqrt[3]{2.7}$, then we can write:

\small \begin{aligned} &\sqrt[3]{2.7} \\ =\ & \sqrt[3]{\tfrac{27}{10}} \\ =\ & \sqrt[3]{27} \div \sqrt[3]{10} \\ \end{aligned}

On the other hand, if the decimal appears in the root order, we change it into an exponent using the formula above and then follow the instructions from the above section. As an example, let's take $\sqrt[3.4]{12.9}$:

\small \begin{aligned} &\sqrt[3.4]{12.9} \\ =\ &12.9^{1/3.4} \\ =\ &12.9^{10/34} \\ =\ &\sqrt[34]{12.9^{10}} \\ \approx\ & 2.1215 \end{aligned}

In itself, finding the (square) root of a decimal may be difficult, but if we keep the above section in mind, it should just be plain sailing.

That leaves us with one last decimal operation to explain: the logarithm.

## Logarithms with decimals

To select this option in the decimal calculator, choose Logarithm in the Operation variable field.

In general, logarithms are difficult. The expression $\log_a b$ tells us to what power we need to raise $a$ so that we get $b$. That seems like a tricky question in itself, and we can further complicate it by introducing decimals.

The good news is that having decimals in a logarithmic expression doesn't change much: the definition stays the same, and, for example, $\log_{2.2} 13.8$ still denotes the exponent to which we should raise $2.2$ to obtain $13.8$. The bad news is that it's hard to calculate that

$\small \log_{2.2} 13.8 \approx 3.3289$

Of course, some tricks get us closer to the result, such as the change of base formula. However, often we simply turn to external tools for help: something like Omni's decimal calculator.

Still, if the decimal appears only in the value under the log and not in its base, there is one easy trick that is quite universal: expand the logarithm.

In essence, if we have a logarithmic expression of a fraction (remember that decimals are, in fact, fractions), then we can expand it into the log of the numerator minus the log of the denominator. For example:

\small \begin{aligned} &\log_3 7.136 \\ =\ &\log_3 \tfrac{7136}{1000} \\ =\ &\log_3 7136 - \log_3 1000 \\ =\ &\log_3 7136 - 10 \end{aligned}

There's still $\log_3 7136$ to find, but at least we got something! Truth be told, logarithms with decimals (or without them, for that matter) can be troublesome!

For completeness, let's have one quick instruction on how to use the decimal calculator.

## Using the decimal calculator

To make your lives easier, we've prepared a nice step-by-step instruction on how to use Omni's decimal calculator.

1. At the top of our tool, choose the operation you'd like to perform. There are seven options:

• Subtraction for the subtracting decimals calculator;
• Multiplication for the multiplying decimals calculator;
• Division for the dividing decimals calculator;
• Exponent for the decimal exponents calculator;
• Root for the decimal root calculator; and
• Logarithm for the logarithms with decimals calculator.
2. Once you select the operation, a symbolic formula appears underneath with a and b as variables.

3. Following the formula, input the values of a and b in the corresponding fields. These can be integers, decimals, etc.

4. Read off the result from underneath.

5. (For the four arithmetic operations) If you'd like to see the calculations described step by step, visit the appropriate Omni tool from the list below the result.

And that's all, folks! Figuring out decimals might be somewhere at the beginning of your journey into mathematics. Remember that will be happy to assist you further along!

## FAQ

### How do I divide decimals without a calculator?

To divide decimals, you need to:

1. Count the number of digits after the dot in both numbers.
2. Take the larger of the two integers from point 1.
3. Multiply both decimals by 10 to the power from point 2.
4. As a result, obtain two integers.
5. Divide the integers the usual way (e.g., with long division).
6. Enjoy the result of dividing the decimals.

### How do I multiply decimals without a calculator?

To multiply decimals, you need to:

1. Count the number of digits after the dot in both numbers.
2. Add the two integers from point 1.
3. Write both decimals as if they didn't have the dot.
4. Multiply the numbers the usual way (e.g., with long multiplication).
5. Count as many digits from the right as given in point 2.
6. Put the decimal dot to the left of those digits.
7. Enjoy the result of multiplying the decimals.

### How do I calculate the square root of a decimal number?

To calculate the square root of a decimal number, you need to:

1. Write the number as a simple fraction.
2. If needed, change the mixed number into an improper fraction.
3. Calculate the square root of the numerator.
4. Calculate the square root of the denominator.
5. Divide the number in step 3 by that in step 4.
6. If you want the irrational exact solution, rationalize the denominator.
7. Enjoy the square root of the decimal number.

### How do I calculate decimal exponents?

To calculate decimal exponents, you need to:

1. Write the exponent as a simple fraction.
2. If needed, change the mixed number into an improper fraction.
3. Raise the number under the exponent to the numerator from step 2.
4. Take the root of order equal the denominator from step 2.
5. Enjoy the result of the decimal exponent.

### How do I calculate the log of a decimal number?

Here are a few options to calculate the log of a decimal number:

• Use an external tool that allows decimal input;
• Apply the change of base formula to have a more attainable basis; or
• Write the number under the log as a simple fraction and expand the logarithm into the log of the numerator minus that of the denominator.

### How do I subtract decimals without a calculator?

To subtract decimals, you need to:

1. Count the digits after the dot in both decimals.
2. Add zeros so that the decimals have an equal number of digits after the dot.
3. Subtract the two numbers as if without the decimal dot.
4. Put the dot as many digits from the right as it was in step 2.
5. Enjoy the result of subtracting the decimals.
Maciej Kowalski, PhD candidate
Operation
a + b = ?
a
b
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