Integer Calculator

Welcome to Omni's integer calculator, where we'll learn all about the four basic arithmetic operations: adding and subtracting integers, multiplying integers, and integer division. Then, we'll move to more difficult operations such as exponents, roots, and logarithms. Remember that the integer definition allows more than just positive integers, so this tool serves as a negative number calculator as well.

So what is an integer in math? What's a negative number squared or the log of a negative number?

The answer to that and so much more are right below!

Colloquially speaking, an integer is a whole number. In other words, they don't include fractions (simple or decimal: if you're not sure what either of these terms means, go quickly to our decimal to fraction calculator) or irrational numbers such as the π from circle calculations. The following are all examples of integers:

2, 2021, 13, -17, -173,029, 0, 1,000,000,000.

Note that:

• All positive integers, however long, are integers: we sometimes call them natural numbers;
• Zero is also an integer; and
• Integers include negative numbers as long as they don't contain fractions.

The formal integer definition is the following:

Let's stress the word "can" in the above integer definition. Observe how the simple fraction 4/2 is equal to 2, so it is an integer by fraction simplification, although it doesn't look like it at first glance.

Before we move on to the operations, let's spare a separate section to the differences between positive integers and negative ones.

Integers (and all other real numbers) appear on an infinite axis called the number line.

In essence, the line tells us where one number lies with respect to the others: is it larger (to the right) or smaller (to the left) of something else? When they introduce us to mathematics, we count to ten on our fingers, so we know that, for example, 2 comes after 1 but before 3.

Negative numbers are the mirror image of positive ones with the mirror put at 0. In other words, if we start at zero and go right, we'll visit 1, then 2, 3, and so on. On the other hand, if we go left, we meet the same numbers but with minuses: -1, then -2, -3, and so on. This way, a number and its opposite are at the same distance from 0 but to the opposite sides (this distance is called the number's absolute value).

Arithmetic and algebraic properties apply to all the values on the negative and positive number line. In particular, we can add, subtract, multiply, divide, raise to a power, take the root, calculate the logarithm, etc., using those numbers. The differences in negative and positive number rules are small, and we point them out in each section below.

Given a positive integer, we can also find the sum of digits in order to determine the divisibility of the number.

When adding and subtracting integers, it's a good idea to keep the negative and positive number line from the above section in mind.

Suppose that we have integers a and b, and let's explain how we can find a + b and a - b.

1. Look for a on the negative and positive number line.
2. To find a + b, move b positions from a:
   • To the right, if b is positive; or
   • To the left, if b is negative.
3. To find a - b, move b positions from a:
   • To the left, if b is positive; or
   • To the right, if b is negative.

See a few examples of adding and subtracting integers below:

• 3 + 4 = 7

• -5 + 9 = 4

• 10 + (-12) = 10 - 12 = -2

• -3 + (-7) = -3 - 7 = -10

• 4 - 5 = -1

• 3 - (-2) = 3 + 2 = 5

• -10 - (-4) = -10 + 4 = -6

Observe how whenever we had two signs next to each other, we have to put the negative number in brackets. Furthermore, it's possible to reduce the two into one in such case according to the following rules:

• + and + gives +;
• + and - gives -;
• - and + gives -; and
• - and - gives +.

The first one doesn't really happen here, but it'll come in handy in the next section.

In essence, the negative and positive number rules for multiplying integers and integer division are almost the same. The only thing we have to keep in mind is the sign. To be precise, the result's sign depends on those of the factors or of the dividend and divisor for multiplication and division, respectively.

On the other hand, the result's value itself, be it positive or negative, doesn't care much about the signs. As such, we can begin our calculations as if both integers were positive, compute what the result would be in that case, and only then fix the sign accordingly. And by "accordingly," we mean the same negative and positive number rules from the above section.

Below, we give a few examples of multiplying integers, followed by some integer division.

• 6 × 8 = 48

• -4 × 5 = -20

• 10 × (-2) = -20

• (-1) × (-8) = 8

• 12 / 4 = 3

• 24 / (-8) = -3

• -7 / (-2) = 3.5

That concludes the four basic operations covered by Omni's integer calculator (or negative number calculator, if you prefer). Now, we move on to more complicated (but still simple!) algebraic expressions.

• Exponents

  For positive integer exponents, the negative, and positive number rules are the same: the result is simply the number multiplied several times. And we've already seen how multiplying integers works in the above section, so let's simply mention a few examples:

  • 4³ = 4 × 4 × 4 = 64

  • (-3)⁴ = (-3) × (-3) × (-3) × (-3) = 81

  • (-2)⁵ = (-2) × (-2) × (-2) × (-2) × (-2) = -32

  Observe how, for negative exponent bases, the result's sign depends on the parity of the power. It's a straightforward consequence of the negative and positive number rules from the adding and subtracting integers section. In particular, a negative number squared will always give a positive value.

  Now, if the exponent is negative, we first get rid of its minus sign by changing the base into its multiplicative inverse: a^-b = (1/a)^b

  From there, we repeat the usual thing while remembering the rules for multiplying fractions. For example:

  4^-3 = (¼)³ = ¼ × ¼ × ¼ = ¹/₆₄

  (-3)^-4 = (-⅓)⁴ = (-⅓) × (-⅓) × (-⅓) × (-⅓) =¹/₈₁

  (-2)^-5 = (-½)⁵ = (-½) × (-½) × (-½) × (-½) × (-½) = -¹/₃₂

• Roots

  Taking the root (also called the radical) is the opposite operation to the exponent. As such, some rules apply to both. Most importantly, note how for exponents, even powers always give a positive result, no matter the base's sign. If we translate that property to roots, we'll get that radicals of even order exist only for positive numbers. In particular, you can't have the square root of a negative number. In fact, such things do exist, but they're no longer real numbers but complex numbers.

  Let's look at a couple of examples of integer roots:

  • ∜256 = 4

  • ∛(-125) = -5

• Logarithms

  Here the case is very simple: logarithms are defined only for positive numbers. In other words, there is no such thing as the log of a negative number. Again, similarly to the roots, in fact, there is, but it goes beyond real numbers, and the story gets quite difficult. If you are curious, check out our complex number calculator

  Let's finish this part of the section with a couple of log examples with positive integers:

  • log₁₀1000 = 3

  • log₉6561 = 4

  • log₂128 = 7

For completeness, let's finish with a quick instruction on how to use the integer calculator (or the negative number calculator, if you prefer).

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

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

   • Addition;
   • Subtraction;
   • Multiplication;
   • Division;
   • Exponent;
   • Root; and
   • Logarithm.

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.

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.

You might call it, "Integer calculator instructions," but we call it, "Five easy steps to happiness and quick maths." Whatever the name, we hope the tool saves you some time and helps with daily homework.

Yes. By definition, integers consist of all positive whole numbers (i.e., 1, 2, 3, and so on), their opposites (i.e., -1, -2, -3, and so on), and zero. The first group are the positive integers, the second are the negative ones, while 0 is neither positive nor negative.

Yes. By definition, we can represent all rational numbers as a simple fraction with integers in the numerator and denominator. As such, an integer a is equal to ^a/₁, which surely satisfies the condition.

Obviously, it doesn't work the other way: not all rational numbers are integers, e.g., ½.

No, unless you're working with complex numbers. Both a positive number and a negative number squared give positive values, so there's no way to get the square root of a negative number (or any other even-ordered root, for that matter).

Yes. By definition, integers consist of all positive whole numbers (i.e., 1, 2, 3, and so on), their opposites (i.e., -1, -2, -3, and so on), and zero. The second group clearly involves negative numbers.

Yes. By definition, an integer is a number that we can express without fractional expressions. For instance, the number 2 is clearly an integer, but we can also write it as ⁴/₂, which is a fraction but a reducible one.

To add integers a and b, you need to:

1. Look for a on the negative and positive number line.
2. Move b positions from a:
   • To the right, if b is positive; or
   • To the left, if b is negative.
3. The point you end up in is the sum.
4. Enjoy your result of adding integers.

To subtract integers a and b, you need to:

1. Look for a on the negative and positive number line.
2. Move b positions from a:
   • To the left, if b is positive; or
   • To the right, if b is negative.
3. The point you end up in is the difference.
4. Enjoy your result of subtracting integers.

No. By definition, integers consist of all positive whole numbers (i.e., 1, 2, 3, and so on), their opposites (i.e., -1, -2, -3, and so on), and zero. Out of those, natural numbers are only the first set. Some people also include zero as a natural number, although not all scientists agree.

To multiply integers a and b, you need to:

1. Multiply a and b as if they didn't have any signs.
2. Fix the result's sign according to these rules:
   • If both factors were positive or both negative, the result is positive; and
   • If one factor was positive and one negative, the result is negative.
3. Enjoy your result of multiplying integers.

For integer division of a and b, you need to:

1. Divide a and b as if they didn't have any signs.
2. Fix the result's sign according to these rules:
   • If both numbers were positive or both negative, the result is positive; and
   • If one number was positive and one negative, the result is negative.
3. Enjoy your result of integer division.