# Arithmetic Basics: Meaning, Operations, Examples & Practice

Whether you’re building a house or starting a new math class, a strong foundation is key. A solid understanding of arithmetic basics will help you not only with tonight’s homework, but also in feeling confident no matter where your math journey takes you next.

We do understand that, sometimes, half the battle is knowing where to begin. That’s why we’re here to walk you through step-by-step.

After all, that’s kind of our thing.

## Start here: What is arithmetic?

Arithmetic is the most fundamental branch of mathematics, which is why it’s so important to learn and really understand. Arithmetic touches every other branch of mathematics, acting kind of like the basic building blocks for more complicated and advanced topics.

It’s also a pretty broad subject. So, depending on grade level, arithmetic can look a little different:

- First & second grade go through the basics of the basics: mainly addition and subtraction, but sometimes multiplication and division, too. Second grade might also dive into rounding numbers or even simple word problems.
- Third, fourth, and fifth grade starts to get more complicated with fractions, decimals, percentages, proportions, and different word problems.
- Sixth grade isn’t just the start of middle school (and all the feelings that go with it!); it can also be the end of basic arithmetic training and the start of algebra — but that’s a whole other discussion 😉

That being said, every school and every student is a little different, so wherever you find yourself on the arithmetic adventure is totally valid!

If you find yourself struggling with a particular example, don’t forget to use your Photomath app to scan the problem so we can walk you through it. For now, let’s get a good picture of arithmetic as a field of study.

### The formal definition of arithmetic

There are a few different ways to define arithmetic as a branch of mathematics, but here’s how we like to define it:

**Arithmetic** is a branch of mathematics that deals with numbers, their properties, and operations with numbers.

### The real-world meaning of arithmetic

Sure, it’s nice to know the true mathematical definition of arithmetic, but it’s also important to know how it applies to your life!

Once you start going through examples, you’ll see that arithmetic is everywhere you look. From adding your shopping cart total to dividing up dinner portions, you’re constantly performing arithmetic whether you know it or not! In a way, you can think of arithmetic as the most basic everyday math.

Fun fact: You can also thank arithmetic for the advancement of society! Arithmetic has been instrumental in getting us from Ancient Egypt to now; farming, economies, technology, and so much more were all made possible by arithmetic. Pretty cool, right?

## Arithmetic basics

Okay, now that we’ve got our bearings, let’s get into some MATH!

Arithmetic is generally thought of as basic operations — addition, subtraction, multiplication, and division — and those operations are the main foundation of this branch of mathematics.

As you progress through arithmetic learning, those basics will combine and coexist to form other concepts like:

- Measuring angles
- Calculating volume
- Calculating area
- Exponentiation
- Converting units of measurement
- Calculating percentages

But right now, we’re focusing on the basics, so let’s start there!

## Arithmetic operators

Arithmetic is all about the building blocks, and the basic arithmetic operators are some of the most important building blocks around!

Operators tell us how one value should relate to another. Here are the four basic arithmetic operators:

Add | $$1 + 1 = 2$$ | The result of addition is the “sum” |

Subtract | $$3 - 2 = 1$$ | The result of subtraction is the “difference” |

Multiply | $$\displaylines{4 × 2 = 8 \\ 2 * 3 = 6 \\ 5 ⋅ 2 = 10}$$ | The result of multiplication is the “product” |

Divide | $$\displaylines{12 ÷ 3 = 4 \\ 10/2=5}$$ | The result of division is the “quotient” |

As arithmetic gets more complicated, you may also see a fifth operator:

Exponentiation | $$10^3$$ | $$\text{ The number } 3 \text{ is the “exponent.” You would read the example expression as } 10 \text{ to the power of } 3 \text{ or } 10 \text{ raised to the power of } 3$$ |

Think of operators like punctuation marks — how we know a sentence ending with “?” is a question, or “!” signals excitement. The operators are our cues for how to act.

## Arithmetic operations

Now that we have a clear picture of what each operator means, let’s dive into their operations!

Essentially, the operator is the symbol shown in the expression, and the operation is the action we perform. Let’s get into them!

$$+$$ | Addition |

$$-$$ | Subtraction |

$$×, *, ⋅$$ | Multiplication |

$$÷ , /$$ | Division |

These operations can apply to:

- Single-digit numbers
- Multi-digit numbers
- Decimals
- Fractions

…and more!

These operations can also exist side-by-side in expressions and equations, which is when you’ll need to know the PEMDAS rule.

“PEMDAS” is a fun acronym to help you remember the order of operations when more than one operator or expression is present:

- Parentheses
- Exponents
- Multiplication (left to right)
- Division (left to right)
- Addition (left to right)
- Subtraction (left to right)

## Arithmetic: Mean

What is a mean in arithmetic? You actually already know what it is — because “mean” is just a fancy word for “average”!

That’s right – we’re not saying that arithmetic is mean (although it may feel like that sometimes!).

In arithmetic, usually around $$6$$th grade, you’ll hear “mean, median, and mode” thrown around quite a bit. These are all in reference to a set of numbers and their tendencies or consistencies.

The median is the midpoint of the data set, while the mode is the most common number in the set. The mean is the average!

### Averages in arithmetic

Need a refresher on taking an average? Let’s do it!

- Identify the group of numbers
*I need to find the average of*$$4, 5, 3, 8,$$*and*$$10$$

- Add all numbers in the group together to find the sum
- $$4 + 5 + 3 + 8 + 10 = 30$$

- Divide the sum by the number of numbers in the group
*There are*$$5$$*numbers in the group, so we’ll divide*$$30$$*by*$$5$$

- Find your average!
- $$30 ÷ 5 = 6$$,
*so the average is*$$6$$

- $$30 ÷ 5 = 6$$,

## Examples of arithmetic in math

Now that we have an idea of the concepts behind arithmetic, let’s take a look at some examples you might see in class, homework, or textbooks! Because arithmetic spans multiple grades, we’ll break it down by what’s traditionally taught at each level:

Grade 1 | $$2+3=5$$ |

Grade 2 | $$64-19=45$$ |

Grade 3 | $$45=9×5, 18÷3=6$$ |

Grade 4 | $$473÷5=94 \text{ with a remainder of } 3$$ |

Grade 5 | $$1.2+4.3=5.5$$ |

Grade 6 | $$1.501=150.1 \%$$ |

Think of that as the roadmap of your arithmetic journey — where you are, where you’ve been, and where you’re headed.

Remember: If you need a detailed explanation for the exact problem in front of you, you can always use your Photomath app for a step-by-step walk-through!

## Arithmetic practice problems

It can be difficult to remember how to do something if you don’t have much experience actually doing it — that’s why practice is so important!

While it might seem like math is all about finding the right answer, we think math is really about finding the right answer. Remember to focus on each step of the process as you work through these practice problems, because the journey matters just as much as the destination.

Problem | Solution |
---|---|

$$2+7$$ | $$9$$ |

$$49-17$$ | $$32$$ |

$$57-29$$ | $$28$$ |

$$2×12$$ | $$24$$ |

$$43×12$$ | $$516$$ |

$$99÷3$$ | $$33$$ |

$$314÷12$$ | $$26 \text{ with a remainder of }2$$ |

$$11.23-2.3$$ | $$8.93$$ |

$$\frac{1}{4}+\frac{3}{4}$$ | $$1$$ |

$$\frac{13}{16}+\frac{3}{4}$$ | $$\frac{25}{16}$$ |

Once you’re finished, scan the problems and check your work with the Photomath app! Do your solving steps match ours?

**Here’s how we solve the first practice problem in the app:**

## Arithmetic vs. Mathematics

As we hope you’ve come to realize, arithmetic is a basic branch of mathematics. That means arithmetic is just one part of the larger field of mathematics.

You may sometimes hear the terms used interchangeably in reference to basic math concepts, but because you’re studying with us, you’ll know that one doesn’t necessarily equal the other!

And just like it’s not “arithmetic vs. mathematics,” it doesn’t have you be YOU vs. mathematics, either! We’re here to help you – every step of the way.

**FAQ**

- What part of math is arithmetic?
- Arithmetic is the base of the mathematical pyramid. It forms the foundation for more complex branches.

- What are the rules of arithmetic?
- The four rules of arithmetic are actually operations: addition, subtraction, multiplication, and division.

- What is an arithmetic sequence?
- An arithmetic sequence is a sequence in which each progressive term increases or decreases by adding or subtracting the same constant, so that the difference of consecutive terms is always equal. Example: “$$3, 6, 9, 12…$$”