Chapters

From elementary math to high school math and beyond, arithmetic is a branch of maths that is present in mathematics courses at **every stage** of our academic life. For many around the world, however, topics in the hard sciences like math and science can often seem like insurmountable obstacles. With basic math becoming ever more important in today’s economy, the word problems and math skills that plagued us in the classroom are playing an increasing role outside of academia.

Mastering basic arithmetic skills is something the UK has historically been unsatisfactory in. In fact, many believe that the UK is undergoing a skills crisis due to low reading and maths scores combined with poor adult training. While understanding arithmetic math concepts is an integral part of **job security**, it can also play a vital role in empowering your decisions.

From politics to housing, maths is deeply entrenched in every important aspect of daily life, which is why many are turning to supplemental instruction in the form of tutoring. Whether you’re looking for an algebra tutor or general maths resources online, here’s what you need to know about arithmetic as well as **some examples** you can try at home.

Finding UK's most sought maths tutors is easy on Superprof.

## Arithmetic Laws and Definitions

Arithmetic is, without a doubt, the oldest branch of mathematics whose written history can be traced as far back as **18,000 BCE**. Derived from the Greek word for numbers, arithmos, arithmetic deals with four basic operations: subtraction, division, addition and multiplication.

While at first **this branch** may seem to only penetrate math lessons at the lower level, arithmetic is involved in everything from 1^{st} grade counting and 7^{th} grade pre-calculus to college level calculus. While all the elements of arithmetic don't just deal with the mathematical aspect, the following equations form the foundation of all maths operations.

**Arithmetic Basic Equations**

In arithmetic, there are three basic laws that govern the branch. Whether you’re working on math courses at the elementary school math level or are studying advanced math, chances are you use these rules everyday and perhaps even without thinking about them. Here are some examples of the **basic equations** that can be solved with these three rules.

#### Commutative Law

The commutative law basically states that, when multiplying or adding, the order by which you complete the operation doesn’t matter. In other words:

**a + b = b + a**

**a x b = b x a**

You can remember this law because of the word “commute,” because no matter the combination of numbers, these groups of numbers always “commute” together to the same place, or result. Here’s a **numerical example**:

2 + 4 = 4 + 2

6 x 3 = 3 x 6

This law also applies to **percentages**. For example, a% of b is always equal to b% of a. Here’s the rule in numbers which you can use to verify the rule:

3% of 60 = 60% of 3

#### Associative Law

The associative law deals with the fact that, no matter how you choose to group an addition or multiplication of numbers, the result is the same. The rule **looks like** this:

(a+ b) + c = a + (b + c)

(a x b) x c = a x (b x c)

Looking above, you can see that it doesn’t matter how we choose to group or “associate” two numbers in a big addition or multiplication operation because the result is the same. Here is a **numerical example**:

(2 + 4) + 3 = 2 + (4+3) which leads to 6 + 3 = 2 + 7

(3 x 3) x 4 = 3 x (3 x 4) which leads to 9 x 4 = 3 x 12

One of the reasons why this law is important is because often, it is easier or sometimes necessary to rearrange these numbers in a different order.

#### Distributive Law

The distributive law is a bit more complicated but is probably the law that is used most often. It deals with the fact that when multiplying an addition of numbers, you can **distribute** the multiplier to each number being added first. In other words:

a x (b + c) = a x b + a x c

As exampled above, the a is “distributed” to both b and c first and then both numbers are added. Again, these kinds of operations can make an operation **easier**. Here’s an example:

3 x (8 + 9) is easier to solve when rearranged as 3 x 8 +3 x 9

Here’s another way of using the same law in **reverse**:

Instead of solving 13 x 6 + 13 x 4 you can solve 13 x (6 + 4)

## Examples Arithmetic at Different Levels

From subjects like trigonometry to concepts like inequalities and linear equations, arithmetic is a vital skill for most math classes at every level. Whether you’re searching for a math tutor or are building your own **progress plan**, understanding the type and which level of arithmetic you’re struggling with can give you a better chance of achieving your academic and skill goals.

*Elementary Level*

Math class at the elementary level involves many basic concepts that are the core standards you will use to build upon in more advanced math. One of the best examples of this can be seen through **decimals**.

Decimals have many practical purposes, but the main way in which people learn the decimal is through fractions. In any maths program, you are **likely to find** some of the following examples.

0.2 = 2/10 = 1/5

0.25 = 2/10 + 50/100 = 25/100 = 1/4

Using the rules and concepts we already know, we can easily see that arithmetic forms the basis of all decimals. Two of the four main arithmetic operations, division and addition, can be used to find the answers. Keep in mind that in the decimal numbers above, the number 2 is in the **tenths** position while the number 5 is in the **hundredths**.

*Middle School Level*

At the middle school level, arithmetic gets another boost. Whether it’s 7^{th} grade or 8^{th} grade **curriculum**, some of the most common concepts involving arithmetic include square roots, linear equations, polynomial equations and inequalities, systems of equations and quadratic equations.

In other words, middle school is filled with algebraic topics that can come in the form of pre-algebra, geometry, and more. Here is an example of one important topic you’re likely to find in middle school arithmetic: **simplifying** algebraic expressions.

3x + 5(x - 6)

The **first step** in this problem is to use the distributive law to get:

3x + 5x - 30

Next, we can add 3x and 5x together. Taking a quick **look again** at the distributive law, we can see why:

x(3+5) is the same thing as 3x + 5x

Combining all the concepts we’ve learned, we get the **final answer**:

8x -30

From here, we cannot simplify any further.

*High School Level*

Maths at the high school level can vary a lot between students because of the fact that many students do not choose to take some higher level maths courses. Examples of the arithmetic you’re **likely to encounter** at this level include maths with rational expressions, math logarithms, polynomials, rational functions, exponents and trigonometric concepts.

One common example of the arithmetic you can use at this level can be seen through matrices. A matrix is a way of arranging numbers in columns and rows and has many practical applications in statistics, project management, computer science and more. Here’s an example of how to find the **determinant** of a matrix.

A = [a b

c d]

To find the determinant of a 2x2 matrix, we use the following equation: **det A = ad – bc**. Try to find the determinant of the following matrix:

A = [2 4

6 3]

*College Level*

Arithmetic at the college level gets a bit more complex, involved in everything from differential equations to probability. A simple example of the arithmetic you’re likely to encounter, here’s how you can **find the probability** of two independent events A and B both happening.

Let’s say the probability you will read this article up to this point is 1/16 (event A). The probability you will win the National Lottery is 1/45,057,474 (event B). What is the probability that you will read this article up to this point *and* will win the national lottery? Try calculating at home:

**Probability (event A and B) = Probability (A) * Probability (B)**

## Online Arithmetic Resources and Private Tutors

If you’re looking for more arithmetic tools and resources, make sure to check out this guide on how to get maths help online.

If you’re looking for a tutor in maths for arithmetic help, there are many different options to check out if you’re living in the UK. Start by checking out ads in your local community as some studies have suggested many students get more motivated when questions like “Why do I have to learn this?” are answered by their peers.

Next, make sure to check out Superprof. With a math teacher community of over 122,700 people offering everything from online group classes to in person one-off sessions, you’ll be able to find the best private tutor for your needs.