The Fundamental Operations: From Addition to Division

11 min readNov 3, 2024

Understanding and Using Arithmetic Operations in Daily Life

Introduction

Arithmetic is the most basic form of math, giving us the ability to manipulate different numbers to achieve a desired result.

It contains the 4 most basic operations — addition, subtraction, multiplication and division — which we’ll be talking about today.

Together, these 4 operations give us the ability to use math as a means of achieving a result. From finding your total when shopping to tallying up electoral votes, these operations are essential for us to use math at all, and they are also the building blocks for more complex but equally as important operations.

Quick Note

I will be using a few different terms — ‘operation’, ‘operand’ and ‘operator’ — throughout this article.

An operation is the action we take, for example adding up different items’ prices from a shopping cart.

An operand is one of the numbers used in the operation

An operator is the mathematical symbol we use to denote the operation, so in the example above it would be the plus sign / +.

The ‘equals’ sign ‘=’ is itself an operator which states that both its left and right sides represent the same value.

I will mention both the operation along with its respective operator as the article progresses.

We will also delve into what PEMDAS — or the order of operations — is.

Addition, Using The Operator + , Read As ‘Plus’

Addition is the process of combining values.

An example would be an election where each vote is added to a total, called sum.

In that scenario, you would have a number that you would repeatedly add 1 to for each vote, but addition allows us to add together any two different numbers.

Perhaps we want to know the total we would be paying at the store, and we have two items — a pound of carrots and a pound of potatoes — which cost 4$ and 3$ respectively.

We can add up the two numbers like so: 4 + 3. The result, once again called a sum, would be equal to 7.

Therefore, we can write up that entire operation as 4 + 3 = 7.

To take a slightly harder example: we have two sets of tomatoes split into 47 and 44 respectively.

This is a bit harder but still manageable, so let’s see: 47 + 44 equals what?

One method I usually use would be to think of 47 and 44 as 40, 40, 7 and 4 respectively.

If you haven’t seen my last article in which I also delved into this, basically, we use a decimal system in which every place value from right to left represents 10 times the amount of stuff as the one before.

This means that any number can be broken up into smaller bits. 47 is the same as 40 + 7, and 44 is the same as 40 + 4.

So, let’s write it like this: 40 + 40 + 7 + 4 = ?.

First, let’s add up the larger numbers, so we end up with: 80 + 7 + 4 = ?.

Now we add 7 + 4 to get 11. Since 11 has two digits, we keep the 1 in the ones place and carry over the 1 to the tens place, giving us 80 + 11 = 91.

Here, 10 + 1 is simple: we just replace the 0 with a 1, giving us 11.

This then leaves us with 80 + 11, which will be: 80 + 11 = 91.

Revisiting each step:

We have 47 + 44 = ?
We split the numbers into their components: 40 + 40 + 7 + 4 = ?
We added up the corresponding numbers together: 80 + 11.
This finally gave us 91, so 47 + 44 = 91.

Simply put, breaking down 47 + 44 can make it easier: we can think of these numbers as (40 + 7) + (40 + 4) = 80 + 11 = 91.

We will delve into the mathematical notation more towards the end, as I want to get the concepts themselves out of the way first.

Properties of Addition

Addition has a few fundamental properties that we always need to keep in mind, they are:

Commutative Property: The order of numbers doesn’t affect the result.

Algebraic Formulation: a + b = b + a
Example: 4 + 6 = 6 + 4 = 10

2. Associative Property: When adding more than two numbers, the grouping of numbers doesn’t affect the result.

Algebraic Formulation: (a + b) + c = a + (b + c)
Example: (2 + 3) + 4 = 2 + (3 + 4) = 9

3. Identity Property: Adding zero to any number doesn’t change its value.

Algebraic Formulation: a + 0 = a
Example: 7 + 0 = 7

4. Additive Inverse: For every number, there exists an opposite number, or “inverse,” that sums to zero.

Algebraic Formulation: a + (−a) = 0
Example: 5 + (−5) = 0

Subtraction, Using The Operator — Read As ‘Minus’

Subtraction is the process by which we take away (or remove) from a number’s value another number. This can also allow us to see the difference between two separate values.

It can be used when trying to figure out how much money you still have after a night out or tracking the stock of an item in a store.

Let’s take as an example 52–29 = ?

Here, we have to go backwards from addition, and we can’t use the method demonstrated above either.

Given that the first digit (remember, it goes from right to left!) of 2 is less than 9, we will have to also take away a 1 from the tens’ place.

Now, let’s start with that first digit. Well, with 2–9, we basically have a disk that loops around when we reach one of its ends. So once we take away enough to get a 0 in place of 2, we remove 2 from 9, we are left with a 7, which we then remove from 10 to get the digit we’ll need for this.

10–7 = 3, so 52–29 = 43–20. This is more manageable. We have a zero there, so all we need to do is remove 2 from 4, which yields 2. So 43–20 = 23.

Let’s go through it once again:

We have 52–29 = ?
We take out 9 from 52, which gives us 43.
We rewrite as 43–20.
We take out 20 from 43, which gives us 23. So: 52–29 = 23

Subtraction is also the reverse operation to addition.

Properties of Subtraction

Subtraction also has some properties, they are:

Non-Commutative: Changing the order of numbers affects the result.

Algebraic Formulation: a − b ≠ b − a
Example: 7 − 3 ≠ 3 − 7

2. Non-Associative: The grouping of numbers affects the result.

Algebraic Formulation: (a − b) − c ≠ a − (b − c)
Example: (10 − 5) − 2 ≠ 10 − (5 − 2)

3. Identity Property: Subtracting zero from a number doesn’t change its value.

Algebraic Formulation: a − 0 = a
Example: 9 − 0 = 9

4. Subtraction as Addition of Inverses: Subtraction can be thought of as adding the opposite of a number.

Algebraic Formulation: a − b = a + (−b)
Example: 6 − 4 = 6 + (−4) = 2

Multiplication, Most Commonly Using × , Read As ‘Times’

Multiplication, although more complex, is very simple, because it is simply repeated addition.

You can use it to calculate how much a certain number of items which all cost the same would cost at once.

For example, if we have 7 bananas in a store for 2$ each. we would write 7 x 2 = 14

By ‘repeated addition,’ I mean that you take a number and add it to itself multiple times. In cases with larger numbers, multiplication makes these calculations faster and more concise.

For example, instead of having to write 7 + 7 + 7 + 7 + 7 + 7 + 7, we can simply say 7 × 7. This is both easier to write and to read.

Properties of Multiplication

Multiplication has a lot of properties which are very similar to addition.

Commutative Property: The order of numbers doesn’t affect the result.

Algebraic Formulation: a × b = b × a
Example: 4 × 5 = 5 × 4 = 20

2. Associative Property: The grouping of numbers doesn’t affect the result.

Algebraic Formulation: (a × b) × c = a × (b × c)
Example: (2 × 3) × 4 = 2 × (3 × 4) = 24

3. Identity Property: Multiplying any number by 1 doesn’t change its value.

Algebraic Formulation: a × 1 = a
Example: 9 × 1 = 9

4. Zero Property: Multiplying any number by zero results in zero.

Algebraic Formulation: a × 0 = 0
Example: 7 × 0 = 0

5. Distributive Property: Multiplication distributes over addition or subtraction.

Algebraic Formulation: a × (b + c) = (a × b) + (a × c)
Example: 3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27

Division, Using The Operator ÷ and / , Read As ‘Divided By’

Division is the process of distributing a quantity into equal parts.

The result of a division is called the quotient. The number being divided (to the left of the division sign) is called the dividend, and the number dividing it (to the right of the sign) is called the divisor.

Think of it like the number of times a certain number goes into another number.

Let’s take an example: we have a box of lemons which we want to sort into equal groups of 4. How many groups will we have?

Well, we can use repeated subtraction and count the number of times we subtract until we reach zero.

Accordingly, we would write 20 / 4 = ?
Then we start subtracting. 20–4 = 16, so we can fit 4 into 20 at least once.
16–4 = 12 so we can fit 4 into 20 at least twice.
12–4 = 8 so we can fit 4 into 20 at least thrice.
8–4 = 4 so we can fit 4 into 20 four times by now
4–4 = 0 so in total, we can fit 4 into 20 a total of 5 times.
Thus, 20 / 4 = 5

If we divide 21 by 4, however, we get a quotient of 5 with a remainder of 1, which we write as 5 R 1.

Division is also the reverse operation to multiplication.

It is useful for splitting bills at restaurants or distributing resources in a classroom.

After middle school, remainders start to become less frequently used, opting for decimals. That is, numbers after a dot placed to the right of the whole number. These decimal numbers represent fractions — or parts — of the number 1.

For example, if we divide 3 / 6, we’ll get either 0 R 3, or, using decimal points, 0.5. This is because just half of 6 fits into 3. 0.5 is, accordingly, called one half, or in its fractional form 1/2.

Decimals are useful because they extend beyond just whole numbers, allowing us to represent any real number.

For the sake of clarify, we won’t go into what the difference between these is, just know that a whole number doesn’t have a decimal place (for example 1, 2, 3…), and real numbers do.

Properties of Division

Non-Commutative: Changing the order of numbers affects the result.

Algebraic Formulation: a / b ≠ b / a
Example: 8 / 4 ≠ 4 / 8

2. Non-Associative: The grouping of numbers affects the result.

Algebraic Formulation: (a / b) / c ≠ a / (b / c)
Example: (12 / 4) / 2 ≠ 12 / (4 / 2)

3. Identity Property: Dividing a number by 1 doesn’t change its value.

Algebraic Formulation: a / 1 = a
Example: 7 / 1 = 7

4. Division by Zero Undefined: Dividing any number by zero is undefined.

Algebraic Formulation: a / 0 is undefined for any real number that a could be. In other words, division by 0 is impossible.

5. Division as Multiplication of Inverses: Division can be represented as multiplication by the reciprocal of a number.

Algebraic Formulation: a / b = a × (1 / b)
Example: 6 / 2 = 6 × (1 / 2) = 3

PEMDAS, what is it?

PEMDAS has been widely used to teach people the order of operations. All calculations are done from left to right, but some take priority over others, resulting in the need for an order of operations.

Let’s take it letter-by-letter:

P stands for Parentheses. These are, in decreasing order of relevance, the normal parentheses ( and ), the square brackets [ and ], and the curly braces { and }. These do not represent operations themselves, but rather group certain operations together, signaling that they need to be done first before continuing.
E stands for Exponents. This is a slightly more complex operation which stands for repeated multiplication. We didn’t go through this operation today, but the syntax looks like a raised to a power n = a times a times a repeating n times.
M and D stand for multiplication and division. They have the same priority as each other, so when we have an expression which contains both, we turn to the default rule of ‘go from left to right’.
A and S stand for addition and subtraction. Like multiplication and division, these have the same priority as each other.

Let’s take a simple example to demonstrate the order or operations: 3 + 6 × (5 + 4) / 3–7 = ?

3 + 6 × (5 + 4) / 3–7 = ?
First, we calculate what the parentheses contains: 5 + 4 is 9 so we’ll write it down
3 + 6 × (5 + 4) / 3–7 = 3 + 6 × 9 / 3–7 = ?
We do not have any exponents here, so we’ll move to multiplication and division. 6 × 9 gives 54, and 54 divided by 3 gives 18.
3 + 6 × 9 / 3–7 = 3 + 18–7 = ?
Now we’re on to addition, as it’s the first in line. 3 + 18 is 21
3 + 18–7 = 21–7 = ?
And lastly for subtraction, 21–7 gives 14.
21–7 = 14

I will clarify again as it is very important: multiplication and division share the same priority, so when we have more than one of these types of operations in an expression, we simply go from left to right and calculate like above. The same applies for addition and subtraction.

Conclusion

To sum up, addition, subtraction, multiplication and division are interconnected with each other, and they are essential to learn in order to study past them, as they are the fundamental building blocks of math.

They are also vital in simple, day to day activities, like buying groceries, splitting bills etc.

I also want to reinforce the fact that understanding PEMDAS is essential in order to solve multi-step problems correctly.

Learning these simple concepts will get you a long way towards understanding harder math concepts.

With consistent practice, they will become second nature, enabling you to handle more complex calculations confidently.

Did you like this tutorial? Make sure to comment what you would like me to cover next!

If you want to delve deeper into these topics, here is a list of materials you could look at.