Addition and Subtraction: Definition, Diagram, Properties, Examples.

Addition and Subtraction: The first official evidence of Addition is that Egyptians and Babylonians used it in \(2000\) B.C. Addition and Subtraction symbols were invented around the \({16^{{\rm{th}}}}\) century, but before that, the equations were written in words, which was making it time-consuming to solve the problems. The Brahmagupta of India was the person who devised the modern methods for four fundamental operations (addition, subtraction, multiplication and division).

The two main arithmetic operations are addition and subtraction, which teach us how to add and subtract two or more integers or other mathematical values. Multiplication and division are the other two basic math operations. The plus sign (‘+’) is used to represent addition, whereas the negative sign (‘-‘) is used to represent subtraction (minus sign). It is essential for students to learn this, as we use them in our day to day lives in almost every activity.

In this article, we will discuss about addition and subtraction, definition, properties, operations, etc. Continue reading to know more.

What is Addition?

Definition: Addition means summing up two or more numbers or values to get another number, and the symbol representing Addition is \(\left( + \right).\)

For example: If you add the numbers \(4\) and \(5,\) we get \(9\) as a result.
\(4 + 5 = 9\)
Here, let us understand the sum of \(4\) and \(5\) practically. Suppose you have \(4\) oranges in a basket and \(5\) more oranges are added in the same basket. So, how many oranges are there altogether?
\(4 \to 1 + 1 + 1 + 1\)
\(5 \to 1 + 1 + 1 + 1 + 1\)
\(4 + 5 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1\)
Now, if we count each orange, then it is equal to \(9.\)

What is Subtraction?

Definition: The term Subtraction means to reduce the value of one number from another to get the required deal (difference), and the symbol representing Subtraction is \(\left( – \right).\)

Example: \(5\) subtracted from \(10\) we get \(5\) as the answer.
\(10 – 5 = 5\)
If we imagine \(10\) peaches in a basket and take out \(5\) peaches from the basket, how many peaches are left?
The answer is \(5\) peaches.
\(10 – 5 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 – 1 – 1 – 1 – 1 – 1 = 5\)

Facts About Addition and Subtraction

The example of addition and subtraction related facts is given below:
\(9 + 3 = 12\)
\(3 + 9 = 12\)
\(12 – 9 = 3\)
\(12 – 3 = 9\)

Specific numbers and facts are related or make up a fact “family”, and there are only three numbers in each family. In the above-given fact family, the members are \(9,\,3\) and \(12.\) These are related because we can add two of the numbers together to get the third number.
\(9 + 3 = 12\)

Here, you can switch the order of the given added numbers above to equal the third number again.
\(3 + 9 = 12\)

As you know, the inverse (opposite) of the Addition is Subtraction, starting with the sum (or the larger number), \(12\) and subtracting one of the addends, then the answer will be the third number.
\(12 – 9 = 3\)
And, the following fact will be \(12 – 3 = 9.\)

Properties of Addition

There are four properties of Addition, and they are given below:

1. Commutative Property
2. Associative Property
3. Distributive Property
4. Additive Identity Property
5. Additive Inverse Property

Let’s see each property in detail.

Commutative Property

The commutative property defines that numbers can be added in any order you want, and you will still get the same answer.

Example:
1. There are two groups, the first group has one \(\left( 4 \right)\) chocolates, and the other group has \(\left( 6 \right)\) chocolates. When you add, you will get a total of \(\left( 10 \right)\) chocolates.
2. Here, what if we changed the distributions. Let’s say the first group now has \(\left( 6 \right)\) chocolate and the second group has \(\left( 4 \right)\) chocolate. How many would be the total?
3. The total would be still the same \(\left( {10} \right).\)
4. In the commutative property of Addition, no matter which group has \(\left( 6 \right)\) chocolates and the \(\left( 4 \right)\) chocolates, the total would be still the same, i.e., \(\left( 10 \right)\) chocolates.

Associative Property

This is another property of addition. The associative property states that regardless of how numbers or objects are grouped, the total sum would still be the same.

Example: \(\left( {A + B} \right) + C,\) we will add the number of teddies Pari and Preet have.
1. Pari has \(4\) teddies while Preet has \(3\) teddies, then they have a total of \(6\) teddies. Then, we will add the number of teddies Sana has, which is \(2.\) The total number of teddies is \(\left( {4 + 3} \right) + 2 = 9.\)
2. Now, let’s try applying the associative property.
3. Following \(A + \left( {B + C} \right),\) we will add the number of teddies Preet and Sana have; we will have \(5\) teddies.
4. Then, add the number of teddies Pari has, which is \(4,\) then we get \(9.\)
5. The total number of teddies are \(9.\)
6. Regardless of whether we added the number of teddies Pari and Preet have, then add Sana’s number of teddies or add Preet and Sana’s teddies first and then add Pari’s teddies. Again, this is due to the associative property.

Distributive Property

This property is entirely different from the Commutative and the Associative property. Here, the sum of two numbers multiplied by the third number equals the sum when each number is multiplied by the third number.
\(A \times \left( {B + C} \right) = A \times B + A \times C\)

In this case, \(A\) is the monomial factor and \(\left( {B + C} \right)\) is the binomial factor.

Example:
Let’s take \(A = 1,\,B = 4\) and \(C = 2\)
L.H.S \( = A \times \left( {B + C} \right) = 1 \times \left( {4 + 2} \right)\)
\( = 1 \times 6\)
\( = 6\)
R.H.S \( = A \times B + A \times C = 1 \times 4 + 1 \times 2\)
\( = 4 \times 2\)
\( = 6\) \({\rm{L}}{\rm{.H}}{\rm{.S}} = {\rm{R}}{\rm{.H}}{\rm{.S}}\)
\(6 = 6\)

In the given above example, you can see, even when distributed \(A\) (monomial factor) to each value of the binomial factor \(B\) and \(C,\) the value remains the same on both sides. Thus, the distributive property is significant as it combines both the addition operation and the multiplication operation.

Additive Identity Property

Another property of Addition is the additive identity property, which refers that the sum of any number and zero is the number itself.

Example:
1. Rahul has \(4\) balloons while his friend does not have any balloon; thus, \(0.\) So, what is the total number of balloons they both have?
2. In regards to the additive identity property of Addition if you add \(4\) and \(0\) you will get \(4 + 0 = 4.\)
3. When you add \(0\) to any number, it does not change the value of the number.

Additive Inverse Property

Additive inverse of a number is a number, which, when added to the original number, results in zero \(\left( 0 \right).\)
Examples: We see that \(7 + \left( { – 7} \right) = 0.\) So, we say \( – 7\) is the additive inverse of \(7.\)
Also, \( – 15 + \left( {15} \right) = 0.\) So, \(15\) is the additive inverse of \( – 15.\)

Properties of Subtraction

The four properties we have in Subtraction are given below:
1. Subtracting a number from itself.
2. Subtraction of \(0.\)
3. Order property.
4. Subtraction of \(1.\)

Subtracting a Number From Itself

When you subtract/remove a number from the same number, your answer is \(0.\)
Example: \(9 – 9 = 0,\,100 – 100 = 0,\,89 – 89 = 0\)are few examples.

Subtraction of 0

When you subtract the number \(0\) from any number, you get the answer as the same number as zero has no value.
Example: \(4 – 0 = 4,\,78 – 0 = 78\) are a few of the samples.

Order Property

When subtracting any two numbers, we cannot change the order of the numbers as we do in Addition because if you change the order, the Subtraction will not be possible in the second case.
Example: \(34 – 23 = 11,\) and when you change the order \(23 – 34\) is not possible.

Subtraction of 1

When you subtract the number \(1\) from any number, you get the previous number.
Example: \(98 – 1 = 97,\,1000 – 1 = 999\) are some of the samples.

Rules of Addition and Subtraction

The rules to addition and subtraction are given below:

1. The Addition of two positive numbers will always be positive.
2. The Addition of two negative numbers is always negative.
3. The Subtraction of two positive numbers can be either positive or negative.
4. The Subtraction of two negative numbers can be either positive or negative.

Solved Examples – Addition and Subtraction

Q.1. Evaluate the given number: \(\left( { – 4} \right) + 8\)
Ans: Given, \(\left( { – 4} \right) + 8\)
We know that minus plus is minus.
So, \(\left( { – 4} \right) + 8 = 4\) [Subtract the numbers and use the sign of greater number]
Hence, the answer is \(4.\)

Q.2. Evaluate the given number: \(\left( { – 1} \right) – \left( { – 3} \right)\)
Ans: Given, \(\left( { – 1} \right) – \left( { – 3} \right)\)
\( = \left( { – 1} \right) + \left( { – 3} \right)\) [Convert subtraction problem into addition problem]
\( = \left( { – 1} \right) + \left( 3 \right)\) [Subtract the numbers and use the sign of greater number]
Hence, \(\left( { – 1} \right) – \left( { – 3} \right) = 2\)

Q.3. Add \( – 20\) and \( – 18.\)
Ans: Given \( – 20\) and \( – 18\) both are negative numbers.
So if we add them, we get the sum is negative as follows:
\(\left( { – 20} \right) + \left( { – 18} \right) = – 20 – 18 = – 38\)
Hence, the answer is \( – 38.\)

Q.4. Subtract \( – 30\) and \( – 19.\)
Ans: Given \(\left( { – 30} \right) – \left( { – 19} \right)\)
Here, we know that two minus signs change to plus.
So, \( – 30 + 19 = 30 – 19 = 11\)

Q.5. Evaluate \(9 – 10 + \left( { – 5} \right) + 6\)
Ans: Given, \(9 – 10 + \left( { – 5} \right) + 6\)
First, we will open the brackets.
\(9 – 10 – 5 + 6\)
Now, we will add the positive and negative integers separately.
\( = 9 + 6 – 10 – 5\)
\( = 15 – 15\)
\( = 0\)

Summary

In this article, we have understood what the term Addition and the term Subtraction means. Also have learnt about the properties of Addition like Commutative Property, Associative Property, Distributive Property and Additive Identity Property, and the properties of Subtraction are Subtracting a number from itself, Subtraction of \(0,\) Order property and Subtraction of \(1.\)

You glanced at facts of Addition and Subtraction, and some rules of Addition and Subtraction are given covered with some solved examples and FAQ.

Frequently Asked Questions

We have provided some frequently asked questions about addition and subtraction here:

Q.1. What is the rule of plus-minus?
Ans: When two plus signs are combined, you get a plus, and when two minus signs are connected, you get a plus. When a plus and a minus is integrated, you get a minus.

Q.2. Why minus and minus is plus?
Ans: Each number has an additive inverse associated with it (opposite number), which gives zero when we add to the original number. In fact, this is why the negative numbers were introduced, so that each positive number would have an additive inverse.

Q.3. What is the difference between Addition and Subtraction?
Ans: When you add or combine any two numbers like \(2 + 2 = 4\) is known as Addition. And, when you take away any number from the other number like \(4 – 2 = 2\) is known as Subtraction.

Q.4. What is the rule for Addition and Subtraction?
Ans: The rules to Addition and subtract are given below:
1. The Addition of two positive numbers will always be positive.
2. The Addition of two negative numbers is always negative.
3. The Subtraction of two positive numbers can be either positive or negative.
4. The Subtraction of two negative numbers can be either positive or negative.

Q.5. Why are Addition and Subtraction essential?
Ans: The terms Addition and Subtraction are used to represent and solve various kinds of problems. We know that different types of problems can be represented by Addition or Subtraction. It is essential to recognize these situations and represent them symbolically, building on counting with whole numbers.

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