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# Column Addition|Definition & Meaning

## Definition

The most common and basic method used for **adding** numbers by writing them in **columns** over each other is called column addition. In this method, numbers are **vertically** stacked such that the digit in **units** place is over each other for all operands and similarly for all the other digits. The numbers are then added 1 digit at a time, starting from the **right** (unit’s place) and moving towards the **left**. Each step may produce a **carry** if the sum is a two-digit number which is carried onto the next addition on the left.

**Figure 1** shows the **column addition** of two one-digit numbers.

## Addends

The word “addend” comes from the Latin word “**addendum**,” which means “to be added.” For example, in the expression given below:

12 + 34 + 16 + 8

The numbers **12**, **34**, **16**, and **8** are the addends as they are being added together.

In column **addition**, the **addends** are written one over the other in a column such that the digits of units, tens, hundreds, etc., place of all the addends **align** in each **column**.

For** example**, in the above equation, the two-digit numbers 12, 34, 16, and 8 have the digits in the **units** place as 2, 4, 6, and 8 and in the **tens** place as 1, 3, 1, and 0, respectively. In the tens place of **8**, 0 can be written, or the **space** can be left empty.

### Augend

The **first** addend while writing an addition expression or **equation** is known as the “augend.” For example, in the expression:

5 + 7 + 10

The augend is **5**; the **augend** can also be **7** or **10** if the order of the addends is changed. This is because the **addition** process holds commutative and associative properties.

## Sum

The **result** obtained after the addition process is known as the **sum**. An addend is also known as a “**summand**” as it includes the word “sum.” For example, the sum of the summands **4**, **7**, and **8** is:

4 + 7 + 8 = 19

The sum of addends is found using **column addition**. For example, in the expression given under the “Addends” heading:

12 + 34 + 16 + 8

The sum of the column addition of 12, 34, 16, and 8 is shown in **figure 2**.

## Properties of Addition

To perform column addition, the **properties** of addition should be known, which are discussed as follows:

### Closure Property of Addition

The closure property of **addition** states that adding two **natural** numbers results in another natural number. The same is **true** for integers, decimals, fractions, and whole numbers. For **example**, adding decimals **1.3** and **2.5** results in another decimal, i.e., **3.8**.

### Commutative Property of Addition

The commutative property of addition states that **changing** the order of two **addends** does not have an effect on the **sum** of the addends. It means the two addends can be written in any **order**. If **p** and **q** are two addends, they can be added in any order, and the result will be the same as:

p + q = q + p

For example, two numbers, **16** and **21**, are to be added. In **column** addition, either **16** can be written on top of **21** or the other way around; the **sum** does not change as addition holds the **commutative** property.

So,

16 + 21 = 21 + 16 = 37

### Associative Property of Addition

The associative law of addition states that the **sum** does not change by changing the order of **grouping** of addends. If **l**, **m**, and **n** are three addends, then according to the associative property of addition:

l + (m + n) = (l + m) + n

It means that if **l** is added to **(m + n)** or** (l + m)** is added to **n**, the result will be the same. For example, the three addends** 17**, **23**, and **33** can be grouped in any order to obtain the same sum, i.e., **73** as:

17 + (23 + 33) = (17 + 23) + 33

17 + 56 = 40 + 33

73 = 73

### Additive Identity

The **additive** identity is a numerical value, when added to any number, gives the **identity** of that number. The additive identity is zero “**0**”, as it is the only number when added to any number, gives the **same** number. If **g** is any number, then

g + 0 = g

For **example**,

21 + 0 = 21

### Additive Inverse

The additive **inverse** is the of a number that is the **same** number but with the opposite **sign**. It is such a number, when added to the **original** number, produces the sum of **zero**. It is the **negation** of the original number. So, the additive inverse of **a** is **-a** and vice versa. For example, in the equation:

10 – 10 = 0

The additive inverse of **10** is **-10**, and that of -10 is 10, as their **sum** produces zero.

## Column Addition of One-digit Numbers

The column addition of a two **one-digit** number is the simplest form of addition. One-digit numbers are from **0** to **9**. Two or more two one-digit numbers are placed separately in a **column** and added together.

For **beginners**, small lines are marked on the **right** side of the column, next to and equal to each number in the column. These **lines** are then **counted** to know the total or sum of all the numbers. The **answer** is written in alignment with the column. Small lines are **skipped** if the student learns the addition operation.

## Column Addition of Two-digit Numbers

The **two-digit** numbers have the digit on the right side in the **units** or one’s place and the digit on the left in the **tens** place. The digits of the numbers in the units place and tens place are **stacked** separately in two columns. **U** or **O** denotes Units or Ones, and** T** denotes Tens. **U** is written on the top of the units column, and **T** is on top of the tens column.

The addition process is started from the **right-most** column(units column), and each column is dealt with separately. After adding the **units** column, similar to one-digit column addition, move towards the **left** to add the digits in the **tens** column. The **total** of every column is written below and in alignment with each column; thus, the final **sum** is obtained.

### Concept of Carry

A carry can be generated if the **sum** of two numbers in a column is a **two-digit** number(greater than **9**). In this case, the number in the **tens** place of the sum in a column is **forwarded** to the top of the next column. This number is known as “**carry**” as it is “**carried**” forward to the next column. This is done in order to **remember** to add the carry when the next column is added with it.

## Column Addition of Three-Digit Numbers

Column addition of **three-digit** numbers is similar to two-digit numbers, with the addition of a hundred’s place. The **hundreds** place column is denoted by** H** and is written on top of the hundreds column. After the tens **column** is added, the hundreds column is added. Thus, column addition is performed from **right** to** left**, always starting from the unit’s place.

Numbers with different numbers of **digits** can also be added through column addition, keeping in mind to **place** the digits of each number correctly in the units, tens, hundreds, thousands, etc., **column** place.

## Column Addition in Multiplication

The process of **multiplication** also involves column addition at the **end** if two **two-digit** numbers or more are being multiplied. At first, the unit place of the **second** number is multiplied by all the digits of the **first** number separately and written in a row from right to left.

Then, in the **next** row, after leaving a **space** in the units place, the tens place of the second number is **multiplied** by all the digits of the first number and written in a row. The **columns** are added to obtain the product. **Figure 3** shows the multiplication of two numbers, **31** and **22**.

## Column Addition of Algebraic Expressions

Algebraic **expressions** can be added using column addition. Like terms are placed on top of one another in different columns so they can be added. **Like terms** have the same variables, the **exponents** of variables should also be the same.

For example, 3y^{2} and 5y^{2} are like terms as the **variable** y and its exponent, i.e., **2** are the same, and the coefficient can be different.

To **add** 4a^{2} + 2ab + 7b^{2} and 2b^{2} + 5a^{2} + 3ab, the **like** terms are placed in different columns, as shown in **figure 4**.

## Solved Examples of Column Addition

### Example 1

Find the **sum** of **27**, **52**, and **369** using column addition.

### Solution

**Figure 5** shows the column** addition** of 27, 52, and 369.

Hence, the sum obtained is 448.

### Example 2

Two **algebraic** expressions are given below:

x^{3} + 3xyz + 4z^{2} + 2y + 4y^{2}

3z^{2} + 5y + 3y^{2} + 5x^{3}

Find the **sum** by using column addition.

### Solution

At first, **like terms** should be separated and placed in columns. The terms x^{3} and 5x^{3}, 4y^{2} and 3y^{2}, 4z^{2} and 3z^{2}, 2y and 5y are like terms. The term 3xyz has no like term, so it is placed separately with no term underneath it. **Figure 6** shows the column addition of the two algebraic expressions.

*All the images are created using GeoGebra.*