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# Base Numbers|Definition & Meaning

## Definition 1

The **base number** is defined as the number multiplied by itself multiple times and can be written with an **exponent**. Different numbers multiplying cannot have a base number as it is only a single number and requires **self-multiplication**.

## Definition 2

A base number is also defined as the **number** of specific or **unique digits** that are present in a **numeral system**. The digits are placed in a **positional** manner that defines the particular value of the digit in the numeral system.

**Figure 1** shows the **base** number **6** multiplied **3** times to get **216**.

## Base Number in Multiplication

The base number or base in the **multiplication** operation is the number multiplied in such a way that it can be written with an** exponent**. The exponent plays a key role while defining the **base number**.

An **exponent** is a number that defines the **number of times** a number is **multiplied** by itself. It is also known as the **power** or** index** of the base number.

Without a **base** number, there is no meaning of exponent. The **exponent** is written in **superscript** to the base number which is multiplied several times.

For example, in $3^2$, **3** is the **base** number. The number** 2** indicates that the base number is **multiplied **twice, hence the **exponent** is **2**. Also, $3^2$ can also be written with the multiplication operator as **3 × 3**.

Consider a multiplication of **4 × 4 × 4 × 4 × 4**. As 4 is multiplied **five **times by itself, so** 4** is the **base** number and **5** is the **exponent**. This multiplication can also be written as **$4^5$**. This is demonstrated in **figure 2**.

## Base Number in Numeral Systems

A **base number** of a number system defines the number of values unique to a particular numeral system. The set of values has **different numerical meanings** for different number systems.

The **base** of a number system is also known as the **radix**. The base number is written in **subscripts** to the specific **number** of a numeral system.

The following **four numeral systems** have different base numbers according to the number of specific digits or symbols present in that system.

### Decimal Number System

The Latin word “**Decimus**” gives the origin to the word decimal meaning “**tenth**”.

The **decimal** number system has the base or** radix 10** as it constitutes ten different **digits**. The unique digits in the decimal number system are from **0** to **9**.

All are **single-digit** numbers and each number has a specific **weight** and position. This is the **most used** system in our daily lives.

The conversion of** $23_{10}$ **from decimal to binary is shown in **figure 3**.

### Binary Digit System

“**Bi**” means** two**. As the name suggests, the binary system consists of two digits **0** and **1**, and has the **base** number **2**. The binary system is primarily used in digital logic design such as **logic gates**.

**Binary coding** is also used in computers to understand instructions from the user and perform relevant tasks.

For example, **${11110}_2$** is converted to a decimal number shown in **figure 4**.

### Octal Number System

Octal means “**eight**”. The octal number system has unique digits from **0 to 7** hence it is a **base-8** numeral system.

It has **fewer digits** and is not as common as the decimal number system. Also, it is easy to convert from **binary **to** octal** as a octal digit requires only three binary numbers.

For example, **${16}_8$** is converted to a binary number as shown in **figure 5**.

### Hexadecimal Digit System

**Hexa** means “**six**”. The hexadecimal number system has the **base 16** as it has sixteen different numbers and symbols from **0** to **15**.

The two-digit numbers **10**, **11**, **12**, **13**, **14**, and **15** should be represented by unique symbols to **differentiate** them from the other numerals. So, they are designated as **A**, **B**, **C**, **D**, **E**, and **F**.

For example, **$(41)_{16}$** is converted into decimal as shown in **figure 6**.

## Calculating a Base Number

The **base number** of a certain number can be identified given the equivalent **number** in some other **numeral system**.

For example, **35** is a number whose **base** number is unknown. But it is given that **35** is equal to **$(23)_{10}$**. So, how to calculate the base number of 35? Let the **unknown** base of **35** be **b**.

First, we need to subtract the **one’s place** of the base b number which is **5** number from **23** which is given. So **23 – 5 = 18**.

Then, we need to divide **18** by the **tens place** of the base b number which is **3**. So, **18 ÷ 3 = 6**. So, the base of the number **35** is **6**.

The result can be **checked** by multiplying the tens place of the number that is **3** by **6** and then adding the result with 5. So, **{3(6)} + 5 = 23**. Hence, **$(23)_{10}$ = $(35)_{6}$**.

## Solved Examples Demonstrating Base Numbers

### Example 1

What is the **base** number in **7 × 7 × 7 × 7 × 7 × 7**? Also, write the multiplication in** exponent** form, find the index and simplify.

### Solution

As **7** is multiplied by itself six times, it can be written as **$7^{6}$**. Here, **7** is the **base** number and **6** is the **exponent** or index. The simplification of this multiplication results in **117,649**.

### Example 2

Convert **$(58)_{10}$** into a **binary** number.

### Solution

To convert **58** to a** binary** number, it is **divided** by **2** repeatedly and the remainder is written beside the quotient.

When the final **quotient** is **1** or **0**, the binary number is the last quotient and the remainders. This process is shown in **figure 7**.

So the binary number for **$(58)_{10}$** is **$(111010)_{2}$**.

*All the images are created using GeoGebra.*