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# Duodecimal|Definition & Meaning

### Definition

The word “Duodecimal” comprises two words, **duo** and **decimal**. “Duo” means “**two**” and “**decimal**” means “**ten**”. Two and ten add up to **twelve**. A duodecimal number system is a **base-12** number system also known as the **dozenal** number system.

**Figure 1** shows the **twelve** basic numerals of a **duodecimal** number system.

## Comparison of Decimal and Duodecimal Systems

A **decimal** number system is a **base-10** system having unique numbers that are **0**, **1**,** 2**, **3**, **4**, **5**, **6**, **7**, **8**, and** 9**.

From **0 to 9**, the **duodecimal** system is similar to a decimal system. For the numbers **10** and **11**, the characters **A** and **B** are used.

These **numerals** from** 0** to **9**, **A**, and** B** make the twelve **unique** notations of a duodecimal system. **Figure 2** shows the **decimal** and **duodecimal** numbers side by side.

The following **terms** are important in the **duodecimal** system.

### Dozen

A “**dozen**” is** twelve** units. It is similar to “**tens**” in the decimal system. So, the number “**12**” in the decimal system is the number “**10**” in the duodecimal system.

Also, “**10**” would be “one **dozen** and zero units” and “**12**” would be “one dozen and two **units”** in the duodecimal system. But a **“one** dozen and **two** units” is **14** in the decimal system.

Also, “**0.1**” in decimal is one-tenth but in duodecimal it is **one-twelfth**.

### Gross and Great Gross

A **hundred** in duodecimal is called a gross and a **thousand** is called a great gross.

## Duodecimal System – An Optimal System

The base** 12** of the **duodecimal** system has the **factors** (**1**, **12**), (**2**, **6**), and (**3**, **4**). It has more factors than the **decimal** system that are (**1**, **10**) and (**2**, **5**).

For the **octal** system, the factors of **8** are (**1**, **8**) and (**2**, **4**). Similarly, the **base-6** system has the factors (**1**, **6**) and (**2**, **3**).

The highest number of **factors** of 12 makes the **duodecimal** system an **optimal** system. It is considered **superior** to the decimal system.

It has **fewer** numerals as compared to a **hexadecimal** system and more factors than any of the systems with a **base** of less than **12**.

## Historical Importance of the Duodecimal System

In ancient times, many **civilizations** used the duodecimal system for the **units** of **time.** Historically, it was established that a **year** has **12** months and there are **12** hours in a **day** but it was later on changed to** 24**.

**Figure 3** shows a **clock** with 12 duodecimal numbers.

The clock shows another **set** of duodecimal **numerals**. Notice that **10** and** 11** are always written with different symbols i.e. **A** and **B** in the duodecimal system.

The number **12** has an essential role in **measurements**. For example, a **foot** is equal to 12 **inches** and a **pound** is made of 12 **ounces.** In Old Britain **currency**, a shilling had 12 pennies in it.

The **12 phalanges** on the four fingers of a **hand** excluding the thumb and the 12 **lunar** cycles per year specify the importance of the **duodecimal** system.

## Notations for Ten and Eleven

Different **notations** can be used for the duodecimal ten and eleven. **Upside-down 2** and **3** are also used but make typewriting difficult.

**Figure 4** shows a different notations used for the numbers 10 and 11.

## Different Base Notations

There are different **ways** to **represent** a duodecimal number. The** base-12** number can be written in **italics** to distinguish it from a **decimal** number.

For example, the **duodecimal** 28 is equal to **decimal** 32 and can be expressed as:

*28* = 32

It can also be written with a** subscript** of the **base** number as:

${(28)}_8$ = ${(32)}_{10}$

Similarly, the **subscripts z** and **d** can be used instead of **8** and **10** respectively.

Also, the notations “**dec**” and “**doz**” are used for decimal and dozenal systems respectively as:

doz 28 = dec 32

A **semicolon** is used for the **decimal point** in a dozenal number such as:

${(20;6)}_8$ = ${(24.5)}_{10}$

## Names of Duodecimal Numbers

The word “**qua**” is used at the end for the whole numbers that have a **positive** power of **12**. Whereas, the “**cia**” ending is used for the fractions having a **negative** power of **12**.

Also, the **prefixes** un, bi, tri, quad, pent, hex, sept, and oct specify the **number** of zeros after the digit** one**.

## Base Conversions

For the base **conversions** given below, the alphabets **T** and **E** are used for **ten** and **eleven** in the duodecimal system.

### Decimal to Duodecimal

The **base-10** number can be converted to a **base-12** number by **dividing** it by **12** until its quotient becomes **zero**.

For example, the decimal number **2146** is divided by **12** to get a quotient of **178** and a **remainder** of 10. **10** is **T** in duodecimal.

The number **178** is again divided by **12** to get **14** as a quotient and a remainder of **10** which is **T**. **14** divided by **12** gives **1** as the quotient with remainder **2**.

The **duodecimal** number starts from the last **quotient** that is less than **12** to the **remainder** going in the **upward** direction. **Figure 5** shows the repeated **divisions** of 2146 by **12**.

So, **base-10** number 2146 in **base-12** is:

$(2146)_{10}$ = $(12TT)_{12}$

### Duodecimal to Decimal

A **duodecimal** number can be converted to a **decimal** number by the **place** values of a **base-12** number.

The **right-most** numeral of the base-12 number holds the 12 to the **zero** power and this **power** increases by **1** as we move towards the **left**.

For **example**, $(T34)_{12}$ needs to be converted to base-10. **Figure 6** shows the place values of the **power** of **12** in $(T34)_{12}$ and its **decimal** conversion.

## An Example of COnversion from Duodecimal to Decimal

Convert the **duodecimal** number 8612 into a **decimal** number.

### Solution

The **weights** of **12** increasing from 0 from right to left are **multiplied** by each digit and **added** to give the desired **decimal** number as follows:

**8612 in Base-10** = [ 8 ${(12)}^3$ ] + [ 6 ${(12)}^2$ ] + [ 1 ${(12)}^1$ ] + [ 2 ${(12)}^0$ ]

**8612 in Base-10** = [ 8(1728) ] + [ 6(144) ] + [ 1(12) ] + [ 2(1) ]

**8612 in Base-10** = 13,824 + 864 + 12 + 2

**8612 in Base-10** = 14,702

So:

**$(8612)_{12}$ = $(14,702)_{10}$**

*All the images are created using GeoGebra.*