Contents

# Senary|Definition & Meaning

## Definition

A **senary** (or heximal) **number** is one that is based on the **Senary** **Number** System, which has a base of 6 (i.e., uses only six **digits** from 0–5). Therefore, **Decimal** **numbers** from 0–5 are equal to the **Senary** **numbers** from 0–5, but a **Decimal** 6 equals a **Senary** 10. From **Senary** 10–15, the equivalent **Decimal**s are 6–11. Then, the **Decimal** 12 equals a **Senary** 20, and so on.

## What Is a Senary Number?

A **senary** **number** is a **number** having a base of **six**. It is also known as base-6 **numbers** or heximal **numbers**. The numeral system based on this **number** is known as the **senary** numeral system.

Usually, the **number** system used in most of our daily routines is the base-10 numeral system, in which **digits** from 0 to 9 are used as the basis. This system is not widely adopted by significant cultures but is rather famous in a **number** of independently small communities. Just like the **decimal** **number** system, it is a product of two prime **numbers** (2 x 3), so it can be referred to as a **semiprime**.

The very first approach to **solve** the 2 and 3 division problems is **senary**, or base-6. **Senary** offers several advantages compared to other systems we have previously thought about, but this is not the only one.

Another benefit of **senary** is that it’s considerably simpler to learn. Every one of its values falls within the range of a person’s intrinsic comprehension, which really is (5). As a result, its **arithmetic** **formulas** are quite simple and very easy to learn.

Counting to base 6 is more clean and easy. Base 10 can generally connect with our 10 binary fingers, while base 6 typically relates mutually with our hand as an abacus – a **one-to-one** image of position counts. To **count** in **hexadecimal**, represent the **number** 1 with one hand and the **number** 6 with the other. Anyone can efficiently **count** to 35 using this approach.

Whereas sometimes the **senary** system may not seem to be entirely perfect. As immaculate as it may sound, it still contains a couple of disadvantages on its own.

The first disadvantage to look into is the length of **numerical** values. It may be easy to **count** the first 35 **senary** **numbers** by hand, but they tend to be far bigger than their **decimal** companions. Although the **senary** **numbers** still require four fewer **numerals** to represent themselves, which holds them to have a slight advantage over other **numeral** **systems**.

The other disadvantage of the **senary** system is that they are entirely resistive towards the division by 4, which is probably the division that is most common as 3.

## What Is a Numeral System?

A symbolic system that is used to express an array of **digits** in a constant fashion is known as a numeral system.

In **mathematics**, we commonly use the base 10 **numeral** **system**, but throughout the world, the most common numeral system used is the **Arabic-Hindu** numeral system. This system first originated in the sub-continent, and it is considered to be the basis for the **development** of the base 10 **system**.

A **number** in one numeral system may possess different meanings in a different numeral system. Such as the **number** ‘11’, which in **decimal** **number**ing is simply eleven, but in a binary system, it is responsible for representing the **decimal** digit 3.

A **number** system is somewhat different from a numeral system, which is a system of expressing **numbers** in different forms. It gives a distinctive illustration of every **number** and depicts the construction of **arithmetic** and **algebraic** expressions.

## The Senary Numeral System

The **senary** system consists of 6 numerals as a standard set which is given by D_{6} = {0, 1, 2, 3, 4, 5}. Every digit in the **senary** set is related to each other in the following **linear** manner 0 < 1 < 2 < 3 < 4 < 5.

When all the prime **numbers** except 2 and 3 are represented in the **senary**, they will have 1 or 5 as their last **digit**. A few prime **numbers** in **senary** are written as:

2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551

If the prime **numbers** are greater than 3, then the modular arithmetic operation comes in handy, where the final digit is either a 1 or a 5. This means whenever a **prime** **number** is divided by 6, the **remainder** is either 1 or 5.

In **addition**, each perfect **number** is represented in form 2$^{p-1}$(2$^p$ – 1) where (2$^p$ – 1) represents a prime **number**. Thus besides 6, every other perfect, consistent **number** will consist of 44 as the last two **numbers** in the **senary** system. Also, being the largest base of **numbers**, the **multiplication** table makes up a huge chunk compare to its size, which helps in memorizing the **products** and requires less effort.

A **number** will contain 3 **digits** 0, 2, and 4 as their last **numbers** when it is divided by 2 in **senary**. Similarly, a numeral will contain 2 **digits** 0 and 3 as their last **numbers** when it is divided by 3 in the **senary**.

## How To Count Senary Numbers

It’s relatively easy to **count** **senary** **numbers** but not as much as compared to **decimal** **numbers**. Where one hand can be used to **count** the **numbers**, the other one is used as a specific position, such as the left hand being used as a unit counter, whereas the right hand is used to **count** the sixes.

Let’s say you hold out 3 fingers of your left hand and 4 fingers of your right hand, which is **equivalent** to 34_{senary} in **senary** but 22_{decimal} in **decimal**. The math behind this counting is quite straightforward, 3 x 6 + 4 = 22 in **decimal**, but the only drawback is that you can only **count** from zero to 35 in **decimal** or 55 in **senary**.

Apparently, this method is the least ideational manner to **count** since it represents the notions of **positional** **notation**, in which you move from one hand to the other hand. Preferably the left hand is used for units and the right for sixes, but it’s up to the counter which hand he or she prefers for the **respected** **role**.

## Solved Examples of Senary Numeral System

### Example 1

Convert **120**_{6} (**senary**) to base 10 (**decimal**).

### Solution

This problem can be solved in two steps. Firstly we will start with the units place in 120.

Multiply **units** place with 6^{0}, then the **tens** **decimal** with 6^{1}, and finally the **hundredth** **decimal** with 6^{2} and keep on going as many **decimals** as there are.

The next step is to add all the products that we just discussed above, which will be the **equivalent** of 120 in **decimal**.

Applying these steps to get our final answer for converting 120_{senary} to **decimal**:

Decimal of **0 = 0** **x 6 ^{0} = 0**

Decimal of **2 = 2** **x 6 ^{1} = 12**

Decimal of **1 = 1 x 6 ^{2} = 36**

Adding all gives us the **decimal** of **120 = 36 + 12 + 0 = 48.**

Thus 48 is the **decimal** equivalent of **120 _{senary}**.

### Example 2

Convert **480**_{10} (**decimal**) to base-6 (**senary**).

### Solution

The first step is to divide the whole **number** by 6, with each remainder and quotient being noted down. The division needs to be completed until the **number** is not further divisible by 6.

Finally, the answer can be achieved by assembling all the remainder in reverse order (down to up).

Applying these steps to get our final answer for converting **480 _{decimal}** to

**senary**.

Divide **480** by **6,** giving **80** with the **remainder** **0.**

Divide **80** by **6** gives **13,** with the **remainder** **2.**

Divide **13** by **6,** giving **2** with the **remainder** **1.**

Divide **2** by **6** gives **0** with the **remainder** **2.**

Now writing the remainder from down to up format, we get the **senary** form of **480 _{10}** which is

**2120**

_{6}.*All images are created using GeoGebra.*