Ancient people used tally marks, sticks, pebbles, and symbols to count objects. As the numbers grew larger, these methods became more difficult and time-consuming.
To represent large numbers efficiently, civilisations developed systems based on grouping.
The Egyptians introduced an important idea:
Instead of counting one by one, count in groups of a fixed size.
This idea later developed into the concept of a base.
I. Egyptian Number System:
In this system, we see the use of landmark numbers to group and represent a given number. However, what makes this system special is its sequence of landmark numbers.
| \(1\) | \(10\) | \(10^2\) | \(10^3\) | \(10^4\) | \(10^5\) | \(10^6\) | \(10^7\) |
 |
 |
 |
 |
 |
 |
 |
 |
| Single Stroke |
Heel bone /
Inverted 'U'
|
Coiled rope | Lotus flower | Pointing finger | Tadpole | Astonished man / Lord of infinity | Sun |
Observe:
\(10 = 10 × 1\)
\(10^2 = 100 = 10 × 10\)
\(10^3 = 1000 = 10 × 100\)
Every landmark number is obtained by multiplying the previous landmark number by \(10\).
Hence, the Egyptian system is called a: Base-10 Number System.
Important!
Landmark numbers are special numbers used to build all other numbers in that system.
Representing Numbers in a Base System:
To represent a number:
- Start with the largest landmark number smaller than the given number.
- Group the numbers using landmark numbers.
- Write the required combinations.
Example:
Represent \(243\) in Egyptian numerals.
\(243 = 100+100+10+10+10+10+1+1+1\)
Thus we get, \(243\) \(=\) 
.
.II Variation on the Egyptian number system and notation of base:
Suppose every landmark number is multiplied by \(5\).
| Power | Landmark Number |
|---|---|
| \(5⁰\) | \(1\) |
| \(5¹\) | \(5\) |
| \(5²\) | \(25\) |
| \(5³\) | \(125\) |
| \(5⁴\) | \(625\) |
Every next landmark number is obtained by multiplying the previous landmark number by \(5\). The newly created base system is called Base-\(5\) Number System.
Example:
Represent \(112\) in Base-\(5\) system.
\(112=25+25+25+25+5+5+1+1\)
\(112\) = 
Important!
The Egyptian number system is a base - \(10\) system, and the number system that we created is a base - \(5\) system. A base - \(10\) number system is also called a \(\text{decimal number system}\).
Base-\(7\) Number System:
Landmark numbers:
| Power | Value |
|---|---|
| \(7⁰\) | \(1\) |
| \(7¹\) | \(7\) |
| \(7²\) | \(49\) |
| \(7³\) | \(343\) |
| \(7⁴\) | \(2401\) |
Example:
Convert \(212\) in base-\(7\) landmark numbers.
Landmark numbers in a base-\(7\) system are the powers of the base \(1, 7, 49, 343, 2401,...\)
\(212 = 49+49+49+49+7+7+1+1\)
\(212 = 4 \times 7^2 + 2 \times 7^1 +1 \times 7^0\)
\(212 = 421\) base-\(7\)
Base - \(2\) System (Binary Number system):
The Binary Number System is a way of writing numbers using only two digits \(0\) and \(1\). Since it uses two symbols, it is called a Base \(2\) number system.
This number is widely used in computers and digital devices.
In the binary system, the place values are powers of \(2\): \(2^0\), \(2^1\), \(2^2\), \(2^3\), \(2^4\), ...
Example:
Convert binary numbers to decimal number.
\((1011)_2 = (1\times 2^3) + (0\times 2^2) + (1\times 2^1) + (1\times 2^0)\)
\(=8 + 0 + 2 + 1 = 11\)
Therefore, the value of \(1011_2\) in decimal number system is \(11\).
General Rule for Any Base-\(n\) System
Number systems having landmark numbers in which the
(a) The first landmark number is \(1\), and
(b) Every next landmark number is obtained by multiplying the current landmark number by some fixed number \(n\), which is said to be \(\text{base-n}\) number system.
Landmark numbers are: \(n⁰\), \(n¹\), \(n²\), \(n³\), \(n⁴\), ...
Advantages of Base-\(n\) System
- Efficient Representation: Large numbers can be written using a few landmark numbers.
2. Easier Addition: Grouping follows a fixed pattern.
- Combine all identical symbols from both numbers.
- Whenever you collect \(10\) of the same symbol, exchange them for a single symbol of the next highest value—just like carrying a digit in modern column addition
3. Easier Multiplication: Products of landmark numbers are again landmark numbers.
- Multiplying by \(10\) is incredibly straightforward: simply swap each symbol for the next highest landmark symbol.
- Because of this base-10 structure, multiplying any two landmark numbers (powers of 10) always shifts them cleanly into a higher-value symbol.
4. Foundation of Modern Number Systems: Our Hindu Number System uses: Base \(10\) & Landmark numbers as powers of \(10\).