What is a Number System?
A number system is a structured way to represent and interpret quantities using a consistent set of symbols and rules. Whether we’re counting apples, programming computers, or transmitting digital signals — number systems give us the language to do it.
Each system is built around a concept called the base (or radix), which defines:
- How many unique symbols it uses?
- How place value works?
- How numbers expand with more digits?
How Many Unique Symbols it Uses?
A base-10 (or radix-10) number system uses ten unique symbols — from \(\mathsf{0}\) to \(\mathsf{9}\) — to represent all possible numbers. In the same way, a radix-5 system uses just five symbols — \(\mathsf{0}\) through \(\mathsf{4}\) — to build its entire number set.
How Place Value Works?
Every number can represent either a complete quantity, a partial quantity, or both. To distinguish between these, we use a special separator — a dot . — known as the radix point (or base point).
- Digits to the left of the radix point represent the whole part of the number.
- Digits to the right represent the fractional part.
For example, \(\mathsf{6.25}\) mean \(\mathsf{6}\) whole units and \(\mathsf{0.25}\) of another unit. Even if the fractional part is not visible, the radix point is always there implicitly. So when we say we have 6 whole apples, mathematically it can also be written as \(\mathsf{6.0}\).
How Numbers Expand with More Digits?
Every number system starts with a set of base symbols — whether it’s \(\mathsf{2}\) in binary, \(\mathsf{8}\) in octal, \(\mathsf{10}\) in decimal, or \(\mathsf{16}\) in hexadecimal. But no matter the base, there comes a point where we run out of single digits.
That’s when we begin forming multi-digit combinations — by adding new digits to the left of the existing ones. These added digits represent higher positional values and contribute more to the overall number.
For example, in decimal:
\[\mathsf{9\rightarrow10,\quad99\rightarrow100}\]
And in binary:
\[\mathsf{1\rightarrow10,\quad11\rightarrow100}\]
Each new digit added on the left increases significance, based on powers of the system’s base.
To express precise fractional values, we use digits to the right of the radix point. In any base system:
- Each position to the right contributes less to the overall value.
- These positions are represented by negative powers of the base.
For example, n decimal:
\[\mathsf{0.1=10^{-1},\quad0.01=10^{-2}}\]
In binary:
\[\mathsf{0.1=2^{-1}=0.5,\quad0.01=2^{-2}=0.25}\]
To summarize, the pattern is consistent across all number systems:
- Leftward expansion means higher magnitude
- Rightward expansion means finer precision
Important Number Systems
For the sake of this course, we would focus on the following number systems:
- Decimal (Base-10): Everyday counting, commerce, education.
- Binary (Base-2): Computing and logic circuits.
- Octal (Base-8): Legacy programming, compact binary representation.
- Hexadecimal (Base-16): Digital systems, memory addressing, color encoding.
- Arbitrary (Base-X): A personal number system.
Using the right system makes data easier to process, store, and display depending on the context.