Decimal Number System
Since the day we were introduced to mathematics, we’ve been using numbers based on the decimal system — a system that relies on ten unique symbols:
\[\mathsf{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}\]
Because it has ten symbols, it’s called a base-10 or radix-10 system or in other words, the base of a decimal number system is \(\mathsf{10}\). A decimal number is typically represented as it is and a subscript is not added even we are dealing with numbers in different systems. A number without a subscript is considered to be in decimal by default while numbers in other systems are represented by adding a subscript to them to identify the base. However, if it should be explicitly mentioned, a \(\mathsf{10}\) or \(\mathsf{D}\) is added in the subscript for clarification, that is, \(\mathsf{10_{10}=10_{D}=10}\) in decimal.
The digit \(\mathsf{0}\) plays a special role in all the number systems, as it represents nothing — so placing it on the left of any number doesn’t affect its value (for example, \(\mathsf{07=007 = 7}\).
What Happens When We Run Out of Symbols?
We start combining them.
After reaching \(\mathsf{9}\), we use the next symbol \(\mathsf{1}\) with \(\mathsf{0}\) to get \(\mathsf{10}\). Then \(\mathsf{11, 12}\), and so on — all the way up to \(\mathsf{19}\). Next comes \(\mathsf{20}\), then \(\mathsf{21}\), and so on until we reach \(\mathsf{99}\).
Enter the Three-Digit Numbers
To go further, we start using three digits, leading to our first 3-digit number — one hundred:
\[\mathsf{100}\]
From there, we count up through the \(\mathsf{100}\textsf{s}\) until we reach:
\[\mathsf{999}\]
Into the Thousands and Beyond
After that comes one thousand:
\[\mathsf{1000}\]
And the pattern continues — \(\mathsf{10000, 100000, 1000000}\), and on. The numbers never end — and it all begins with just ten simple symbols.
Place Value in the Decimal System
Every time we read or write a multi-digit number, there’s an invisible system at play — one that tells us how much each digit is worth based on where it sits. This is called place value.
Start from the rightmost digit — this is the least significant digit, because it contributes the smallest amount to the overall number. As we move left, each digit becomes more important — more “significant” — until we reach the leftmost digit, known as the most significant digit.
In the decimal system, these positions go like this:
… Ten Thousands, Thousands, Hundreds, Tens, Ones
Each step to the left represents a value ten times bigger than the one before. That’s why it’s called a base-10 system — each position is a power of 10:
\[\textsf{Ones }\mathsf{=10^0,} \textsf{ Tens }\mathsf{=10^1,} \textsf{ Hundreds }\mathsf{=10^2,} \textsf{ and so on}\]
What About Numbers Smaller Than 1?
When we move to the right of the decimal point, things flip. Each new digit now has less value:
Tenths, Hundredths, Thousandths, …
And just like before, each place represents a power of 10 — but this time we’re dividing:
\[\mathsf{10^{-1},10^{-2},10^{-3},}\textsf{ …}\]
So:
\[\mathsf{10^{-1}=0.1=\frac{1}{10}} \textsf{ (one-tenth)}\]
\[\mathsf{10^{-2}=0.01=\frac{1}{100}} \textsf{ (one-hundredth)}\]
\[\mathsf{10^{-3}=0.001=\frac{1}{1000}} \textsf{ (one-thousandth)}\]
Each step to the right of the decimal makes a digit’s value ten times smaller.
Can we Convert Decimal Numbers to Other Systems?
Of course, we can. In order to learn to convert numbers between different systems, go directly to the conversion between numbers.