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. The digit \(\mathsf{0}\) plays a special role 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.