How Humans count (Decimal System)

Decimal

Decimals are a base-10 numbering system used to represent numbers between whole numbers or fractions of a whole, utilizing a decimal point to separate the whole number part (left) from fractional parts like tenths, hundredths, and thousandths (right).

Why 10?

If we were evolved with eight fingers instead of ten? We use Base-10 (Decimal) simply because of our anatomy.

The Logic of "Place Value"

In a decimal system, the position of a digit determines its power. We only have ten unique symbols to work with: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

But what happens when we reach ten? We don't have a single symbol for "ten," so we move to the next "column" and start over.

The Mathematical Foundation

Every digit in a decimal number represents a specific power of 10. If we have a number $d_2 d_1 d_0$ (where each $d$ is a digit), its value is:

$$d_2 \times 10^2 + d_1 \times 10^1 + d_0 \times 10^0$$

In general, for any position $n$ (counting from 0 on the right):

$$\text{Value} = \sum_{i=0}^{n} d_i \times 10^i$$

Breaking Down a Number

When we see a number like 342, our brain actually performing a quick calculation of columns:

Hundreds	Tens	Ones
3	4	2
($3 \times 100$)	($4 \times 10$)	($2 \times 1$)
($3 \times 10^2$)	($4 \times 10^1$)	($2 \times 10^0$)

Total: $300 + 40 + 2 = 342$

Or more explicitly: $$(3 \times 10^2) + (4 \times 10^1) + (2 \times 10^0) = 300 + 40 + 2 = 342$$

Fractions

The same principle extends to fractions. Positions to the right of the decimal point use negative exponents:

$$0.75 = (7 \times 10^{-1}) + (5 \times 10^{-2}) = \frac{7}{10} + \frac{5}{100} = 0.7 + 0.05 = 0.75$$

Tenths ($10^{-1}$)	Hundredths ($10^{-2}$)
7	5

Why matters?

The decimal system is a "Positional Notation" system. Once we understand that a digit's value is just (Symbol) × (Base^Position), we've unlocked the secret to understanding how computers count (Binary/Base-2) or how programmers shortcut colors (Hexadecimal/Base-16). Which we will explore in the future blogs.