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Have you ever wondered why your microwave clock never glitches when stepping from 09 seconds to 10 seconds, even though computer chips naturally count in powers of 2?
Instead of treating a multi-digit number as one giant binary value, digital displays often use Binary Coded Decimal (BCD) to encode every base-10 digit into its own 4-bit binary chunk.
What happens when binary patterns produce values larger than 9 that don't correspond to any single decimal digit?

The Anatomy of BCD and Invalid States

In standard 8421 BCD, each decimal digit from 00 to 99 is represented by an independent 4-bit binary group (called a nibble).
Since 4 bits can create 24=162^4 = 16 unique combinations (000020000_2 to 111121111_2), the patterns from 101021010_2 (101010_{10}) through 111121111_2 (151015_{10}) are strictly invalid states in BCD.
How do we convert a large multi-digit decimal number into BCD?

Decimal to BCD Conversion

To convert from decimal (denary) to BCD, write each decimal digit as its own 4-bit binary nibble, preserving all leading zeros within each nibble.
Let's convert 39510395_{10} into BCD: convert 3ightarrow001123 ightarrow 0011_2, 9ightarrow100129 ightarrow 1001_2, and 5ightarrow010125 ightarrow 0101_2 to get 0011 1001 0101BCD0011\ 1001\ 0101_{\text{BCD}}.
To convert backwards from BCD to decimal, split the bitstream into groups of 4 from right to left, verify that none are invalid states, and translate each nibble to a base-10 digit.