Part 02 of 12
Numbers made from bits
Learn to see patterns such as 101101 as structured values rather than mysterious strings.
The story before the details
Our decimal system feels natural only because we practiced it. In reality, “724” is three symbols given their values by their positions. The rightmost symbol is multiplied by one, the next by ten, and the next by one hundred. Once that construction makes sense, binary is no longer a strange computer-only language; it is the same positional idea with a different base.
A common stumbling block is to confuse a symbol with its value. Written binary 10 does not have the quantity ten. It has one group of two and zero groups of one. A pattern can be written in binary, decimal, or hexadecimal without changing the underlying quantity. The written representation is a choice made by the reader or the programming language.
Fixed width is the next crucial idea. Four switches can represent sixteen combinations, not every integer. The range of numbers we can assign depends on whether we treat those combinations as unsigned magnitudes or signed values. Bit width is a hardware and software contract that follows us into every circuit and program.
What you will discover
These six lessons are arranged so each one answers a question created by the previous one. Do not rush to the last lesson. If a concept feels opaque, use the checkpoint before continuing.
- 02.01Decimal is already a code↗Why is 724 equal to 724?
- 02.02Binary positional weights↗What changes when the base is two?
- 02.03Binary to decimal↗How do you read an unfamiliar bit pattern?
- 02.04Decimal to binary↗How do you write 45 with only 0 and 1?
- 02.05Fixed-width numbers↗Why does bit width matter?
- 02.06Hexadecimal as shorthand↗Why do engineers use A–F?
Learn to see patterns such as 101101 as structured values rather than mysterious strings. You should be able to explain each concept using only an input, a rule, an output, and any state that must be retained.