2.1 Positional Number Systems

In Chapter 1, memory and buses carried patterns of bits. A pattern has no useful numerical meaning until we choose an interpretation rule. This section begins with the rule used by ordinary decimal numbers and then applies it to other bases.

A digit's position changes its contribution

In decimal, the two 5 digits in 505 do not contribute the same amount:

A positional number system assigns each place a weight. The rightmost integer place has weight $b^0=1$, the next has weight $b^1=b$, and each move left multiplies the weight by the base or radix $b$.

For a four-digit numeral $(d_3d_2d_1d_0)_b$:

Each digit must satisfy $0\le d_i<b$. Base 2 therefore permits only 0 and 1; base 8 permits 0 through 7; base 16 needs sixteen symbols, conventionally 0–9 and A–F.

In hexadecimal, A means ten, B means eleven, through F meaning fifteen. They are digit symbols, not variables.

Lab 1 — build a numeral from its contributions

Choose a base and rotate four digit wheels. Each tile displays its place weight and current contribution, so you can see why changing a left digit usually matters more than changing a right digit.

Solve every base mission. Then increase only the leftmost digit once and predict the change before reading the updated total.

Worked example: the same symbols under different bases

The numeral 1011 does not name one universal value. The base is part of its meaning.

In binary:

In decimal:

Writing a small base subscript prevents ambiguity. If the surrounding context makes the base obvious, engineers sometimes omit it—but a careful derivation should state it.

Leading zeros do not change the value:

They can still matter operationally because 001011 explicitly shows a six-bit storage width while 1011 does not.

The radix point extends weights to the right

The dot in a decimal fraction is a radix point. “Decimal point” is the base-10 special case. Places to its right use negative powers:

For example:

The first place to the right is worth one half in binary, one eighth in octal, one tenth in decimal, and one sixteenth in hexadecimal.

Precision comes from available places

A fixed number of fractional places creates a smallest step. With three binary fractional places, that step is:

Such a format can represent 5.625 exactly, but it cannot represent 5.6 exactly because 5.6 is not an integer multiple of 0.125. A system must round, truncate, or use more places.

This is not a failure of binary. Decimal with a fixed number of places has the same limitation: two decimal places cannot represent one third exactly.

Lab 2 — control both value and precision

Construct targets with three integer places and three fractional places. The live error tells you whether the current value is too small or too large, while every digit shows its exact contribution.

After an exact match, set the rightmost digit to zero and measure the lost precision. Compare the smallest step in bases 2, 8, 10, and 16.

Checkpoint

A positional representation is determined by three things:

1. a base $b$;

2. allowed digit symbols from $0$ through $b-1$;

3. place weights $b^i$, including negative exponents to the right of the radix point.

The next section turns this definition into practical conversion workflows among binary, octal, decimal, and hexadecimal.