ELECTENG 101 Chap.5 Number Systems and Representation Limits
Number Systems and Representation Limits
Positional notation represents value as digits weighted by powers of a base. Binary, octal, decimal and hexadecimal conversions become reliable when every digit is paired with its place value. Finite bit width limits the number of available codes, so range and minimum-width questions require careful distinction between code count and largest unsigned value.
Base subscripts prevent the same written digits from being assigned different values. Conversion to decimal expands weighted powers, while conversion from decimal can use repeated division and correctly ordered remainders. Binary and hexadecimal align in groups of four bits, giving a compact reversible translation. Fixed width matters: leading zeros preserve field length even though they do not change numerical value.
Representation limits connect directly to logic truth tables, counters and ADC output codes.
What this chapter covers
- 01
Positional place value
- 02
Expansion into decimal
- 03
Repeated-division conversion
- 04
Binary-octal-hex grouping
- 05
Unsigned code ranges
- 06
Minimum bit width
Worked example · free
Size and encode a state counter
- +2Ten bits are required because 2⁹ is too small and 2¹⁰ is enough.
- +1The 10-bit unsigned maximum is 1023.
- +345 is 101101₂ and 2D₁₆.
Key terms
- Bit width
- The number of binary digits allocated to one representation.
- Place value
- The base power multiplying a digit at a given position.
- Unsigned range
- The non-negative values represented by n bits, from zero through 2ⁿ−1.
Number Systems and Representation Limits FAQ
Why is the unsigned maximum one less than 2ⁿ?
There are 2ⁿ patterns, but the first represents zero. Counting from zero makes the last value 2ⁿ−1. For example, three bits create eight patterns from 000 through 111, representing unsigned values zero through seven. Keep code count and greatest value on separate lines so the subtraction is applied only to the requested quantity.
How do I convert binary to hexadecimal?
Group bits from the right in sets of four, pad the left group with zeros if necessary and translate each group into one hexadecimal digit. Preserve the grouping on paper and check by expanding the hexadecimal powers afterward. Padding on the left changes display width but not value; padding on the right would multiply an integer binary value.
How do I find the minimum number of bits?
Choose the smallest integer n for which 2ⁿ is at least the required number of distinct states. Do not round a logarithm downward. Test the neighbouring powers explicitly: one must be too small and the chosen one sufficient. If states are numbered from zero, the largest label is one less than the number of required states.
Why keep base subscripts during a calculation?
A digit string such as 101 has different values in base two and base ten. Subscripts keep the interpretation visible while place values are expanded or groups are formed. Remove them only after the target base is unambiguous. This small notation habit prevents a correct-looking arithmetic line from combining numbers expressed in incompatible bases.
How do repeated-division remainders become binary digits?
Divide the decimal integer by two, record the remainder, and continue with each quotient until it reaches zero. The first remainder is the least significant bit, so read the recorded remainders upward or reverse their order at the end. Verify by summing the powers of two selected by the resulting one bits.
Do leading zeros matter?
They do not change an unsigned numerical value, but they matter whenever a register, data field, ADC or logic interface has a declared width. Writing decimal five as 0101 in a four-bit field communicates all bit positions. Dropping the first zero can hide a width mismatch even though the represented value remains five.
Exam move
Practise conversions in both directions and keep base subscripts visible. Show place-value expansions and repeated-division remainders rather than relying on mental shortcuts. Separate three recurring questions: number of codes, largest value and minimum width. Then connect representation to ADC and logic chapters by writing fixed-width codes with leading zeros.
Prepare a conversion grid with binary, decimal and hexadecimal columns for values near power-of-two boundaries such as 15, 16, 31 and 32. For each row, verify one method with a second route: place-value expansion after repeated division, or decimal evaluation after four-bit grouping. Write code count, valid unsigned interval and required bit width as distinct statements for several device capacities.
Include one case where the requested states exceed a power of two by exactly one, since it exposes downward rounding. Finish by encoding the same value at two widths to see why leading zeros carry interface information without changing magnitude.
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