FIT1047 Chap.1 Binary, Decimal and Hexadecimal Representation
Binary, Decimal and Hexadecimal Representation
A stored bit pattern has no numeric meaning until a representation is named. This chapter builds the positional model behind binary, decimal and hexadecimal: each digit is a coefficient multiplying a power of the base. It then develops two decimal-to-binary routes—power decomposition and repeated division—and uses them as independent checks.
Binary and hexadecimal are exact alternate views because four binary positions represent the same range as one hexadecimal digit. The chapter also separates unrestricted mathematical results from fixed-width stored results, shows why a carry needs an interpretation before it is called overflow, and treats invalid digits, leading zeros and radix-point grouping explicitly.
The published Week 1 material supports this number-system foundation; every value and drill here is newly authored and does not reproduce the live Written task. The reusable method is: name base and width, validate digits, preserve position, convert, attach a base marker and verify through a different route.
A strong representation answer also separates syntax validation from value conversion: an invalid digit should be rejected before any weighted sum is attempted. Width must remain visible because leading zeros, carry-out and field boundaries may matter even when the unrestricted value is unchanged.
This material extends the verified Week 1 foundation through original examples only; the personalised A1 task, its chosen values and its required reflection remain outside this guide.
What this chapter covers
- 01
Representation as an agreement
- 02
Positional weights and radix points
- 03
Binary-to-decimal evaluation
- 04
Decimal-to-binary decomposition
- 05
Repeated division and remainder direction
- 06
Four-bit hexadecimal grouping
- 07
Fixed-width binary addition
- 08
Boundary tests and error signatures
- 09
Digit validation before conversion
- 10
Predicting magnitude and required width
- 11
Round-trip checks across three bases
AskSia-authored practice weighting (not an official mark scheme): Fresh three-way conversion
- methodDecompose 156 as 128+16+8+4, giving binary coefficients 10011100.
- groupGroup the binary pattern as 1001 1100.
- representationMap the nibbles to hexadecimal digits 9 and C, giving 9C.
- checkCheck 9×16+12=156 and 128+16+8+4=156.
Key terms
- Base
- The number of digit values in a positional system and the multiplier between adjacent positions.
- Radix point
- The boundary separating integer and fractional positional weights.
- Most significant bit
- The highest-weight bit in the represented field.
- Nibble
- A group of four bits that maps exactly to one hexadecimal digit.
- Carry-out
- A carry leaving a fixed-width field; its overflow meaning depends on interpretation.
- Horner evaluation
- Left-to-right repeated multiply-and-add evaluation of a positional pattern.
- Representation contract
- An explicit statement of base, width, digit order and interpretation that gives a stored pattern meaning. Without this contract, the same bits can support different values or roles, so conversion arithmetic alone cannot settle the question.
- Round-trip verification
- A check that converts a result through a different representation and returns to the starting value. It should reuse as few decisions as possible from the main route so an exponent, grouping or reading-direction error is less likely to survive.
Binary, Decimal and Hexadecimal Representation FAQ
Why does the rightmost integer digit have exponent zero?
Its place weight must be one, and every non-zero base raised to exponent zero equals one. Each move left multiplies that weight by the base.
Why are repeated-division remainders reversed?
Division by the base discovers the least-significant digit first. Later divisions discover increasingly significant digits, so encounter order is opposite written order.
Why do four bits map to one hexadecimal digit?
Four bits represent values from zero through fifteen, exactly the coefficient range of one base-sixteen digit. Grouping changes notation without changing the stored pattern.
Do leading zeros matter?
They do not change an unsigned value, but they can preserve required field width, align nibbles or identify storage format. Remove them only when width is irrelevant.
How should I test a conversion method?
Use zero, one, values just below/at/above a power of the base, and a value containing interior zeros. Verify by a distinct method and keep base labels visible.
What should be checked before evaluating a digit string?
Confirm the declared base, reject digits outside its allowed set, identify the radix point and note any required field width. This separates invalid syntax from a valid representation whose arithmetic later turns out to be wrong.
How can I tell whether a carry-out is actually overflow?
State the stored width and the numerical interpretation first. Carry-out describes an event at a field boundary; overflow is a claim that the mathematical result is outside the representable range under a particular scheme. They must not be treated as synonyms without that context.
Assessment move
Create a two-column practice ledger: main conversion route and independent check. For base-to-decimal work, align digits and weights, then use Horner evaluation. For decimal-to-binary work, use decomposition, then verify with a quotient/remainder table. For binary–hexadecimal work, group from the radix point and expand every digit back to four bits. Include boundary triples around powers so width changes are predictable.
When an answer differs, classify the signature: exponent shifted, remainders not reversed, zero omitted, invalid digit accepted or nibble boundary misplaced. The Week 1 Written specification prohibits generative-AI-produced assessment content; keep this fresh practice separate and complete the live task independently under Moodle instructions.
Build a weekly conversion set with one integer, one finite binary fraction, one invalid digit string and one fixed-width addition. Before calculating, predict required width and the structural form of the answer. After calculating, perform a bit-for-bit round trip and a magnitude check. Preserve the first incorrect attempt beside the correction and write the smallest rule that would have prevented it.
For example, a reversed remainder list calls for a reading-direction annotation; a shifted value calls for marking the rightmost exponent zero; a wrong hexadecimal digit calls for expanding that nibble by weights 8, 4, 2 and 1. Finish by explaining one pattern in three contexts—unsigned data, an instruction field and a character-code fragment—to reinforce that storage alone does not select meaning.
Keep base labels on intermediate work, not only the final line. When using a calculator for verification, compare the first divergent step rather than replacing your derivation. This process makes conversion a transparent representation argument that transfers to every later chapter.
Add a provenance check to every practice page: mark it as original study work, keep the A1 personalised prompt out of the ledger, and use the supplied textbook only to confirm concepts and notation. Do not copy its prose, diagrams, worked sequences or answers into revision notes.
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