Data representation: text and numbers

Learn the concept

Information has meaning for a person; data are recorded symbols that can be interpreted under a rule. The bits 01000001 can represent the letter A in UTF-8 or the number 65 in unsigned binary. The bits alone do not announce which interpretation is intended. A file format and context tell a program how to decode them.

A representation lets a system store, transmit or process a feature of the world. A temperature sensor reading may use a number and unit; a picture uses pixel or shape data; text uses character codes. Every representation simplifies or selects details, so ask what information may be lost or misinterpreted.

When two systems exchange data, they need agreement on the representation. Sending a number without unit, encoding or field label can produce a plausible but wrong result. Write the rule alongside an original small example and test that a second person decodes it as intended.

Learn the concept

Digital systems store binary values, so text needs an agreed mapping. Unicode assigns code points to characters across many writing systems. An encoding such as UTF-8 converts those code points to bytes for storage and transmission. The printed letter is not literally a row of ink stored inside the computer.

In UTF-8 the English capital A uses the byte 01000001, equal to decimal 65. Some characters need more than one byte: 'é' is not guaranteed to occupy the same number of bytes as A. Do not use 'one character equals one byte' as a general rule. The font then decides how a decoded character looks on screen.

If sender and receiver interpret bytes with different encodings, text can appear corrupted. Record the encoding when exchanging files and test a sample with accents or other scripts. Encoding does not prove that a displayed message is accurate or safe; it only specifies how symbols are represented.

Learn the concept

A binary digit has two possible values, 0 or 1. Two independent bit positions allow 00, 01, 10 and 11: four patterns. Three positions allow 2×2×2 = 8 patterns. In general n bits allow 2ⁿ patterns if each position is independently available.

A pattern count is different from the numeric value of a particular pattern. Three unsigned bits allow eight values, 0 through 7, but 111₂ has value 7. If a two-bit code is used for four fictional symbols, the designer must write a one-to-one mapping so the receiver can decode them.

Physical systems may represent two states with different voltage ranges, magnetic states or light levels under a design. The abstract 0 and 1 are not themselves tiny printed digits inside every device. Real signals need thresholds and error handling. A classroom table shows the logic, not a full engineering specification.

Learn the concept

Decimal uses powers of ten; binary uses powers of two. Read 101101₂ by assigning weights 32, 16, 8, 4, 2, 1 from left to right. The 1 bits contribute 32 + 8 + 4 + 1 = 45. The subscript ₂ indicates the numeral's base and is not another digit.

To encode unsigned decimal 13, choose the largest power of two not exceeding it: 8, remainder 5; then 4, remainder 1; then 1. In places 8,4,2,1 the bits are 1101₂. Check by reversing: 8 + 4 + 0 + 1 = 13.

A fixed bit width limits an unsigned range. Four bits represent 0 through 15, because the greatest pattern 1111₂ sums to 8 + 4 + 2 + 1. Signed numbers use a different representation and should not be inferred from this unsigned model.

Add binary columns from right to left, carrying 1 when a column totals two or three. For 1011₂ + 0110₂, the ones column gives 1; the twos gives 1+1 = 10₂, write 0 and carry 1; the fours gives 0+1+1 = 10₂; the eights gives 1+0+1 = 10₂; the final carry occupies the sixteens place. The result is 10001₂, or 17 in decimal, matching 11 + 6. A four-bit register would overflow, so keep the fifth bit.

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Eight bits make one byte. If a message is represented by 80 bits, its raw bit count is 10 bytes. A file may also contain metadata or formatting, so the final saved file can be larger than the text payload alone. Keep the units attached to every calculation.

Storage labels use more than one convention. A decimal kilobyte is 1,000 bytes; a kibibyte (KiB) is 1,024 bytes. The difference grows at larger scales. When a device displays 'KB', check its stated convention before comparing advertised capacity or transferring a size into a calculation.

Network speeds are commonly in bits per second while file sizes are commonly in bytes. A theoretical 8 megabits per second corresponds to 1 megabyte per second before protocol overhead and real-world variation. Do not promise an exact download time from a headline speed alone.

Learn the concept

A code can map fixed bit patterns to symbols. In an original classroom puzzle, use three bits to encode eight letters: 000=A, 001=B, 010=C, 011=D, 100=E, 101=F, 110=G, 111=H. The receiver needs the mapping and a way to split the stream into three-bit groups; without the grouping, a long run of bits is ambiguous.

Decode 010 000 001 as C A B, then check every group against the table. This is encoding, not secrecy: anyone with the published mapping can read it. A cipher aims to conceal meaning using a key or transformation. Alan Turing's name in the source lesson title does not make this simple code equivalent to wartime cryptography.

A puzzle can use a secret substitution key, but its security must not be overstated. Simple classroom substitution is easy to break and unsuitable for protecting real messages or passwords. Work only with invented words, state the key and test by re-encoding the decoded result.