Number bases, prefixes and BCD

What you need to be able to do

Binary and decimal prefixes

A computer stores everything as bits (binary digits, 0 or 1). Eight bits make a byte. Large amounts are written with a prefix, and there are two families of prefix because memory is built in powers of 2 while everyday measurement uses powers of 10.

How to convert between prefixes

1. Write the amount in bytes first: multiply by 1000 for each decimal step, or 1024 for each binary step.
2. Then divide by 1000s or 1024s to reach the unit the question asks for.
3. To compare amounts in different prefixes ("which file is largest?"), convert them all to bytes and compare.

Number bases and place value

A number base is how many different digits a system uses. Every system works the same way: each column is worth the base times the column to its right.

The same value, 45, in all three:

How many values do n bits hold?

Binary and denary

Binary to denary

1. Write the place values (1, 2, 4, 8 …) above the bits, starting with 1 at the right.
2. Add up the place values that have a 1 under them.
3. Check: an odd number ends in 1; the answer can't be more than \(2^n-1\).

Denary to binary: two methods, same example (156)

Hexadecimal

Hexadecimal (hex) has 16 digits, 0-9 then A-F. The reason it is used in computing: one hex digit is exactly four bits (a nibble), because \(16=2^4\). So a byte is always two hex digits, and converting is just grouping.

Binary to hex

1. Split the bits into groups of four from the right. If the left group is short, pad it with 0s.
2. Turn each group into one hex digit (8 4 2 1 place values, 10-15 become A-F).
3. Write the digits in the same order.

Hex to binary

Reverse it: write every hex digit as exactly four bits, zeros included. A04 = 1010 0000 0100. Dropping the middle 0000 (giving 1010 0100 = 164) was a common mistake in June 2023.

Hex to denary: two methods, same example (C0F)

Denary to hex: two methods, same example (251)

Binary Coded Decimal (BCD)

BCD does not convert the whole number into binary. It converts each digit. Compare 87 both ways:

Where hexadecimal and BCD are used

Benefits of BCD to quote (November 2022): straightforward to convert to and from denary, so less complex to encode and decode; easy for digital equipment to display; monetary values are represented exactly. Its drawback: it wastes bits (4 bits hold only 10 of the 16 possible patterns), so it needs more storage than plain binary.

Mark-scheme focus

“State one difference between a tebibyte and a gigabyte” (1 mark)

Credited

• A tebibyte = 1024 gibibytes (\(2^{40}\) bytes) whereas a gigabyte = 1000 megabytes (\(10^9\) bytes)
• Tebi is a binary prefix, giga is a denary (decimal) prefix

Not credited

• “A tebibyte is bigger than a gigabyte” (true, but not a difference in what they are)
• “A tebibyte is 1024 gigabytes” (must be gibibytes)
• “a kilobyte is 1000 bits”-style slips: bits instead of bytes

Why

The mark is for the base each prefix counts in (1024 vs 1000, or binary vs denary). A comparison of size alone does not show you know why they differ.

Also accepted

Any correct pair of values with both units right, e.g. kibibyte = 1024 bytes and megabyte = 1 000 000 bytes.

From: 9618/11/O/N/24 Q1(a)

“Give one application where BCD is used and justify its use” (2 marks)

Credited

• Application: an application performing financial / banking calculations B1
• Justification: values must be exact, no accumulating / rounding errors; it is difficult to represent decimal fractions exactly in normal binary B1
• or: electronic displays B1 because displays only need to show individual digits / conversion is straightforward B1

Not credited

• “Calculators” with “because it is easier” (easier than what? for which part?)
• A justification that doesn't match the application (e.g. banking … because it shows digits)

Why

The second mark is a corresponding justification: it has to explain why BCD suits that application.

Also accepted

Date and time in the BIOS (conversion with denary is easier); barcode systems.

From: 9618/12/M/J/25 Q2(c)

“State why the value in the register cannot be a BCD value” (1 mark)

Credited

• The denary value in each nibble / group of 4 bits is greater than 9 (1100 = 12, 1101 = 13)

Not credited

• “The register holds a value greater than 9” (the whole register, not each nibble)

Why

BCD is judged nibble by nibble: each 4-bit group must be one denary digit 0-9. The report noted many answers talked about the whole register.

From: 9618/12/O/N/21 Q4(d)

“State the number of unique binary values that can be represented in 16 bits” (1 mark)

Credited

• \(2^{16}\) or 65 536

Not credited

• 65 535 (that is \(2^{16}-1\), the largest value, not the number of values)

Why

Zero is one of the values. 16 bits run from 0 to 65 535, which is 65 536 different values.

From: 9618/12/M/J/23 Q4(a)

Worked examples

Example 1 · Matching binary values to prefixes (Paper 1, 5 marks) ★ Easy

1 Read it with me what is actually happening?

Five amounts on the left, seven on the right: two right-hand boxes are spare. Each left box equals exactly one right box.

• one line from each5 lines, one per left box; two right boxes stay empty
• equivalent (same)exactly equal, so kilo vs kibi matters

2 What is it asking for? the target, part by part

• (a)five correct lines5 marks, 1 each

3 What do we know? everything the question gives us

• 8 bits
• 8000 bits
• 1000 kilobytes
• 1024 mebibytes
• 8192 bits

• 1 byte = 8 bits
• decimal steps × 1000, binary steps × 1024

4 Plan the route work backwards from what we want

FindKey ideaAlready have itWork out firstTrap

• Matcheach left box
  • Bits÷ 8 to get bytes first
  • Prefix1000 → decimal, 1024 → binary

8000 bits is how many bytes?

1000 bytes = 1 kilobyte.

8192 bits?

1024 bytes = 1 kibibyte.

5 Now solve it have a go on paper first, then open the worked solution

1. 8 bits → 1 byte B1
2. 8000 bits = 1000 bytes → 1 kilobyte B1
3. 1000 kilobytes → 1 megabyte B1
4. 1024 mebibytes → 1 gibibyte B1
5. 8192 bits = 1024 bytes → 1 kibibyte B1

6 Check it does the answer make sense?

1 gigabyte and 1 mebibyte are the unused distractors: neither equals any left box. A decimal amount (1000) always went to a decimal prefix and 1024 to a binary one ✓.

Example 2 · Largest file size, then hex to denary (Paper 1, 3 marks) ★ Easy

1 Read it with me what is actually happening?

(a) mixes binary and decimal prefixes, so the numbers in front can't be compared directly. (c) wants hex place values, with working.

• tick one box onlytwo ticks scores nothing
• show your workinga working mark: write the place-value sum

2 What is it asking for? the target, part by part

• (a)the largest1
• (c)C0F in denary2

3 What do we know? everything the question gives us

• 3300 KiB, 0.3 MB, 3 MiB, 3300 kB(a)
• C0F(c)

• C = 12, F = 15
• hex place values 256, 16, 1

4 Plan the route work backwards from what we want

FindKey ideaAlready have itWork out firstTrap

• (a)largest
  • Bytesconvert all four
• (c)denary
  • Place values12 × 256 + 0 × 16 + 15

3300 KiB vs 3300 kB?

Same number, binary prefix bigger: 3300 KiB wins over 3300 kB.

Is 3 MiB bigger than 3300 KiB?

3 MiB = 3072 KiB, so no.

5 Now solve it have a go on paper first, then open the worked solution

1. (a) 3300 KiB = 3 379 200 B; 0.3 MB = 300 000 B; 3 MiB = 3 145 728 B; 3300 kB = 3 300 000 B
2. Largest: 3300 kibibytes B1
3. (c) \(C0F=12\times16^2+0\times16+15\) M1
4. \(=3072+15=\)3087 A1

Also accepted for the working mark: 1100 0000 1111 // 2048 + 1024 + 8 + 4 + 2 + 1.

6 Check it does the answer make sense?

3087 is less than 4096 = 16³, as a 3-digit hex number must be ✓.

Example 3 · Five conversions in one table (Paper 1, 5 marks) ★★ Medium

1 Read it with me what is actually happening?

Five separate 1-mark conversions. Read each row's target carefully: two of the reported errors were answering in the wrong form.

• represented in denaryanswer in base 10
• unsignedno negatives: starts at 0
• BCDdigit by digit, not plain binary
• two's complementnegative numbers: see the next page

2 What is it asking for? the target, part by part

• Row 111 hex → denary1
• Row 2smallest unsigned 8-bit1
• Row 387 → BCD1
• Row 4240 → hex1
• Row 5−20 → 8-bit two's complement1

3 What do we know? everything the question gives us

• hex 11
• unsigned 8-bit
• 87
• 240
• −20

4 Plan the route work backwards from what we want

FindKey ideaAlready have itWork out firstTrap

• Row 117?
  • Place values1 × 16 + 1
• Row 3BCD
  • Trap0101 0111 is plain binary
• Row 5−20
  • +200001 0100, flip, add 1

Smallest unsigned 8-bit: write it as?

The question says “denary number”: write 0, not 00000000.

240 as hex?

240 = 15 × 16 + 0 → F0.

5 Now solve it have a go on paper first, then open the worked solution

1. Hex 11 = 1 × 16 + 1 = 17 B1
2. Smallest unsigned 8-bit integer = 0 (in denary, as asked) B1
3. 87 in BCD = 8 → 1000, 7 → 0111 = 1000 0111 B1
4. 240 = 15 × 16 → F0 B1
5. −20: +20 = 0001 0100 → flip 1110 1011 → add 1 = 1110 1100 B1 (how this works is on the next page)

6 Check it does the answer make sense?

Row 3 starts with 1000 because the first digit is 8 - in plain binary 87 would start 0101. Row 5 is negative, so its first bit is 1 ✓.

Example 4 · Hex, then the BCD trap (Paper 1, 3 marks) ★★★ Hard

Why it is hard: the input looks like BCD. Only reading the stem (“unsigned binary integer”) stops you converting it the wrong way.

1 Read it with me what is actually happening?

Both inputs are unsigned binary. (ii) is pure grouping. (iii) looks like BCD already (1001 0101 reads as “9 5”) but it isn't: it is a binary number.

• unsigned binary integera normal positive binary number
• into BCDdenary digits first, then 4 bits each
• show your workingthe denary value is the working mark

2 What is it asking for? the target, part by part

• (a)(ii)hex1
• (a)(iii)BCD2

3 What do we know? everything the question gives us

• 1001 0110(ii)
• 1001 0101(iii)

4 Plan the route work backwards from what we want

FindKey ideaAlready have itWork out firstTrap

• (iii)BCD
  • Denary128 + 16 + 4 + 1
  • Theneach digit to 4 bits
  • Trapanswering 95

What does 1001 0101 mean as BCD?

95 - but the question told you it is binary, not BCD.

How many nibbles will 149 need?

Three: one per digit.

5 Now solve it have a go on paper first, then open the worked solution

1. (ii) 1001 | 0110 → 9 | 6 → 96 B1
2. (iii) 1001 0101 = 128 + 16 + 4 + 1 = 149, the working mark B1
3. 1 → 0001, 4 → 0100, 9 → 1001: 0001 0100 1001 B1

In (ii), converting to denary first and then to hex is allowed, but the report says it introduced errors: group the bits directly.

6 Check it does the answer make sense?

BCD of a 3-digit number has 12 bits, and every nibble is ≤ 1001 ✓. The report: “a very common incorrect response was 95”.

Summary

• Decimal prefixes (kilo, mega, giga, tera) go up in 1000s; binary prefixes (kibi, mebi, gibi, tebi) in 1024s.
• n bits: \(2^n\) values, largest unsigned \(2^n-1\).
• Hex digit = 4 bits: group binary in fours from the right; write every hex digit as four bits.
• BCD: each denary digit as its own nibble (0000-1001). Plain binary → BCD goes through denary.
• Hex: short, readable, maps to binary. BCD: exact decimal values, easy display.

Can I do this?

Quiz

Concept checks

Answer each one in your head first, then open it.

Exam-style practice

New questions, not used above. Do them on paper, then mark your work against the official mark scheme.