← IGCSE Computer Science Paper 1

16-bit binary place values

The 16 place values of a 16-bit binary number. Each is double the one on its right, and the right-hand byte uses 128 to 1. All 16 bits set to 1 make the largest value, 65535.

Part of: IGCSE Computer Science 0478 — 1.1 Number Systems — Module 10

Understanding 16-bit binary place values in IGCSE Computer Science Paper 1

What it does: The lowest 8 bits of a 16-bit number have the place values 128 down to 1, just like an 8-bit number. Moving one place to the left doubles the value, so 256 doubled gives 512. This pattern continues up to 32768. Place values double each step going left (1, 2, 4, 8 ...), they do not go up by 2. Double 4096 to get 8192. Bits from the right are 1, 2, 4, 8, 16, 32, 64, 128 for the first 8, so the 9th is 256. After 512 the next place value to the right is 256. 128 belongs to the right-hand byte.

Common mistakes: Each place value is double the one on its right, so the next one down is half of 32768. It should be next = top // 2, which gives 16384, not top - 2. The place values keep halving: 128, 64, 32, 16. The value 36 is not a power of 2, so the line should be c = [128, 64, 32, 16].

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This module's key terms are also in a printable crossword (Modules 3–17) and word search (Modules 3–27).

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Learning
Duo
What is the place value of the leftmost bit in a 16-bit binary number?
3276865535
Matches
32768 16384 8192 4096 2048 1024 512 128 256
↓
The place values are out of order. 256 must come before 128, because each place value is double the one on its right.
Applying
Meanings
Term
32768 16384 8192 4096 2048 1024 512 256 128 64 32 16 8 4 2 1
Meaning
The 16 place values of a 16-bit binary number. Each is double the one on its right, and the right-hand byte uses 128 to 1. All 16 bits set to 1 make the largest value, 65535.
Practice
32768 16384 8192 4096 2048 1024 512 256
Testing
Gaps
32768 16384 4096 2048 1024 512 256
Check Answer
Spots
2top = 32768
3next = top - 2

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Quick Questions

1. What is the place value of the leftmost bit in a 16-bit binary number?
A. 65535
B. 32768 ✓
The leftmost bit is worth 32768. 65535 is the largest value you can make with all 16 bits, not a place value.
2. Which is true about 16-bit place values?
A. Each is double the one to its right ✓
B. Each is 2 more than the one to its right
Place values double each step going left (1, 2, 4, 8 ...), they do not go up by 2.
3. What place value comes straight after 4096 when moving left?
A. 6144
B. 8192 ✓
Double 4096 to get 8192.
4. The right-hand byte of a 16-bit number uses which place values?
A. 128 64 32 16 8 4 2 1 ✓
B. 32768 16384 8192 4096 2048 1024 512 256
The right-hand 8 bits are the same as an 8-bit number: 128 down to 1. The larger values belong to the left-hand byte.
5. What is the place value of the 9th bit from the right in a 16-bit number?
A. 256 ✓
B. 512
Bits from the right are 1, 2, 4, 8, 16, 32, 64, 128 for the first 8, so the 9th is 256.

Match the Pairs

32768 16384 8192 4096 2048 1024 512 256 → The place values of the left-hand byte (the top 8 bits) of a 16-bit binary number
1111111111111111 → The largest 16-bit binary number, which is 65535 in denary
The right-hand byte of a 16-bit number uses the same place values as an 8-bit number → 128 64 32 16 8 4 2 1
Each place value is double the one to its right, so the place value left of 256 is 512 → 256 x 2 = 512
32768 16384 8192 4096 2048 1024 512 128 256 → The place values are out of order. 256 must come before 128, because each place value is double the one on its right.
Largest 16-bit value = 65536 → The largest 16-bit value is 65535, not 65536. 65536 is the number of different values, because 0 counts as one of them.

Key Term

32768 16384 8192 4096 2048 1024 512 256 128 64 32 16 8 4 2 1
The 16 place values of a 16-bit binary number. Each is double the one on its right, and the right-hand byte uses 128 to 1. All 16 bits set to 1 make the largest value, 65535.

Fill in the Blanks

1. 32768 16384 8192 4096 2048 1024 512 256
Hint: Type the place value that is half of 16384, the next one to the right.
2. Right-hand byte: 128 64 32 16 8 4 2 1
Hint: Type the largest place value in the right-hand byte, which is double 64.
3. Largest 16-bit value: 65535
Hint: Type the denary total when all 16 bits are set to 1.
4. 1000000000000000 = 32768
Hint: Type the denary value of the single 1 in the leftmost place.
5. 0000000100000000 = 256
Hint: Type the place value of the only bit set to 1, the first place left of the right-hand byte.

Spot the Error

Question 1
A bits = 16
B top = 32768
C next = top - 2
D print(next)
Line C — Each place value is double the one on its right, so the next one down is half of 32768. It should be next = top // 2, which gives 16384, not top - 2.
Question 2
A a = [32768, 16384, 8192, 4096]
B b = [2048, 1024, 512, 256]
C c = [128, 64, 36, 16]
D d = [8, 4, 2, 1]
Line C — The place values keep halving: 128, 64, 32, 16. The value 36 is not a power of 2, so the line should be c = [128, 64, 32, 16].
Question 3
A bits = 16
B values = 2 ** bits
C print(values) # 65536
D print(values + 1) # 65535
Line D — With all 16 bits set to 1 the largest value is 65535, which is one less than the 65536 possible patterns. It should be values - 1, not values + 1.
Question 4
A place = 128
B for i in range(8):
C print(place)
D place = place * 2
Line D — Moving right along the byte, each place value is half the previous one (128, 64, 32 ... 1). It should be place // 2, not place * 2.
Question 5
A place = 1
B for i in range(8):
C print(place)
D place = place * 2
Line B — A 16-bit number has 16 place values, so the loop must run 16 times: range(16). Using range(8) stops at 128 and misses the left-hand byte.

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