← IGCSE Computer Science Paper 1

Converting 16-bit binary to denary

To convert 16-bit binary to denary, write the place values (32768 down to 1) over the bits, skip the 0 columns, then add the values that have a 1 under them. Leading zeros add nothing.

Part of: IGCSE Computer Science 0478 — 1.1 Number Systems — Module 11

Understanding Converting 16-bit binary to denary in IGCSE Computer Science Paper 1

What it does: The left byte has a 1 in the 256 column and the right byte has 2 + 1 = 3. Adding the two byte totals gives 259. Only two bits are 1: the 16384 column and the 32 column. All the 0 columns are skipped, so the total is 16416. Skip the 0 columns because they add nothing. Only add the values that have a 1 under them. The 16 place values start at 32768 on the left and halve each time down to 1. 16384 is the second column. The only 1 is in the 256 column. The 128 column is the one to its right and holds a 0.

Common mistakes: In 1010 0101 the 64 column has a 0 under it, so it is skipped. The 1s are under 128, 32, 4 and 1, so the line should be Add: 128 + 32 + 4 + 1. The left 1 is the lowest bit of the top byte, which is worth 256 in 16 bits, not 128. The line should be Ones at: 256 and 1.

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Learning
Duo
Which place values do you add together when converting 16-bit binary to denary?
Only the place values that have a 1 under themEvery place value, including the 0 columns
Matches
0000000000001010 = 8 + 4 + 2 = 14
↓
The 4 column has a 0 under it, so it should be skipped. The correct sum is 8 + 2 = 10.
Applying
Meanings
Term
add the place values with a 1
Meaning
To convert 16-bit binary to denary, write the place values (32768 down to 1) over the bits, skip the 0 columns, then add the values that have a 1 under them. Leading zeros add nothing.
Practice
0000000100101100
Testing
Gaps
32768 16384 4096 2048 1024 512 256
Check Answer
Spots
2Leading zeros add nothing
3Add: 128 + 64 + 4 + 1

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Quick Questions

1. Which place values do you add together when converting 16-bit binary to denary?
A. Every place value, including the 0 columns
B. Only the place values that have a 1 under them ✓
Skip the 0 columns because they add nothing. Only add the values that have a 1 under them.
2. A 16-bit number has a 1 in the leftmost column only. What is that column worth?
A. 32768 ✓
B. 16384
The 16 place values start at 32768 on the left and halve each time down to 1. 16384 is the second column.
3. What is 0000000000000101 in denary?
A. 5 ✓
B. 4
Only the 4 column and the 1 column have a 1 under them, so 4 + 1 = 5.
4. What is 0000000100000000 in denary?
A. 128
B. 256 ✓
The only 1 is in the 256 column. The 128 column is the one to its right and holds a 0.
5. Splitting 00000001 00001010 into two bytes, what is the denary value?
A. 266 ✓
B. 11
The high byte 00000001 is worth 256, not 1. The low byte 00001010 is 8 + 2 = 10. So 256 + 10 = 266.

Match the Pairs

0000000000000101 → Leading zeros add nothing, so only the 4 and 1 columns count: 4 + 1 = 5
1000000000000001 = 32768 + 1 = 32769 → Only the first and last columns have a 1, so you add 32768 and 1 and skip all the 0 columns
Split the 16-bit number into two bytes, find the value of each byte's 1s, then add the two totals → 00000001 00000011 = 256 + 3 = 259
Skip the 0 columns and add only the place values that have a 1 under them → 0100000000100000 = 16384 + 32 = 16416
0000000000001010 = 8 + 4 + 2 = 14 → The 4 column has a 0 under it, so it should be skipped. The correct sum is 8 + 2 = 10.
0000000100000000 = 128 → The 1 is in the 256 column, not the 128 column. The place values must continue up to 32768, so the answer is 256.

Key Term

add the place values with a 1
To convert 16-bit binary to denary, write the place values (32768 down to 1) over the bits, skip the 0 columns, then add the values that have a 1 under them. Leading zeros add nothing.

Fill in the Blanks

1. 32768 16384 8192 4096 2048 1024 512 256
Hint: Type the missing place value, which is half of the one before it.
2. 0000000000000101 = 4 + 1 = 5
Hint: Type the place value of the third column from the right.
3. 0000000100000000 = 256
Hint: The leading zeros add nothing. Type the place value of the single 1, in the ninth column from the right.
4. High byte 00000001 = 1 x 256 Low byte 00000011 = 2 + 1 Total = 256 + 3 = 259
Hint: Type the value of the low byte, found by adding its place values that have a 1.
5. 1000000000000001 = 32768 + 1 = 32769
Hint: Type the place value of the right-most column.

Spot the Error

Question 1
A Binary: 0000 0000 1010 0101
B Leading zeros add nothing
C Add: 128 + 64 + 4 + 1
D Denary: 165
Line C — In 1010 0101 the 64 column has a 0 under it, so it is skipped. The 1s are under 128, 32, 4 and 1, so the line should be Add: 128 + 32 + 4 + 1.
Question 2
A Binary: 0000000100000001
B Ones at: 128 and 1
C 256 + 1 = 257
Line B — The left 1 is the lowest bit of the top byte, which is worth 256 in 16 bits, not 128. The line should be Ones at: 256 and 1.
Question 3
A Top byte: 00000011
B Top byte = 128 + 64 = 192
C Bottom: 00001010 = 8 + 2 = 10
D 768 + 10 = 778
Line B — The top byte uses the place values 32768 down to 256, so its last two bits are worth 512 and 256, not 128 and 64. It should be Top byte = 512 + 256 = 768.
Question 4
A Binary: 0000000010000001
B Add only the columns with a 0
C 128 + 1 = 129
Line B — We skip the 0 columns and add only the place values that have a 1 under them. The line should say columns with a 1.
Question 5
A Binary: 0000000000000111
B Ones: 8, 4, 2, 1
C 8 + 4 + 2 + 1 = 15
Line A — This binary number only has 1s under 4, 2 and 1, so it does not match the 8 that is added later. The program should start with Binary: 0000000000001111.

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