← ACARA Digital Technologies Years 7–8

Converting numbers 0 to 31 with five bits

The five binary place values. Use a 1 to count a place value and a 0 to skip it. Together they make any whole number from 0 to 31 as five bits, so 31 is 11111.

Part of: ACARA Digital Technologies Years 7–8 — Binary Number Representation — Module 17

Understanding Converting numbers 0 to 31 with five bits in ACARA Digital Technologies Years 7–8

What it does: Put a 1 under 16, 2 and 1 and a 0 under 8 and 4. That gives 10011, which is 19. Count only the place values that have a 1: 16, 4 and 2. 16 + 4 + 2 = 22. 31 = 16 + 8 + 4 + 2 + 1, so every place value is counted. 11110 is 30 because it skips the 1. All five place values together make 16 + 8 + 4 + 2 + 1 = 31. Five bits can show 32 different values, but they run from 0 to 31. 19 = 16 + 2 + 1, so put a 1 under 16, 2 and 1 and a 0 under 8 and 4. 10101 is 21.

Common mistakes: 19 = 16 + 2 + 1, so the 16, 2 and 1 columns are 1 and the 8 and 4 columns are 0. That gives 10011. 10101 is 16 + 4 + 1 = 21. The five place values are 16 8 4 2 1. 32 is the sixth place value and belongs to a six-bit number. The line should read 16 8 4 2 1.

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This module's key terms are also in a printable crossword (Modules 189–203) and word search (Modules 189–236).

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Learning
Duo
What is 10110 in denary?
26 (16 + 8 + 2)22 (16 + 4 + 2)
Matches
13 in five bits is 11010
↓
11010 is 16 + 8 + 2 = 26, not 13. To make 13 use 8 + 4 + 1, so it should be 01101.
Applying
Meanings
Term
16 8 4 2 1
Meaning
The five binary place values. Use a 1 to count a place value and a 0 to skip it. Together they make any whole number from 0 to 31 as five bits, so 31 is 11111.
Practice
16 8 4 2 1
Testing
Gaps
Five place values: 16, , 4, 2, 1
Check Answer
Spots
33 = 2 + 1
419 is 10101

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Quick Questions

1. What is 10110 in denary?
A. 26 (16 + 8 + 2)
B. 22 (16 + 4 + 2) ✓
Count only the place values that have a 1: 16, 4 and 2. 16 + 4 + 2 = 22.
2. Which five-bit number is 31?
A. 11110
B. 11111 ✓
31 = 16 + 8 + 4 + 2 + 1, so every place value is counted. 11110 is 30 because it skips the 1.
3. What is the largest whole number that five bits can show?
A. 31 ✓
B. 32
All five place values together make 16 + 8 + 4 + 2 + 1 = 31. Five bits can show 32 different values, but they run from 0 to 31.
4. Which is 19 written in five bits?
A. 10011 (16 + 2 + 1) ✓
B. 10101 (16 + 4 + 1)
19 = 16 + 2 + 1, so put a 1 under 16, 2 and 1 and a 0 under 8 and 4. 10101 is 21.
5. Which list shows the five binary place values from left to right?
A. 16, 8, 4, 2, 1 ✓
B. 10, 8, 6, 4, 2
Each binary place value is double the one on its right: 1, 2, 4, 8, 16. Written left to right, the biggest comes first.

Match the Pairs

10110 → Five bits worth 16 + 4 + 2, which makes the number 22
11111 → All five place values switched on: 16 + 8 + 4 + 2 + 1 = 31, the biggest five-bit number
To write 19 in five bits: 19 = 16 + 2 + 1, so use the 16, 2 and 1 columns → 10011
Numbers below 16 need no 16 column, so the first bit is 0. For example, 9 = 8 + 1 → 01001
13 in five bits is 11010 → 11010 is 16 + 8 + 2 = 26, not 13. To make 13 use 8 + 4 + 1, so it should be 01101.
32 in five bits is 100000 → 100000 has six bits. Five bits only reach 31 (11111), so 32 does not fit in five bits.

Key Term

16 8 4 2 1
The five binary place values. Use a 1 to count a place value and a 0 to skip it. Together they make any whole number from 0 to 31 as five bits, so 31 is 11111.

Fill in the Blanks

1. Five place values: 16, 8, 4, 2, 1
Hint: Type the place value that sits between 16 and 4.
2. 10110 = 16 + 4 + 2 = 22
Hint: Add up the place values that have a 1 and type the decimal total.
3. 31 in five bits is 11111
Hint: Type the five-bit pattern where every place value is counted.
4. 19 = 16 + 2 + 1, so 19 is 10011
Hint: Type the five bits, using a 1 for each place value that is added and a 0 for each one skipped.
5. 25 - 16 = 9 9 = 8 + 1, so 25 is 11001
Hint: Type what is left after the 16 column has been counted.

Spot the Error

Question 1
A Convert 19 to five bits
B 19 - 16 = 3
C 3 = 2 + 1
D 19 is 10101
Line D — 19 = 16 + 2 + 1, so the 16, 2 and 1 columns are 1 and the 8 and 4 columns are 0. That gives 10011. 10101 is 16 + 4 + 1 = 21.
Question 2
A Place values: 32 16 8 4 2
B Five bits: 5 columns
C Smallest number: 00000 = 0
D Largest number: 11111 = 31
Line A — The five place values are 16 8 4 2 1. 32 is the sixth place value and belongs to a six-bit number. The line should read 16 8 4 2 1.
Question 3
A Convert 25 to five bits
B 25 - 16 = 8, 8 - 8 = 1
C Columns: 16 and 8 and 1
D 25 is 11001
Line B — 25 - 16 is 9, not 8. Then 9 - 8 = 1, which leaves the 1 column. The line should be 25 - 16 = 9, 9 - 8 = 1.
Question 4
A Convert 12 to five bits
B 16 is too big, so put 0
C 8 fits: 12 - 8 = 5
D 4 fits, so 12 is 01100
Line C — 12 - 8 is 4, not 5. The leftover 4 fills the 4 column, giving 01100. The line should be 8 fits: 12 - 8 = 4.
Question 5
A Binary to decimal: 01101
B 16 8 4 2 1
C 0 1 1 0 1
D 8 + 4 + 2 = 14
Line D — The bits 0 1 1 0 1 have 1s under 8, 4 and 1. The 2 column holds a 0, so it is not added. The line should be 8 + 4 + 1 = 13.

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