Working out how many bits a number needs
Name:
Date:
Class:
A
Quick Questions
Shade in the A or B circle for each question.
1. Can the number 20 be stored in 4 bits?
2. How many bits does the number 25 need?
3. Which rule finds a big enough bit length for a number?
4. What is the largest number 5 bits can hold?
5. You want to store the number 9. Which bit length is the smallest that is big enough?
B
Match the Pairs
Draw a line connecting each item on the left to its match on the right.
| Left column | Right column | |
|---|---|---|
| 12^4 - 1 = 15 | A20 <= 2^4 - 1 is False | |
| 22^5 - 1 = 31 | BForgot to subtract 1. Counting starts at 0, so the largest 4-bit number is 2^4 - 1 = 15. | |
| 3To test if a number fits in a bit length, compare it with the largest value for that length | C4 bits can hold at most 15, so 20 is too big for 4 bits | |
| 4Choose the smallest bit length n whose largest value is at least your number | Dn = 5 because 2^5 - 1 >= 25 | |
| 5Largest 4-bit number = 2^4 = 16 | E5 bits can hold up to 31, so 25 fits in 5 bits | |
| 631 needs 6 bits because 31 is not bigger than 2^5 - 1 | F31 equals the largest 5-bit value (2^5 - 1 = 31), so it fits in 5 bits. A number only needs more bits if it is bigger than the maximum. |
C
Definitions
2^n - 1 >= number
D
Fill in the Blanks
Complete each line of code by filling in the missing parts.
1.
4 bits: 2^4 - 1 = __
2.
2^_ - 1 = 31, which is at least 25
3.
Is 2^5 - 1 __ 25? Yes, so 5 bits is enough
4.
n = 1
WHILE 2^n - 1 < 20:
n = _____
5.
9: 2^3 - 1 = 7 is too small, so use 2^_ - 1
E
Spot the Error
Shade in the letter of the line that contains an error.
Question 1
A
max_value = 2**4
B
number = 20
C
if number > max_value:
D
print('Needs more than 4')
Question 2
A
number = 25
B
bits = 1
C
while 2**bits - 1 < number:
D
bits = bits - 1
E
print(bits)
Question 3
A
number = 25
B
bits = 5
C
largest = 2*bits - 1
D
print(largest >= number)
Question 4
A
number = 20
B
largest_4 = 2**3 - 1
C
print(number <= largest_4)
Question 5
A
largest ← 5^2 - 1
B
IF 25 <= largest THEN
C
OUTPUT '5 bits is enough'
D
ENDIF
Answer Key
Working out how many bits a number needs
Section A: Quick Questions
Q1
A — No, 4 bits only go up to 15
Q2
B — 5 bits (largest value 31)
Q3
A — Pick the smallest n where 2^n - 1 is at least the number
Q4
A — 31
Q5
B — 4 bits, because 3 bits only reach 7
Section B: Match the Pairs
Q1
2^4 - 1 = 15 → 4 bits can hold at most 15, so 20 is too big for 4 bits
Q2
2^5 - 1 = 31 → 5 bits can hold up to 31, so 25 fits in 5 bits
Q3
To test if a number fits in a bit length, compare it with the largest value for that length → 20 <= 2^4 - 1 is False
Q4
Choose the smallest bit length n whose largest value is at least your number → n = 5 because 2^5 - 1 >= 25
Q5
Largest 4-bit number = 2^4 = 16 → Forgot to subtract 1. Counting starts at 0, so the largest 4-bit number is 2^4 - 1 = 15.
Q6
31 needs 6 bits because 31 is not bigger than 2^5 - 1 → 31 equals the largest 5-bit value (2^5 - 1 = 31), so it fits in 5 bits. A number only needs more bits if it is bigger than the maximum.
Section C: Definitions
Q1
2^n - 1 >= number — Find the smallest number of bits n whose largest value, 2^n - 1, is at least your number. Example: 20 is too big for 4 bits (max 15), so it needs 5 bits (max 31).Section D: Fill in the Blanks
Q1
4 bits: 2^4 - 1 = 15Q2
2^5 - 1 = 31, which is at least 25Q3
Is 2^5 - 1 >= 25? Yes, so 5 bits is enoughQ4
n = 1
WHILE 2^n - 1 < 20:
n = n + 1Q5
9: 2^3 - 1 = 7 is too small, so use 2^4 - 1Section E: Spot the Error
Q1
A — The largest value n bits can hold is 2^n - 1, not 2^n. With 4 bits the largest value is 15, so the line should be max_value = 2**4 - 1.
Q2
D — While the largest value is still too small, we need MORE bits, so bits must go up by 1. Subtracting would never reach 5 bits for 25. It should be bits = bits + 1.
Q3
C — Each extra bit doubles the number of values, so we need a power: 2**bits. Using 2*bits gives 9 instead of 31. It should be largest = 2**bits - 1.
Q4
B — The variable is for 4 bits, so the power must be 4: 2**4 - 1 = 15. Using 2**3 - 1 gives the largest 3-bit value, which is 7.
Q5
A — The base must be 2 and the number of bits is the power. 5^2 - 1 is 24, but 5 bits hold up to 2^5 - 1 = 31. It should be largest ← 2^5 - 1.