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The largest number n bits can hold

The biggest whole number n bits can hold. Counting starts at 0, so n bits give 2^n values and the largest is one less. Example: 4 bits hold 0 to 15 and 8 bits hold 0 to 255.

Part of: ACARA Digital Technologies Years 7–8 — Binary Number Representation — Module 10

Understanding The largest number n bits can hold in ACARA Digital Technologies Years 7–8

What it does: 2^4 is 16 values (0 to 15), so the biggest number is 16 - 1 = 15. Zero uses up one of the 2^n patterns, so the top number is 2^n - 1. 4 bits give 2^4 = 16 values, but counting starts at 0, so the biggest is 16 - 1 = 15 (1111). 3 bits hold 2^3 = 8 values. They are 0, 1, 2, 3, 4, 5, 6, 7, so the largest is 7 (111). The biggest value has every bit set to 1. 1111 is 15, while 1000 is only 8. 2^8 = 256 different values, because 0 counts as a value. The largest of them is 255.

Common mistakes: The number of values n bits can store is 2 to the power of n, written 2 ** bits. Multiplying (2 * bits) gives 8 instead of 16 values. The base is 2 and the number of bits is the power, so 8 bits give 2 ** 8 = 256 values. 8 ** 2 swaps them and gives 64.

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This module's key terms are also in a printable crossword (Modules 173–188) and word search (Modules 153–188).

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Learning

Duo

What is the largest whole number that 4 bits can hold?

1615

Matches

Largest 8-bit number = 2^8 = 256

↓

2^8 is the number of values, not the biggest number. The largest is 2^8 - 1 = 255.

Applying

Meanings

Term

2^n - 1

Meaning

The biggest whole number n bits can hold. Counting starts at 0, so n bits give 2^n values and the largest is one less. Example: 4 bits hold 0 to 15 and 8 bits hold 0 to 255.

Practice

bits = 4

Testing

Gaps

Largest value with n bits = 2^n 1

Check Answer

Spots

1bits = 4

2values = 2 * bits

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Quick Questions

1. What is the largest whole number that 4 bits can hold?

A. 15 ✓

B. 16

4 bits give 2^4 = 16 values, but counting starts at 0, so the biggest is 16 - 1 = 15 (1111).

2. What is the largest whole number that 8 bits can hold?

A. 256

B. 255 ✓

8 bits give 2^8 = 256 values from 0 to 255. The largest is 256 - 1 = 255 (11111111).

3. Which formula gives the largest whole number n bits can hold?

A. 2^n

B. 2^n - 1 ✓

2^n is how many values there are. Because counting starts at 0, the biggest number is one less: 2^n - 1.

4. Why is the largest 3-bit number 7 and not 8?

A. Counting starts at 0, so 8 values run from 0 to 7 ✓

B. Counting starts at 1, so 8 values run from 1 to 8

3 bits hold 2^3 = 8 values. They are 0, 1, 2, 3, 4, 5, 6, 7, so the largest is 7 (111).

5. Which bit pattern is the largest number 4 bits can hold?

A. 1111 ✓

B. 1000

The biggest value has every bit set to 1. 1111 is 15, while 1000 is only 8.

Match the Pairs

2^n - 1 → The largest whole number that n bits can hold

1111 1111 → The biggest 8-bit number, which is 255 in decimal

4 bits can show 16 different values, but the largest number is 15 → 2^4 - 1 = 15

The largest number is one less than the number of values because counting starts at 0 → values = 2^n, largest = values - 1

Largest 8-bit number = 2^8 = 256 → 2^8 is the number of values, not the biggest number. The largest is 2^8 - 1 = 255.

4 bits hold the numbers 1 to 16 → Counting starts at 0, not 1. 4 bits hold 0 to 15.

Key Term

2^n - 1

The biggest whole number n bits can hold. Counting starts at 0, so n bits give 2^n values and the largest is one less. Example: 4 bits hold 0 to 15 and 8 bits hold 0 to 255.

Fill in the Blanks

1. Largest value with n bits = 2^n - 1

Hint: Type the operator that makes the biggest value one less than the number of values.

2. 4 bits: 0 to 2^4 - 1 = 15

Hint: Type the largest whole number that 4 bits can hold.

3. 8 bits: 0 to 2^8 - 1 = 255

Hint: Type the largest whole number that 8 bits can hold.

4. 3 bits hold 2^3 = 8 values, so max is 7

Hint: Type how many different values 3 bits can store, counting 0.

5. The largest 4-bit pattern is 1111

Hint: Type the four-bit pattern where every bit is switched on.

Spot the Error

Question 1

A bits = 4

B values = 2 * bits

C biggest = values - 1

D print(biggest)

Line B — The number of values n bits can store is 2 to the power of n, written 2 ** bits. Multiplying (2 * bits) gives 8 instead of 16 values.

Question 2

A bits = 8

B values = 8 ** 2

C biggest = values - 1

D print(biggest)

Line B — The base is 2 and the number of bits is the power, so 8 bits give 2 ** 8 = 256 values. 8 ** 2 swaps them and gives 64.

Question 3

A biggest = 2 ** 4 + 1

B print(biggest)

C print("4 bits: 0 to", biggest)

Line A — The biggest number is one less than the number of values, because counting starts at 0. It should be 2 ** 4 - 1 = 15, not +1.

Question 4

A bits = 5

B biggest = 2 ** bits

C print(biggest)

Line B — 2 ** bits is how many values 5 bits can store (32), not the biggest value. The biggest is one less, so it needs - 1 to give 31.

Question 5

A lowest = 0

B values = 2 ** 4

C biggest = values

D print(lowest, "to", biggest)

Line C — 4 bits store 16 values, from 0 up to 15. The biggest number is values - 1, because counting starts at 0 and not at 1.

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