← ACARA Digital Technologies Years 7–8

Powers of 2 as binary place values

Binary place values: 1, 2, 4, 8, 16, 32, 64, 128 and so on. Each place is double the one before it, which is why binary is called base 2.

Part of: ACARA Digital Technologies Years 7–8 — Binary Number Representation — Module 12

Understanding Powers of 2 as binary place values in ACARA Digital Technologies Years 7–8

What it does: The place after 64 is found by doubling it, which gives 128. Each place value is 2 raised to a power, so the number 2 is the base. This gives 1, 2, 4 and 8. Each binary place is double the one before it: 1, 2, 4, 8. Counting 1, 2, 3, 4 adds one each time, which is how a normal count works, not how binary places work. 2^3 means 2 x 2 x 2 = 8. A common slip is 2 x 3 = 6, but a power means multiplying 2 by itself 3 times. Doubling 32 gives 64, and doubling 64 gives 128. 96 is halfway between 64 and 128, but it is not a power of 2.

Common mistakes: The first binary place value is 1, not 0. Doubling 0 would give 0 forever. It should be place = 1. Each binary place value is double the one before, so we multiply by 2. Adding 2 gives 3, which is not a power of 2. It should be place = place * 2.

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Free printable worksheet with full answer key — or add this module to a class and your students practise it interactively.

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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
Which list shows the first four binary place values, starting from the right?
1, 2, 3, 41, 2, 4, 8
Matches
Binary place values: 1, 2, 3, 4, 5, 6, 7, 8
↓
The values go up by 1 each time, but binary place values double each time. It should be 1, 2, 4, 8, 16, 32, 64, 128.
Applying
Meanings
Term
powers of 2
Meaning
Binary place values: 1, 2, 4, 8, 16, 32, 64, 128 and so on. Each place is double the one before it, which is why binary is called base 2.
Practice
value = 1
Testing
Gaps
2^0 =
Check Answer
Spots
1place = 0
2print(place)

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Quick Questions

1. Which list shows the first four binary place values, starting from the right?
A. 1, 2, 4, 8 ✓
B. 1, 2, 3, 4
Each binary place is double the one before it: 1, 2, 4, 8. Counting 1, 2, 3, 4 adds one each time, which is how a normal count works, not how binary places work.
2. How do you find the next binary place value after 16?
A. Add 10 to 16 to get 26
B. Double 16 to get 32 ✓
Every binary place is double the one before it, so after 16 comes 32. Binary places never go up by adding 10.
3. What is 2^3 (2 to the power of 3)?
A. 8 ✓
B. 6
2^3 means 2 x 2 x 2 = 8. A common slip is 2 x 3 = 6, but a power means multiplying 2 by itself 3 times.
4. What is the value of the place 2^0?
A. 1 ✓
B. 0
2^0 = 1. The right-most binary place is worth 1, which is why every whole number can be built from the places.
5. Which place value comes between 32 and 128 in the binary table?
A. 96
B. 64 ✓
Doubling 32 gives 64, and doubling 64 gives 128. 96 is halfway between 64 and 128, but it is not a power of 2.

Match the Pairs

2^3 = 8 → The fourth binary place value from the right, found by multiplying 2 by itself three times
1, 2, 4, 8, 16, 32, 64, 128 → The first eight binary place values, each double the one before
To extend the place value table, double the last value → 64 x 2 = 128
Binary is called base 2 because every place value is a power of 2 → 2^0, 2^1, 2^2, 2^3
Binary place values: 1, 2, 3, 4, 5, 6, 7, 8 → The values go up by 1 each time, but binary place values double each time. It should be 1, 2, 4, 8, 16, 32, 64, 128.
2^3 = 6 → 2^3 was worked out as 2 x 3. It means 2 x 2 x 2, so the answer is 8.

Key Term

powers of 2
Binary place values: 1, 2, 4, 8, 16, 32, 64, 128 and so on. Each place is double the one before it, which is why binary is called base 2.

Fill in the Blanks

1. 2^0 = 1
Hint: Type the value of the first (rightmost) binary place, which is 2 to the power of 0.
2. 1, 2, 4, 8, 16, 32, 64, 128
Hint: Type the place value that comes after 8 by doubling it.
3. place ← 1 place ← place * 2
Hint: Type the operator that makes each place value double the one before.
4. 2^7 = 128
Hint: Type the power. Counting from 0, the 128 place is the eighth place.
5. Binary is base 2: each place doubles
Hint: Type the base number, which is the number you multiply by to get each new place.

Spot the Error

Question 1
A place = 0
B print(place)
C place = place * 2
D print(place)
Line A — The first binary place value is 1, not 0. Doubling 0 would give 0 forever. It should be place = 1.
Question 2
A place = 1
B place = place + 2
C print(place)
Line B — Each binary place value is double the one before, so we multiply by 2. Adding 2 gives 3, which is not a power of 2. It should be place = place * 2.
Question 3
A values = [1, 2, 4, 8]
B last = values[3]
C next = last * 3
D print(next)
Line C — To extend the table you double the last value (8 x 2 = 16). Multiplying by 3 gives 24, which is not a binary place value. It should be next = last * 2.
Question 4
A places = []
B p = 1
C for i in range(8):
D places.append(i)
E p = p * 2
Line D — The list should store the place values 1, 2, 4, 8 and so on, held in p. Adding i stores 0, 1, 2, 3 instead. It should be places.append(p).
Question 5
A bits = 8
B biggest = 2 ** bits
C print(biggest)
Line B — With 8 bits the places run from 2^0 up to 2^7, so the biggest place value is 2 ** (bits - 1), which is 128. 2 ** bits would give 256.

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