← ACARA Digital Technologies Years 7–8

Converting numbers from 0 to 127 using seven place values

The seven binary place values. Put a 1 under each value you need and a 0 under the rest to write any whole number from 0 to 127. Check by adding them back up. 1111111 = 127.

Part of: ACARA Digital Technologies Years 7–8 — Representing Text, Images and Sound — Module 11, ACARA Digital Technologies Years 7–8 — Binary Number Representation — Module 18

Understanding Converting numbers from 0 to 127 using seven place values in ACARA Digital Technologies Years 7–8

What it does: Put a 1 under 64, 8, 2 and 1 and a 0 under 32, 16 and 4. That gives 1001011, which adds back up to 75. Seven 1s mean 64+32+16+8+4+2+1 = 127. 128 needs an eighth place. Take 64 from 100 to leave 36. Take 32 to leave 4. Then use 4. Putting 1s under 64, 32 and 4 gives 1100100. The 1s sit under 64, 16, 4 and 1. Adding them gives 64+16+4+1 = 85. 32+8+4+1 = 45. The wrong choice adds up to 53. Work from the biggest place value down. 64 fits in 70, which leaves 6 (4+2). So 70 is 1000110.

Common mistakes: 100 - 64 is 36, not 46. Subtracting wrongly would give the wrong remainder and the wrong binary number. The line should be 100 - 64 = 36. 13 - 8 is 5, not 6. The remainder 5 is then used for the 4 place and the 1 place. The line should be 13 - 8 = 5.

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Learning
Duo
What is the largest whole number you can write using seven binary places?
127, written as 1111111128, written as 10000000
Matches
40 = 1010000
↓
The 1s are under 64 and 16, which add up to 80, not 40. Forty is 32 + 8, so it should be 0101000.
Applying
Meanings
Term
64 32 16 8 4 2 1
Meaning
The seven binary place values. Put a 1 under each value you need and a 0 under the rest to write any whole number from 0 to 127. Check by adding them back up. 1111111 = 127.
Practice
64 32 16 8 4 2 1
Testing
Gaps
Place values: 32 16 8 4 2 1
Check Answer
Spots
1Places: 64 32 16 8 4 2 1
2100 - 64 = 46, so 64 gets 1

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Quick Questions

1. What is the largest whole number you can write using seven binary places?
A. 127, written as 1111111 ✓
B. 128, written as 10000000
Seven 1s mean 64+32+16+8+4+2+1 = 127. 128 needs an eighth place.
2. Which is 100 written in seven-place binary?
A. 1100010 (64+32+2)
B. 1100100 (64+32+4) ✓
Take 64 from 100 to leave 36. Take 32 to leave 4. Then use 4. Putting 1s under 64, 32 and 4 gives 1100100.
3. What is 1010101 as a decimal number?
A. 85 ✓
B. 75
The 1s sit under 64, 16, 4 and 1. Adding them gives 64+16+4+1 = 85.
4. Which place values add up to 45?
A. 32 + 8 + 4 + 1 ✓
B. 32 + 16 + 4 + 1
32+8+4+1 = 45. The wrong choice adds up to 53.
5. To convert 70, which place value do you try first?
A. 32, the middle place value
B. 64, the biggest value that fits in 70 ✓
Work from the biggest place value down. 64 fits in 70, which leaves 6 (4+2). So 70 is 1000110.

Match the Pairs

1111111 → The largest 7-bit number: 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127
1100100 → The number 100, because 64 + 32 + 4 = 100
Write 75 with seven place values: 64 + 8 + 2 + 1 = 75 → 1001011
Work down from 64 to write 50: no 64, then 32, 16 and 2 add up to 50 → 0110010
40 = 1010000 → The 1s are under 64 and 16, which add up to 80, not 40. Forty is 32 + 8, so it should be 0101000.
1111111 = 128 → Seven 1s add up to 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127, not 128. The largest 7-bit number is 127.

Key Term

64 32 16 8 4 2 1
The seven binary place values. Put a 1 under each value you need and a 0 under the rest to write any whole number from 0 to 127. Check by adding them back up. 1111111 = 127.

Fill in the Blanks

1. Place values: 64 32 16 8 4 2 1
Hint: Type the largest of the seven place values.
2. 100 = 64 + 32 + 4 100 in binary is 1100100
Hint: Put a 1 under each place value used in the sum and a 0 under the rest.
3. 1111111 = 64+32+16+8+4+2+1 = 127
Hint: Type the total when all seven place values are added up.
4. 50 is less than 64, so the 64 place gets a 0
Hint: Type the digit for a place value that is too big to fit.
5. Largest 7-bit number: 1111111 = 127
Hint: Type the binary number that has a 1 in every one of the seven places.

Spot the Error

Question 1
A Places: 64 32 16 8 4 2 1
B 100 - 64 = 46, so 64 gets 1
C 36 - 32 = 4, so 32 gets 1
D 4 needs only the 4 place
E Answer: 1100100
Line B — 100 - 64 is 36, not 46. Subtracting wrongly would give the wrong remainder and the wrong binary number. The line should be 100 - 64 = 36.
Question 2
A Places: 64 32 16 8 4 2 1
B 77 - 64 = 13, so 64 gets 1
C 13 - 8 = 6, so 8 gets 1
D 5 - 4 = 1, so 4 gets 1
E 1 - 1 = 0, so 1 gets 1
F Answer: 1001101
Line C — 13 - 8 is 5, not 6. The remainder 5 is then used for the 4 place and the 1 place. The line should be 13 - 8 = 5.
Question 3
A Binary: 1010101
B Places: 64 32 16 8 4 2 1
C Add: 64 + 32 + 4 + 1
D Total: 64+16+4+1 = 85
Line C — In 1010101 the 32 place has a 0, so 32 must not be added. The 1s are under 64, 16, 4 and 1, so the line should be Add: 64 + 16 + 4 + 1.
Question 4
A Largest 7-bit number: 1111110
B 64+32+16+8+4+2+1 = 127
C So 7 bits store 0 to 127
Line A — The largest 7-bit number has a 1 in every place, including the 1 place. It is 1111111, which equals 127. 1111110 is only 126.
Question 5
A Places: 64 32 16 8 4 2 1
B 45 < 64, so the 64 place is 1
C 45 - 32 = 13, so 32 gets 1
D 13 = 8 + 4 + 1
E Answer: 0101101
Line B — 45 is smaller than 64, so 64 does not fit into 45. The 64 place must be 0, not 1.

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