Lesson 5 of 5
Add bit patterns, follow the carries and check the total
One idea at a time. Try the example, then keep going.
Big ideas · 1 of 12
Big ideas
Do this: Read the column facts. Say what you write and what you carry.
Place valueWhat a digit is worth because of WHERE it sits.Exactly: In binary, the columns from right to left are worth 1, 2, 4, 8, 16, 32, 64 and 128. Each column is worth double the one to its right.Mix-up alert: A 1 in the 8s place is worth 8. A 1 in the 1s place is worth only 1.See this in Key words tells you where each bit belongs. Put the ones under the ones, twos under the twos, and so on. Start adding at the right.
BinaryA way of writing numbers with only TWO digits: 0 and 1.Exactly: A number system where each place value is a power of two: 1, 2, 4, 8 and so on. Computers store all data in binary.Mix-up alert: 1010 in binary is ten. It is not one thousand ten.See this in Key words uses only 0 and 1. Column facts: 0 + 0 = 0 and 0 + 1 = 1. Write that digit; there is no carry.
A carry moves into the next bigger place. In binary, 1 + 1 = 10: write 0 and carry 1 one column left. The 10 means two, not ten.
A carry counts in the column it enters. In binary, 1 + 1 + 1 = 11: write 1 and carry 1 one column left. The 11 means three.
Overflow means the full sum does not fit in the chosen width. If a carry leaves the leftmost place, write it in an extra place for the full sum. A fixed-width box keeps only its own places, not the full total.
Use Place valueWhat a digit is worth because of WHERE it sits.Exactly: In binary, the columns from right to left are worth 1, 2, 4, 8, 16, 32, 64 and 128. Each column is worth double the one to its right.Mix-up alert: A 1 in the 8s place is worth 8. A 1 in the 1s place is worth only 1.See this in Key words to check the sum in decimal. Read both starting patterns, add their decimal values, then read the full binary answer. The totals must agree.
The CPUThe part that carries out the instructions.Exactly: The processor that executes program instructions, using working data held in RAM.Mix-up alert: RAM holds current work. The CPU carries out instructions; RAM does not do the calculations for it.See this in Key words executes instructions and does arithmetic on bit patterns. Adding two counts is processing; showing the total on a screen is output.
Step-through
Do this: Step through the picture. Predict the next answer digit before each step.
Watch 0110 + 0111 = 1101. The places are 8, 4, 2, 1 from left to right.
Whole picture, 6 steps
6 steps. Tap a number or Show me how it works.
Try it
Do this: Predict the total, then step through each column. Change the switches and try again.
Start with four bits. Watch where each carry goes, then try eight bits. Use the decimal check to test your prediction.
Start on the right. Press Enter or Space to flip a bit.
Number A: 0110 = 6
Number B: 0111 = 7
Rows: carry, A, B, then answer. The extra place on the left holds the final carry.
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Remember: Each column adds two bits and any incoming carry.
Do this: Use the place table. Cover the answer and add from the right.
| 8 | 4 | 2 | 1 |
|---|---|---|---|
| 0 | 1 | 0 | 1 |
Add 0101 and 0010 under places 8, 4, 2, 1.
Carry row: 0000. Answer: 0101 + 0010 = 0111. Decimal check: 5 + 2 = 7.
Do this: Point to the column that receives each carry.
| 8 | 4 | 2 | 1 |
|---|---|---|---|
| 0 | 0 | 1 | 1 |
Add 0011 and 0001 under places 8, 4, 2, 1.
Carry row: 0110. Answer: 0011 + 0001 = 0100. Decimal check: 3 + 1 = 4.
Do this: Find the column with two 1s and an incoming carry.
| 8 | 4 | 2 | 1 |
|---|---|---|---|
| 0 | 1 | 1 | 1 |
Add 0111 and 0011 under places 8, 4, 2, 1.
Carry row: 1110. Answer: 0111 + 0011 = 1010. Decimal check: 7 + 3 = 10.
Do this: Use all eight places. Say each incoming carry before adding.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
Add 00101101 and 00011011 under places 128, 64, 32, 16, 8, 4, 2, 1.
Carry row: 01111110. Answer: 00101101 + 00011011 = 01001000. Decimal check: 45 + 27 = 72.
Do this: Distinguish the full sum from the four bits a box keeps.
| 8 | 4 | 2 | 1 |
|---|---|---|---|
| 1 | 1 | 0 | 0 |
Add 1100 and 0101 under places 8, 4, 2, 1.
Carry row: 1000. Full sum: 1100 + 0101 = 10001. Decimal check: 12 + 5 = 17. This is overflow in four bits. Kept bits: 0001. Those bits show only 1, not the full total of 17.
Do this: Say how binary writes two.
Writing a 2 in an answer box is not binary. 1 + 1 makes two, written 10 in binary: write 0 and carry 1 left. For three, write 1 and carry 1; never put a 3 in a bit box.
Do this: Point to all three inputs a column may have.
Adding only the two starting bits can miss an incoming carry. Look above the column before you add. In the 2s column of 0011 plus 0001, include the carried 1. The correct sum is 0011 + 0001 = 0100.
Do this: Trace the direction with your finger.
Starting on the left can make you finish a column before its carry arrives. Start at the 1s place on the right. A carry moves one place left, never right or back into the column that made it.
Do this: Read the two digits with binary place values.
Reading 10 as ten uses decimal places. In BinaryA way of writing numbers with only TWO digits: 0 and 1.Exactly: A number system where each place value is a power of two: 1, 2, 4, 8 and so on. Computers store all data in binary.Mix-up alert: 1010 in binary is ten. It is not one thousand ten.See this in Key words, the two places are 2 and 1, so 10 means two. That is why two 1s become a 0 here and a carried 1 in the next place.
Ready when you are.
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Try it
Do this: Fill the carry row and answer row, then check. Work from the right.
Add these patterns in four bits. These first sums fit; include every incoming carry.
Start on the right. Press Enter or Space to flip a bit.
Sum 1 of 3. Set the answer bits. Carries are optional.
Number A: 0010 = 2
Number B: 0011 = 3
Rows: carry, A, B, then answer. The extra place on the left holds the final carry.
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Remember: A column may need two bits plus a carry.
Try it
Do this: Keep all eight answer places, including zeros on the left.
Use the same column rules with a whole byte. Check the decimal total after each sum.
Start on the right. Press Enter or Space to flip a bit.
Sum 1 of 3. Set the answer bits. Carries are optional.
Number A: 00010101 = 21
Number B: 00001011 = 11
Rows: carry, A, B, then answer. The extra place on the left holds the final carry.
Left places
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128
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16
Right places: start here
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Try it
Do this: Write the full sum. Use the extra place on the left when it is needed.
A four-bit box cannot hold every sum of two four-bit numbers. Do not drop the final carry. Later rounds may or may not overflow.
Start on the right. Press Enter or Space to flip a bit.
Sum 1 of 3. Set the answer bits. Carries are optional.
Number A: 1111 = 15
Number B: 0001 = 1
Rows: carry, A, B, then answer. The extra place on the left holds the final carry.
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Remember: The full sum and the kept bits are different when overflow happens.
Try it
Do this: Work out the sum, check, then try another.
Practice fresh four-bit and eight-bit sums. Each needs a carry and fits its width. Show the working if you get stuck.
Streak: 0 correct in a row. This session only.
Write it
Do this: Write two or three sentences, then compare your reasoning.
Start here if it helps: I start in the 1s column because...
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Flashcards
Do this: Say the answer and explain the carry before you flip.
Card 1 of 16
0 Know it0 Still learning
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On the card: Enter or Space flips. Left and Right move. After revealing, 1 marks Still learning and 2 marks Know it. Tab reaches every control.
What does a column containing 0 and 0 write?
Write 0. There is no carry.
What does a column containing 0 and 1 write, with no incoming carry?
Write 1. There is no carry.
What do you do when a column makes two?
Two is 10 in binary. Write 0 and carry 1 one column left.
What do you do when two 1s and an incoming carry make three?
Three is 11 in binary. Write 1 and carry 1 one column left.
Which column do you add first?
The 1s column on the right, so carries reach each next column in time.
Where does a carry from the 4s column go?
Into the 8s column, one place left.
What must you count in each column?
Both starting bits and any incoming carry.
| 8 | 4 | 2 | 1 |
|---|---|---|---|
| Blank | Blank | Blank | Blank |
Work it out on paper if it helps.
Add 0001 + 0100.
0001 + 0100 = 0101. Decimal check: 1 + 4 = 5. No column needs a carry.
| 8 | 4 | 2 | 1 |
|---|---|---|---|
| Blank | Blank | Blank | Blank |
Work it out on paper if it helps.
Add 0101 + 0001.
0101 + 0001 = 0110. Decimal check: 5 + 1 = 6. The 1s column carries into the 2s column.
| 8 | 4 | 2 | 1 |
|---|---|---|---|
| Blank | Blank | Blank | Blank |
Work it out on paper if it helps.
Add 0111 + 0001.
0111 + 0001 = 1000. Decimal check: 7 + 1 = 8. Follow the carries into the 8s column.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| Blank | Blank | Blank | Blank | Blank | Blank | Blank | Blank |
Work it out on paper if it helps.
Add 00001111 + 00000001.
00001111 + 00000001 = 00010000. Decimal check: 15 + 1 = 16. Keep all eight places.
What is overflow in a fixed-width addition?
The full sum does not fit. A carry leaves the leftmost place of the chosen width.
| 8 | 4 | 2 | 1 |
|---|---|---|---|
| Blank | Blank | Blank | Blank |
Work it out on paper if it helps.
Add 1001 + 1000. Give the full sum.
1001 + 1000 = 10001. Decimal check: 9 + 8 = 17. This overflows four bits. Kept bits: 0001.
How can decimal help you check a binary sum?
Read both starting numbers in decimal and add them. Read the full binary answer in decimal too. The totals must agree.
Why keep a leading zero in an eight-bit answer?
It keeps the requested width. The zero adds no value.
Which computer part carries out arithmetic instructions?
The CPU executes instructions and does arithmetic on bit patterns.
True or false
Do this: Choose True or False, then explain your choice.
Self-test
Do this: Answer all eight questions. Read the explanation before moving on.
8 questions. One at a time.
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Match
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Connect the ideas
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