← IGCSE Computer Science Paper 1

Shifts as multiplying and dividing by powers of 2

Each left shift multiplies a binary number by 2 and each right shift divides it by 2, so n shifts multiply or divide by 2^n. Bits shifted off the end are lost.

Part of: IGCSE Computer Science 0478 — 1.1 Number Systems — Module 27

Understanding Shifts as multiplying and dividing by powers of 2 in IGCSE Computer Science Paper 1

What it does: 15 / 4 is 3 remainder 3. The two 1s shifted off the right end are lost, so the result is 3 (00000011). Each left shift multiplies by 2, so 3 shifts multiply by 2 x 2 x 2 = 2^3 = 8. Shifting left 3 places multiplies by 8, and 5 x 8 = 40. The bits move 3 places left to give 00101000. Two right shifts divide by 4, and 28 / 4 = 7. 00001110 (14) would be a shift of only 1 place. An 8-bit register cannot grow. A 1 shifted off the left end is lost, so the answer 192 x 2 = 384 is not stored correctly.

Common mistakes: 5 x 8 = 40, not 48. Three left shifts multiply by 2^3 = 8, so 5 becomes 40 (00101000). The result 3 comes from dividing by 4, which needs a RIGHT shift of 2 places. A left shift would multiply by 4. It should read 'Shift right 2 places'.

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Learning
Duo
A logical left shift of 3 places multiplies a binary number by what?
68
Matches
00000011 << 3 = 00001100 (3 x 8 = 12)
↓
Only shifted left 2 places, which multiplies by 4. Shifting left 3 places multiplies by 8, so the answer is 24 (00011000).
Applying
Meanings
Term
n shifts = 2^n
Meaning
Each left shift multiplies a binary number by 2 and each right shift divides it by 2, so n shifts multiply or divide by 2^n. Bits shifted off the end are lost.
Practice
Left shift 3: 00000101 -> 00101000
Testing
Gaps
Left shift by 3 multiplies by
Check Answer
Spots
3Multiply by 2^3 = 8
4Result: 5 x 8 = 48

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Quick Questions

1. A logical left shift of 3 places multiplies a binary number by what?
A. 8 ✓
B. 6
Each left shift multiplies by 2, so 3 shifts multiply by 2 x 2 x 2 = 2^3 = 8.
2. What is 00000101 (5) after a logical left shift of 3 places?
A. 00101000 (40) ✓
B. 00001111 (15)
Shifting left 3 places multiplies by 8, and 5 x 8 = 40. The bits move 3 places left to give 00101000.
3. What is 00011100 (28) after a logical right shift of 2 places?
A. 00001110 (14)
B. 00000111 (7) ✓
Two right shifts divide by 4, and 28 / 4 = 7. 00001110 (14) would be a shift of only 1 place.
4. A right shift of 2 places is applied to 00001111 (15). What is the result?
A. 00000011 (3) ✓
B. 3.75
15 / 4 = 3 remainder 3. The two 1s shifted off the right end are dropped, so the remainder is lost and the result is a whole number.
5. Which statement about a right shift by 2 places is correct?
A. It divides by 2 and keeps the remainder
B. It divides by 4 and drops any remainder ✓
Two right shifts divide by 2^2 = 4. Bits shifted off the right end are lost, so there is no remainder or fraction.

Match the Pairs

00000101 << 3 → Shifts left 3 places, multiplying 5 by 2^3 (8) to give 40 (00101000)
00010100 >> 2 → Shifts right 2 places, dividing 20 by 2^2 (4) to give 5 (00000101)
A right shift by 2 divides by 4 and drops the remainder, so 15 becomes 3 → 00001111 >> 2 = 00000011
A left shift loses the answer if a 1 is shifted off the end of the 8-bit register → 11000000 << 1 = 10000000
00000011 << 3 = 00001100 (3 x 8 = 12) → Only shifted left 2 places, which multiplies by 4. Shifting left 3 places multiplies by 8, so the answer is 24 (00011000).
00001101 >> 2 = 00000011.25 (13 / 4) → Binary shifts cannot make fractions. The remainder is dropped, so 13 / 4 gives 3 (00000011), not 3.25.

Key Term

n shifts = 2^n
Each left shift multiplies a binary number by 2 and each right shift divides it by 2, so n shifts multiply or divide by 2^n. Bits shifted off the end are lost.

Fill in the Blanks

1. Left shift by 3 multiplies by 8
Hint: Type the number you get from 2 to the power of 3
2. Right shift by 2 divides by 4
Hint: Type the number you get from 2 to the power of 2
3. 00000110 << 3 = 00110000
Hint: Type the 8-bit result of shifting left three places (6 times 8 = 48)
4. 13 >> 2 is 13 / 4 = 3 remainder 1
Hint: Type the remainder that is dropped by the right shifts
5. 10000001 << 1 = 00000010, so the 1 is lost
Hint: Type what happens to a 1 that is shifted off the left end

Spot the Error

Question 1
A Start: 00000101 (5)
B Shift left 3 places
C Multiply by 2^3 = 8
D Result: 5 x 8 = 48
Line D — 5 x 8 = 40, not 48. Three left shifts multiply by 2^3 = 8, so 5 becomes 40 (00101000).
Question 2
A Start: 00001100 (12)
B Shift left 2 places
C Divide by 2^2 = 4
D Result: 00000011 (3)
Line B — The result 3 comes from dividing by 4, which needs a RIGHT shift of 2 places. A left shift would multiply by 4. It should read 'Shift right 2 places'.
Question 3
A x = 00000011 (5)
B Shift left 3 places
C Multiply by 2^3 = 8
D Result: 00011000 (24)
Line A — 00000011 is 2 + 1 = 3, not 5. The start value must be 3 so that 3 x 8 = 24 matches the result 00011000.
Question 4
A Right shift 2: multiply by 4
B Start: 00010100 (20)
C Result: 00000101 (5)
Line A — A right shift divides the number. Two right shifts divide by 2^2 = 4, so 20 becomes 5. It should say 'divide by 4'.
Question 5
A Start: 00001101 (13)
B Shift right 2 places
C 13 / 4 = 3 remainder 1
D Result: 00000100 (4)
Line D — A right shift drops the remainder, it does not round up. 13 / 4 = 3 remainder 1, so the result is 00000011 (3).

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