Stretch tasks Binary Arithmetic and Shifts

Round 1: Quiz Blitz

1. 10100011 109 + 54 = 163, which is 10100011.
2. 0 carry 1 There is no digit 2 in binary.
3. 1 carry 1 Three is 11 in binary: write 1 and carry 1.
4. False It usually stores a wrong value with no error reported, which is what makes it dangerous.
5. 01000000, which is 64 192 + 128 = 320 needs nine bits, so the carry out is lost and 64 is stored.
6. 00010110 11 becomes 22: a left shift doubles.
7. 00001100 25 becomes 12: the 1 that falls off the right is the remainder, and it is lost.
8. 8 2 x 2 x 2 = 8.
9. A shift is a single, cheap hardware operation It does less work than a general multiply.
10. True Adding the first two, then the third, is safe and fully acceptable.
11. The rightmost column Exactly as in denary, from the least significant bit.
12. Its remainder The bit that falls off the right is the remainder of the division by 2.

Round 2: Quick Fire

1. • The result needs more bits than are available, so a carry is lost
   • A wrong value is stored and used, with no error reported
   Model answer

   Overflow happens when the result of an addition needs more bits than are available, so the carry out of the most significant bit is lost. It is dangerous because a wrong value is stored and used, and usually no error is reported.

2. • Every bit moves one place right, and a zero comes in on the left
   • The rightmost 1 is discarded, so 25 becomes 12
   Model answer

   Every bit moves one place to the right and a zero comes in on the left. The 1 on the right-hand end is discarded, so 25 becomes 12: the remainder is lost.

3. • Shifting left 2 places multiplies by 2 twice
   • A shift is a single, fast hardware operation
   Model answer

   Shifting left 2 places multiplies by 2 twice, which is 4, and a shift is a single, fast hardware operation, far cheaper than a general multiply.

4. • 01010101 + 00110011 = 10001000
   • 10001000 + 00000111 = 10001111
   Model answer

   01010101 + 00110011 = 10001000. Then 10001000 + 00000111 = 10001111.

Round 3: Big Think

1. • Every bit moves a number of places
   • A left shift multiplies by 2 for each place
   • A right shift divides by 2 for each place
   • Zeros come in at the other end
   • Bits shifted out are discarded, not wrapped round
   • A right shift loses the remainder, so precision is lost
   • A left shift overflows if a 1 falls off the left
   Model answer

   A binary shift moves every bit a number of places left or right. A left shift multiplies by 2 for each place, with zeros coming in on the right; a right shift divides by 2 for each place, with zeros coming in on the left. Bits shifted out of the end are discarded, not wrapped round. So a right shift of an odd number loses the remainder: 25 shifted right once gives 12, not 12.5, so precision is lost. And if a 1 falls off the left of a left shift, the answer is wrong: that is overflow.

2. • The full answer is 101000000
   • That is 320
   • It needs nine bits, so the carry out is lost: overflow
   • 01000000, which is 64, is stored: a wrong value
   Model answer

   Adding column by column from the right gives 101000000, which is 192 + 128 = 320. That needs nine bits, but the register only has eight, so the carry out of the most significant bit is lost: this is overflow. The value stored is 01000000, which is 64, a wrong answer, and no error is reported.

This or That

1. No: 1 + 1 is 0 carry 1 Binary only has 0 and 1.
2. Zeros The empty places are always filled with 0.
3. 12 6 doubled is 12.
4. 3 6 halved is 3.
5. The most significant bit The leftmost bit is the most significant.
6. It earns method marks Examiners can only credit a method they can see.
7. Are discarded They are gone, which is why shifts can lose information.
8. 12 There is nowhere to store the half.

Memory Match

• A 2 0 + 0: 0
• B 3 1 + 0: 1
• C 1 1 + 1: 0 carry 1
• D 6 1 + 1 + 1: 1 carry 1
• E 4 Left shift by 1: Multiply by 2
• F 5 Right shift by 1: Divide by 2, losing any remainder

Sort It

1. Left shift 00001011 becomes 00010110 11 becomes 22.
2. Right shift 00011001 becomes 00001100 25 becomes 12.
3. Overflow 192 + 128 stored in 8 bits gives 64 320 needs nine bits, so the stored value is wrong.
4. Right shift An odd number loses its remainder The bit that falls off the right is the remainder.
5. Overflow A carry out of the most significant bit is lost There is no bit left to hold it.
6. Left shift Zeros come in on the right A left shift fills the right-hand end with zeros.
7. Right shift Divides by 2 for each place Moving bits right halves the value.
8. Overflow A wrong value is stored with no error reported That is what makes overflow dangerous.
9. Overflow The result needs more bits than the register has That is the definition of overflow.
10. Right shift Zeros come in on the left A right shift fills the left-hand end with zeros.
11. Left shift Multiplies by 2 for each place Moving bits left doubles the value.