Book 2B: Chapter 9 – Different Numeral Systems
Revision 1. (a) Numerals in the system Numeral system Numerals Denary 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 Binary 0 and 1 Hexadecimal 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (i.e. 10), B (i.e. 11), C (i.e. 12), D (i.e. 13), E (i.e. 14) and F (i.e. 15) (b) In a number, the position of each digit has a fixed place value. The value of a number can be expressed in an expanded form.
2. Conversions between denary system and other numeral systems (a) Converting a binary number (or hexadecimal number) into a denary number Key step: Express the number in the expanded form. The value of the expression obtained gives the required denary number. (b) Converting a denary number into binary number (or hexadecimal number) Key step: Divide the denary number successively by 2 (or 16) until the quotient becomes zero. The remainders we get in the above process form the required binary number (or hexadecimal number).
Example 1. Write the following expressions as binary numbers. 1 (a) 25 + 23 + 2 + 1 (b) 1× 4 +1×1+1× 2 1. Solution:
(a) 25 + 23 + 2 +1 = 1× 25 + 0 × 24 +1× 23 + 0 × 22 +1× 21 +1× 20 = 101011 2
1 − (b) 1× 4 +1×1+1× = 1× 22 + 0× 21 +1× 20 +1× 2 1 2 = 101 1. 2
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Book 2B: Chapter 9 – Different Numeral Systems
2. (a) Write down the place value of each digit in CAB 16 and 5DA6.F3 16 . (Give your answers in index notation.)
(b) Hence, express CAB 16 and 5DA6.F3 16 in the expanded form. 2. Solution:
(a) For CAB 16 : Digit C A B Place value 16 2 16 1 16 0
For 5DA6.F3 16 : Digit 5 D A 6 F 3 Place value 16 3 16 2 16 1 16 0 16 -1 16 -2
(b) CAB = 12 ×16 2 +10 ×16 1 +11 ×16 0 16 = × 3 + × 2 + × 1 + × 0 + × −1 + × −2 5DA6.F3 16 5 16 13 16 10 16 6 16 15 16 3 16
3. Write the following expressions as hexadecimal numbers. (a) 9 × 16 5 + 10 × 16 4 + 12 × 16 3 + 16 (b) 15 × 16 2 + 11 × 16 1 + 5 × 16 0 + 1 × 16 −1 3. Solution: (a) 9× 165 +× 10 16 4 +× 12 16 3 + 16 =×9165 +× 1016 4 +× 1216 3210 +× 016 +× 116 +× 016 = 9AC010 16
× 2 + × 1 + × 0 + × −1 = (b) 15 16 11 16 5 16 1 16 FB 1.5 16
4. Convert 110101 2 into a denary number. 4. Solution:
= × 5 + × 4 + × 3 + × 2 + × 1 + × 0 110101 2 1 2 1 2 0 2 1 2 0 2 1 2 = 32 +16 + 0 + 4 + 0 +1 = 53 10
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Book 2B: Chapter 9 – Different Numeral Systems
5. Convert D9A8 16 into a denary number. 5. Solution:
= × 3 + × 2 + × 1 + × 0 D9A8 16 13 16 9 16 10 16 8 16 = 53 248 + 2304 +160 + 8 = 720 55 720 10
6. Convert 59 10 into a binary number. 6. Solution: Dividing 59 successively by 2, we have = 59 10 111011 2
2 59 remainder 2 29 ……1 2 14 ……1 2 7 ……0 2 3 ……1 2 1 ……1 0 ……1
7. Convert 815 10 into a hexadecimal number. 7. Solution: Dividing 815 successively by 16, we have = 815 10 32 F16
16 815 remainder 16 50 ……F 16 3 ……2 0 ……3
8. (a) Rearrange the digits 0, 0, 3, 5 and 7 to form the largest and the smallest denary numbers without decimal places. (b) Express each of the numbers obtained in (a) in the expanded form. 8. Solution:
(a) Largest denary number: 75 300 10 .
Smallest denary number: 30 057 10 .
(b) 75 300 = 7 ×10 4 + 5×10 3 + 3×10 2 + 0 ×10 1 + 0×10 0 10 = × 4 + × 3 + × 2 + × 1 + × 0 057 30 057 10 3 10 0 10 0 10 5 10 7 10
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Book 2B: Chapter 9 – Different Numeral Systems
9. (a) Convert 1011001 2 and AF 16 into denary numbers.
(b) Hence, arrange 1011001 2, 123 10 and AF 16 in descending order. 9. Solution: = × 6 + × 5 + × 4 + × 3 + × 2 + × 1 + × 0 (a) 1011001 2 1 2 0 2 1 2 1 2 0 2 0 2 1 2 = 64 + 0 +16 + 8 + 0 + 0 +1 = 89 10
= × 1 + × 0 AF 16 10 16 15 16 = 160 +15 = 175 10
> > (b) AF 16 123 10 1011001 2
Exercise Level 1 1. Express each of the following decimal numbers in the expanded form.
(a) 8723 10 (b) 20 694 10
2. Express each of the following binary numbers in the expanded form.
(a) 101 2 (b) 11101 2
3. Express each of the following hexadecimal numbers in the expanded form.
(a) 326 16 (b) 3A8E 16
4. Express the following numbers in the expanded form. (a) 2006 10 (b) 1001 2 (c) ABCD 16
5. Write down the place value of the digit ‘0’ in each of the following numbers. (Give your answers in index notation.) (a) 3049 10 (b) 1101 2 (c) AF0 16
6. Write each of the following expressions as a denary number. (a) 4 × 1000 + 3 × 100 + 9 × 10 + 2 (b) 5 × 10 000 + 1 × 1000 + 8 × 100 + 6 (c) 2 × 10 4 + 10 2 + 3 × 10 1 + 5
7. Write each of the following expressions as a binary number. (a) 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 (b) 1 × 24 + 1 × 22 + 1 × 21
8. Write each of the following expressions as a binary number. (a) 1 × 23 + 1 × 22 + 1 × 25 + 1 (b) 1 × 32 + 1 × 16 + 1 × 2 + 1 4
Book 2B: Chapter 9 – Different Numeral Systems
9. Write each of the following expressions as a hexadecimal number. (a) 14 × 16 2 + 5 × 16 1 + 15 × 16 0 (b) 2 × 16 4 + 6 × 16 3 + 9 × 16 2 + 12 × 16 1 + 13
10. Write each of the following expressions as a hexadecimal number. (a) 7 × 16 5 + 11 × 16 3 + 4 × 16 2 (b) 10 × 16 2 + 3 × 16 4 + 8
11. Convert the following binary numbers into denary numbers. (a) 111 2 (b) 1101 2 (c) 11010 2
12. Convert the following hexadecimal numbers into denary numbers. (a) 74 16 (b) F2 16 (c) 3CD 16
13. Convert the following numbers into denary numbers. (a) 110.01 2 (b) 9A8B 16
14. Convert the following numbers into denary numbers.
(a) 1102 (b) CD 16
Level 2 1. Write down the place value of every underlined digit in each of the following numbers. (Give your answers in index notation.)
(a) 30 269.5710 (b) 10111.0012 (c) BE056.F4916
2. Express each of the following numbers in the expanded form.
(a) 17.061 10 (b) 11011.01 2 (c) 7DE5.9C 16
3. Write each of the following expressions as a denary number. (a) 8 × 10 3 + 7 × 10 1 + 2 × 10 0 + 1 × 10 –2 1 1 (b) 1 × 10 000 + 2 × 1 + 3 × + 5 × 10 100
4. Write each of the following expressions as a binary number. (a) 1 × 24 + 1 × 2–1 + 1 × 2–2 + 1 × 2–3 1 (b) 1 × 8 + 1 × 4 + 1 × 1 + 1 × 4
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Book 2B: Chapter 9 – Different Numeral Systems
5. Write each of the following expressions as a hexadecimal number. (a) 10 × 16 3 + 1 × 16 –1 + 8 × 16 0 + 15 × 16 2 1 1 (b) 7 × 16 2 + 5 × 16 + 10 + 11 × + 14 × 16 16 2
6. Convert the following denary numbers into binary numbers.
(a) 19 10 (b) 26 10 (c) 40 10
7. Convert the following numbers into binary numbers.
(a) 111 10 (b) DC 16
8. Convert the following denary numbers into hexadecimal numbers.
(a) 1754 10 (b) 9805 10 (c) 38 888 10
9. Convert the following numbers into hexadecimal numbers.
(a) 2007 10 (b) 101101 2
10. Convert 150 10 into a (a) binary number, (b) hexadecimal number.
11. Find the difference between the values represented by the two ‘1’s in each of the following numbers. (Express your answers as denary numbers.)
(a) 18 513 10 (b) 100010 2 (c) 1B8.1 16
12. Arrange the following numbers in descending order.
180 10 , 10111110 2, AF 16
13. Arrange the following numbers in ascending order.
11111111 2, 235 10 , F00D16
14. (a) Convert 3910 and 22 16 into binary numbers.
(b) Using the result of (a), arrange 3910 , 101101 2 and 22 16 in ascending order.
15. (a) Convert 170 10 and AC 16 into binary numbers.
(b) Using the result of (a), arrange 10101011 2, 170 10 and AC 16 in descending order.
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