Numeral system Numerals

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1 Book B: Chapter 9 Different Numeral Systems Revision. (a) Numerals in the system Numeral system Numerals Denary,,,,,, 6, 7, 8 and 9 Binary and Hexadecimal,,,,,, 6, 7, 8, 9, A (i.e. ), B (i.e. ), C (i.e. ), D (i.e. ), E (i.e. ) and F (i.e. ) 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.. 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. Converting a denary number into binary number (or hexadecimal number) Key step: Divide the denary number successively by (or ) until the quotient becomes zero. The remainders we get in the above process form the required binary number (or hexadecimal number). Example. Write the following expressions as binary numbers. (a) Solution: + + (a) = + + = + = =

2 Book B: Chapter 9 Different Numeral Systems. (a) Write down the place value of each digit in CAB and DA6.F. (Give your answers in index notation.) Hence, express CAB and DA6.F in the expanded form.. Solution: (a) For CAB: Digit C A B Place value For DA6.F: Digit D A 6 F Place value - - CAB DA6.F = = Write the following expressions as hexadecimal numbers. (a) Solution: (a) = = 9AC = FB.. Convert into a denary number.. Solution: = = = + + +

3 Book B: Chapter 9 Different Numeral Systems. Convert D9A8 into a denary number.. Solution: D9A8 = = 7 = Convert 9 into a binary number. 6. Solution: Dividing 9 successively by, we have 9 = 9 remainder Convert 8 into a hexadecimal number. 7. Solution: Dividing 8 successively by, we have 8 = F 8 remainder F 8. (a) Rearrange the digits,,, and 7 to form the largest and the smallest denary numbers without decimal places. Express each of the numbers obtained in (a) in the expanded form. 8. Solution: (a) Largest denary number: 7. Smallest denary number: = 7 =

4 Book B: Chapter 9 Different Numeral Systems 9. (a) Convert and AF into denary numbers. Hence, arrange, and AF in descending order. 9. Solution: (a) AF = = 89 = 7 6 = = + = AF > > Exercise Level. Express each of the following decimal numbers in the expanded form. (a) Express each of the following binary numbers in the expanded form. (a). Express each of the following hexadecimal numbers in the expanded form. (a) 6 A8E. Express the following numbers in the expanded form. (a) 6 (c) ABCD. Write down the place value of the digit in each of the following numbers. (Give your answers in index notation.) (a) 9 (c) AF 6. Write each of the following expressions as a denary number. (a) (c) Write each of the following expressions as a binary number. (a) Write each of the following expressions as a binary number. (a)

5 Book B: Chapter 9 Different Numeral Systems 9. Write each of the following expressions as a hexadecimal number. (a) Write each of the following expressions as a hexadecimal number. (a) Convert the following binary numbers into denary numbers. (a) (c). Convert the following hexadecimal numbers into denary numbers. (a) 7 F (c) CD. Convert the following numbers into denary numbers. (a). 9A8B. Convert the following numbers into denary numbers. (a) CD Level. Write down the place value of every underlined digit in each of the following numbers. (Give your answers in index notation.) (a) (c) BE6.F9. Express each of the following numbers in the expanded form. (a) 7.6. (c) 7DE.9C. Write each of the following expressions as a denary number. (a) Write each of the following expressions as a binary number. (a)

6 Book B: Chapter 9 Different Numeral Systems. Write each of the following expressions as a hexadecimal number. (a) Convert the following denary numbers into binary numbers. (a) 9 6 (c) 7. Convert the following numbers into binary numbers. (a) DC 8. Convert the following denary numbers into hexadecimal numbers. (a) 7 98 (c) Convert the following numbers into hexadecimal numbers. (a) 7. Convert into a (a) binary number, hexadecimal number.. Find the difference between the values represented by the two s in each of the following numbers. (Express your answers as denary numbers.) (a) 8 (c) B8.. Arrange the following numbers in descending order. 8,, AF. Arrange the following numbers in ascending order.,, FD. (a) Convert 9 and into binary numbers. Using the result of (a), arrange 9, and in ascending order.. (a) Convert 7 and AC into binary numbers. Using the result of (a), arrange, 7 and AC in descending order. 6