# Exploring Number System

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1 Exploring Number System Mun Chou, Fong QED Education Scientific Malaysia LEVEL High schools after students have been introduced to concepts of number systems. OBJECTIVES To explore converting between binary, octal, decimal and hexadecimal number systems by hand and also by using the Base-N mode. OVERVIEW Alternative to the decimal number system such as the binary has proven to be vitally important in computer sciences. We explore this binary system together with the octal (base-8) number system and the hexadecimal number system (base-16) as they are natural conversions from binary and are just as important in computer sciences. The decimal system is the widely used number system and as such we shall explore a lot on the relation between the three aforementioned systems with the decimal system. EXPLORATORY ACTIVITIES [Note] (a) Unless otherwise specified, we choose MthIO mode in the SETUP menu by tapping (b) The system each number represented will be described with a subscript of 2, 8, and 16 for binary, octal and hexadecimal respectively; except for number in base-10, which will be written just as it is. Binary, Octal and Hexadecimal Systems The decimal number system (base-10) is a position-value system which uses the numeral symbols of 0, 1, 2 9. In similar fashion, the binary uses only 1 s and 0 s for representation of number. For example the number 55 in decimal system will be represented as and can be easily illustrated using the binary table as follow: 2005 CASIO Computer Co., LTD. All Right Reserved. Page A

2 In power Of Input in 1 and So based on the table above, if we are to convert into decimal, the calculation is = = 55. On the calculator this can be done fairly easy in the Base-N mode. Each time you enter Base-N the calculator by default is in the Dec, or decimal system. Switch to the Bin, or binary system Enter the number Register into the calculator Switch to Dec to convert The calculator displays 55 verifying our hand calculation. How about expressing in octal (base-8) or hexadecimal (base-16)? Switch to Oct for conversion Or switch to Hex for base-16 Both these outputs of 67 8 and can be easily illustrated with the base tables. For example the octal representation of or 55: In power Of Input from 0 till = = And to explain the hexadecimal representation of or 55: In power Of = = Input from 0 till Hexadecimal in Focus Let s explore further on the hexadecimal (base-16) system. We begin by entering 10 in the Dec mode CASIO Computer Co., LTD. All Right Reserved. Page A

3 Enter Base-N/Dec mode Input 10 and register it Now tap once on to switch to the base-16 mode. Do you notice the output is A? The reason of this is for avoiding confusion when using the base-16 system. Since we use 0 till 15 in hexadecimal for representation, we may get confused over and 10, therefore 10 is represented as A in hexadecimal. How about the representation for 11, 12, 13, 14 and 15? Key these numbers into Dec mode and then switch to Hex mode to find out. The actual representation is as below. The digit In Hexadecimal A B C D E F Conversion from Base-10 to the Other 3 Systems As shown above if we want to do a conversion from the other three systems to decimal we can use the base table. Converting from decimal to the other systems is slightly more challenging. Suppose we are to express 13 in binary, an effective method is the repeated-division technique: 13 2 = 6 remainder is = 3 remainder is = 1 remainder is 1 The division stops when the quotient is < 2, then we read the binary conversion following the curved arrows, i.e. 13 = Similarly, conversion from decimal to octal or hexadecimal is along the same line too. For example to express 742 in hexadecimal: = 46 remainder is = 2 remainder is 14 We stop when the quotient is < 16; in this case it is 2. Reading in direction of the curved arrows, 742 is expressed as 2(14)6 16, but recalling the table just now, 14 is written as E in hexadecimal, therefore we have 742 = 2E6 16. The repeated-division algorithm could take a while if we convert a larger number such as 175. The calculator can be used to process the conversion CASIO Computer Co., LTD. All Right Reserved. Page A

4 While in Dec mode, input 742 Then switch to Hex mode And the result of the conversion is displayed immediately. Conversion between the 3 Systems Conversion between the binary, octal and hexadecimal is much easier as the octal or the hexadecimal system is naturally closer to the binary than decimal. Consider the number Since the octal system has power index of 3 (2 3 = 8) as relative to the binary system, conversion can be performed as follow: Partitions into bits of 3 starting from right: / 010 / Then convert each part to its decimal equivalent: In decimal equivalent Reading the numbers in the direction of the shaded arrow, we have = To convert into hexadecimal, we partition the binary number to bits of 4 from the right, as hexadecimal system has power index of 4 (2 4 = 16) relative to the binary system: In decimal equivalent 13 5 Therefore = (13)5 16 or more appropriately, D5 16. Use the calculator to verify your conversion for all the above. Switch to Bin mode Now input the binary number Now switch to Hex or Oct mode It seems that the calculator outputs verify the manual calculation. Conversion from octal or hexadecimal into binary is the reverse of the above algorithm. Consider the number 1A5D 16. Again using the partition technique: Split all the digits in 1A5D 16 into 4 parts (because it has 4 digits): 1A5D A D CASIO Computer Co., LTD. All Right Reserved. Page A

5 Then convert each digit to its 4-bit binary equivalent: 1 16 A D 16 In 4-bit binary equivalent And in the direction of the shaded arrow, we have 1A5D 2 = or more appropriately, 1A5D 2 = , excluding the zeroes at front. Again use the calculator to verify the conversion for all the above. Switch to Hex mode, input 1A5D 16 Switch to Bin mode for result The output verifies that 1A5D 2 is equivalent to EXERCISES These exercises should enhance your understanding on the subject matter of number systems while using the calculator to explore further on conversion between different number systems. Calculate each one manually before verifying the solution with 991-ES. Exercise 1 Use the base table to express each of the following in decimal, then verify with the Base-N mode. (i) (ii) (iii) (iv) E3AC 16 Exercise 2 Express 167 in (i) binary, (ii) Octal, and finally (iii) Hexadecimal form. Exercise 3 Express in base-8 and base-16 using the 3-bit and 4 bit conversions. Exercise 4 Express (i) 37A 16 and (ii) in binary. Exercise 5 The number system of base-5 should observe all the techniques and properties we discussed in the exploration also. Referring to the number system of base-5: (i) Express 389 in base-5; (ii) Convert into decimal and binary. Hint: You might not able to use the calculator for verification. SOLUTIONS to Exercises Exercise 1 (i) 45 (ii) 175 (iii) 41 (iv) CASIO Computer Co., LTD. All Right Reserved. Page A

6 Exercise 2 (i) (ii) (iii) A7 16 Exercise 3 174D 16 and Exercise 4 (i) (ii) Exercise 5 (i) 389 = (ii) 1482 and CASIO Computer Co., LTD. All Right Reserved. Page A

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