9.6.2: Converting from decimal to non-decimal bases. info About A number N is given below in decimal format. Compute the representation of N in the indicated base. (a) N = 217, binary. (b) N = 99, hex. (c) N = 344, hex. (d) N =136, base 7. (e) N = 542, base 5. (f) N = 727, base 8. (g) N = 171, hex. (h) N = 91, base 3. (i) N = 840, base 9.

Answer:

g

Step-by-step explanation:

a) A good method to convert a decimal number to binary is dividing it by 2 and using the remainder of the division as the converted number, starting by the most significant bit (the right one). We we can't divide anymore. So we have:

217/2 = 108 + 1

108/2 = 54 + 0

54/2 = 27 + 0

27/2 = 13 + 1

13/2 = 6 + 1

6/2 = 3 + 0

3/2 = 2 +1

2/2 = 1

The binary equivalent to 217 is 11011001

b) To convert a number from decimal to hex we can divide the number by 16, taking out the decimal part and multiplying it by 16 using that as our most significant number while using the result of the original division to continue our conversion. So we have:

99/16 = 6.1875

The decimal part is 0.1875, we multiply it by 16 and obtain 3 as our most significant number. Since we can't divide 6 by 16 we have that as our least significant number then the hexadecimal equivalent is 63.

c) We follow the same steps as in item b:

344/16 = 21.5

The most significant number is 0.5*16 = 8

21/16 = 1.3125

The next number is 0.3125*16 = 5

Since we can't divide it anymore we have our result wich is 158 in hex.

d) To convert from decimal to base 7 we'll use the same method as to hex, but this time dividing and multiplying by 7.

136/7 = 19.428571

The most significant number is 0.428571 * 7 = 3

19/7 = 2.71428571

The next number is 0.71428571*7 = 5

Since we can't divide it anymore we have our result wich is 253.

e) To convert from decimal to a base 5 we'll use the same method as before but dividing and multiplying by 5.

542/5 = 108.4

The most significant number is 0.4*5 = 2

108/5 = 21.6

The next number is 0.6*5 = 3

21/5 = 4.2

The next number is 0.2*5 = 1

Since we can't divide it anymore we have our result that is 4132.

f) To convert from decimal to a base 8 we'll use the same method as before but dividing and multiplying by 8.

727/8 = 90.875

The most significant number is 0.875*8 = 7

90/8 = 11.25

The next number is 0.25*8 = 2

11/8 = 1.375

The next number is 0.375*8 = 3

Since we can't divide anymore we have our result wich is 1327

g) Following the same steps as before:

171/16 = 10.6875

The most significant number is 0.6875*16 = 11

Since we can't divide anymore we have our result wich is 1011

h) Following the same steps as before:

91/3 = 30.333333333

The most significant number is 0.333333*3 = 1

30/3 = 10

The next number is 0

10/3 = 3.3333333333

The next number is 0.333333*3 = 1

3/3 = 1

Since we have the final value remainder as 0 the least significant number is 1

Since we can't divide anymore we have our result that is 10101.

i) Following the same steps as before:

840/9 = 93.333333

The most significant number is 0.33333*9 = 3

93/9 =  10.3333333

The next number is 0.333333*9 = 3

10/9 = 1.11111111

The next number is 0.11111111*9 = 1

Since we cant divide anymore we have our result that is 1133

Final answer:

To convert decimal numbers to non-decimal bases, divide the number by the base, note the remainders, and read the remainders in reverse order to get the converted number. This applies to binary, hexadecimal, and other bases like base 7 or base 5. The process continues until the quotient is zero.

Explanation:

The process of converting numbers from decimal to non-decimal bases involves division for integer parts and multiplication for fractional parts, using the target base until the dividends are zero or an acceptable level of fractional precision is reached. For whole numbers, divide the number by the new base, note the remainder, and continue dividing the quotient until it equals zero. The remainders noted, read in reverse order (from last to first), give the resulting non-decimal number.

To convert 217 to binary, you would start by dividing 217 by 2, the base for binary numbers, and noting the remainders. This division continues until the quotient is zero. Read the remainders backwards to get the binary representation.

For conversion to hexadecimal, which is base 16, you divide the number by 16, note the remainders, continue with the quotient, and read the remainders in reverse order to get the hexadecimal equivalent. Alphabetic characters are used for remainders of 10-15, representing 'A' to 'F'.

When converting to other bases like base 7 or base 5, you follow a similar division method but use the base you're converting to. For example, in base 7, you divide by 7 and use the remainders to construct the new base representation. Always read the remainders in reverse to obtain the correct number in the new base.

In cases like base 8 or base 9, the same principles apply. Dividing the number by 8 or 9, respectively, and noting the remainders until you reach a quotient of zero will give you the representation in the new base.

Learn more about Converting Decimal Numbers here:

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