1. What is the significance of the use of the word "digit" to represent a numerical quantity?

 

2. Which radices are "natural" (i.e., they correspond to some aspect of human anatomy)? What is their anatomical tie-in?

 

3. What is the principal advantage of the (Zoroastrian) sexagesimal (base 60) number system?

 

4. How long ago did the concept of zero originate?

 

5. (a) How many different numbers (or different anythings) can be represented by n bits?

(b) How many different items are representable with 24 bits? 32? 48?

(c) How could you rapidly determine how many different items are representable with any given number of bits that you might be asked about?

 

6. How should you go about converting a number from decimal representation to another radix?

 

7. Sample Problems: Convert each of the following decimal numbers to the radices indicated:

1. 2,346.1359 to radix 3, radix 9, and radix 16.
2. 7,924.9483 to radix 2, radix 5, and radix 11.
3. 1,781.1953 to radix 7, radix 8, and radix 13.

1. How should you go about converting from other radices to decimal?

 

2. Sample Problems: Convert each of the following numbers to decimal:

1. 111001010111.000101110111₂
2. 123.321₄
3. 123.4₇
4. 7123.4567₈
5. 123.4₉
6. 123.4₁₁
7. E56.D78₁₆

1. How should you interconvert among binary, octal, and hex?

 

2. Sample Problems: Each of the following numbers is written either in binary or in octal or in hexadecimal. Convert it to the other two radices.

1. 10101010101.0101010101₂
2. 10111011101.1101110111₂
3. 10101.0101₈
4. 11101.11011₈
5. 101.01₁₆
6. 1101.11₁₆
7. 12345.67124₈
8. 12345.67124₁₆
9. DEAD.ACE₁₆
10. BAD.BEEF₁₆

Question 2:

Question 3: Radix is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.

Question 4: About 2,300 years ago in primitive form, and about 1400 years ago in its full-blown form.

Question 5a: 2ⁿ

Question 5b: 16M 4G 256T

1. Deal separately with the integer part and the functional part.
2. Integer part: Divide successively by the new radix. Each division produces a quotient and a remainder. The remainder from the n^th division is the n^th significant digit, starting from the least significant digit immediately to the left of the radix point. N.B.: If the remainder is greater than 9, be sure to convert it to a single digit appropriate to the new radix.
3. Fractional part: Multiply successively by the new radix. Each successive multiplication produces a new product with an integer part and a fractional part. The integer part from the n^th multiplication provides the n^th significant fractional digit, starting from the radix point; the fractional part only is used for the next multiplication. N.B.: If the integer part of any multiplication is greater than 9, be sure to convert it to a single digit in the new radix.
4. Stopping point for the fractional part:

1. There is nothing more left to multiply.
2. A repeating group is recognized (be sure to indicate clearly in your answer both that there is a repeating group and what it is).
3. The required level of precision is reached. For CS-350, the required level of precision is three hex digits, four octal digits, or twelve bits; six digits in any other radix.

Question 7a: 2346.1359

 Answer = 10012220.0102000012+₃

 Answer = 3186.120056743+₉

 Answer = 92A.22C+₁₆

Question 7b: 7924.9483

Answer = 1111011110100.111100101101+₂ 

 Answer = 223144.43323320432+₅

Answer = 5A54.A4822₁₁

Question 7c: 1781.1953

 Answer = 5123.12366256₇

 Answer = 3365.143774+₈

 Answer = A70.2700C6+₁₃

1. 3671.091552646875
2. 27.890625
3. 66.731428+
4. 3667.591554734375894.444. . .
5. 146.3636. . .
6. 3670.841796875000

Partial Answer to 10: Always outwards from the radix point, and add zeroes when needed to the most significant end of the integer part a d to the least significant end of the fractional part. NEVER add a zero between the least significant integer digit and the radix point, or between the most significant fractional digit and the radix point..

1. 2525.2524₈ = 555.554₁₆
2. 2735.6734₈ = 5DD.DDC₁₆
3. 001 000 001 000 001.000 001 000 001₂ = 1241.2412₁₆
4. 001 001 001 000 001.001 001 000 001 001₂ = 1241.2412₁₆
5. 0001 0000 0001.0000 0001₂ = 401.002₈
6. 0001 0001 0000 0001.0001 0001₂ = 10401.042₈
7. 001 010 011 100 101.110 111 001 010 100₂ = 14E5.DCA8₁₆
8. 0001 0010 0011 0100 0101.0110 0111 0001 0010 0100₂ = 221505.316111₈
9. 1101 1110 1010 1101.1010 1100 1110₂ = 157255.5316₈
10. 1011 1010 1101.1011 1110 1110 1111₂ = 5655.575674₈

Last updated 03 Jan 2000