Problem 9 - A Gentle Introduction to the Art of Mathematics

Question

Try the following conversions between various number systems: (a) Convert 30 (base 10 ) to binary. (b) Convert 69 (base 10 ) to base 5 . (c) Convert \(1222_{3}\) to binary. (d) Convert \(1234_{7}\) to base 10 . (e) Convert \(E E E D_{15}\) to base 12. (Use \(\\{1,2,3 \ldots 9, d, e\\}\) as the digits in base \(12 .)\) (f) Convert \(678_{9}\) to hexadecimal.

Step-by-Step Solution

Answer

30 (base 10) = 11110 (binary); 69 (base 10) = 234 (base 5); 1222 (base 3) = 110101 (binary); 1234 (base 7) = 466 (base 10); EEEE (base 15) = 25368 (base 12); 678 (base 9) = 22D (hexadecimal).

Step 1: Convert 30 (base 10) to binary

Divide 30 by 2, keep track of the quotient and the remainder. Continue dividing the quotient by 2 until you get a quotient of 0. The binary number is formed by the remainders, read from bottom to top.30 ÷ 2 = 15 remainder 015 ÷ 2 = 7 remainder 17 ÷ 2 = 3 remainder 13 ÷ 2 = 1 remainder 11 ÷ 2 = 0 remainder 1Thus, 30 (base 10) = 11110 (binary).

Step 2: Convert 69 (base 10) to base 5

Divide 69 by 5, keep track of the quotient and the remainder. Continue dividing the quotient by 5 until you get a quotient of 0. Form the base 5 number by reading the remainders from bottom to top.69 ÷ 5 = 13 remainder 413 ÷ 5 = 2 remainder 32 ÷ 5 = 0 remainder 2Thus, 69 (base 10) = 234 (base 5).

Step 3: Convert 1222 (base 3) to binary

First convert the base 3 number to base 10. Then convert the base 10 result to binary.For base 3 to base 10: 1*(3^3) + 2*(3^2) + 2*(3^1) + 2*(3^0)= 27 + 18 + 6 + 2 = 53 (base 10)Then convert 53 (base 10) to binary:53 ÷ 2 = 26 remainder 126 ÷ 2 = 13 remainder 013 ÷ 2 = 6 remainder 16 ÷ 2 = 3 remainder 03 ÷ 2 = 1 remainder 11 ÷ 2 = 0 remainder 1Thus, 1222 (base 3) = 110101 (binary).

Step 4: Convert 1234 (base 7) to base 10

Convert the base 7 number to base 10 using positional values.1*(7^3) + 2*(7^2) + 3*(7^1) + 4*(7^0)= 343 + 98 + 21 + 4 = 466 (base 10).Thus, 1234 (base 7) = 466 (base 10).

Step 5: Convert EEEE (base 15) to base 12

First, convert EEEE (base 15) to base 10, then convert from base 10 to base 12.For base 15 to base 10: E = 14, so EEEE = 14*(15^3) + 14*(15^2) + 14*(15^1) + 14*(15^0)= 14*3375 + 14*225 + 14*15 + 14*1 = 47250 + 3150 + 210 + 14 = 50624 (base 10)Then convert 50624 (base 10) to base 12:50624 ÷ 12 = 4218 remainder 84218 ÷ 12 = 351 remainder 6351 ÷ 12 = 29 remainder 329 ÷ 12 = 2 remainder 52 ÷ 12 = 0 remainder 2Thus, EEEE (base 15) = 25368 (base 12).

Step 6: Convert 678 (base 9) to hexadecimal

First convert the base 9 number to base 10, then convert from base 10 to hexadecimal.For base 9 to base 10: 6*(9^2) + 7*(9^1) + 8*(9^0)= 486 + 63 + 8 = 557 (base 10)Then convert 557 (base 10) to hexadecimal:557 ÷ 16 = 34 remainder 13 (D)34 ÷ 16 = 2 remainder 22 ÷ 16 = 0 remainder 2Thus, 678 (base 9) = 22D (hexadecimal).

Key Concepts

Base 10 to Binary Conversion

Converting a number from base 10 (decimal) to binary (base 2) involves repeatedly dividing the number by 2 and recording the remainders. This sequence of remainders, when read in reverse (from bottom to top), forms the binary equivalent.
For example, to convert 30 (base 10) to binary:

• 30 ÷ 2 = 15 remainder 0
• 15 ÷ 2 = 7 remainder 1
• 7 ÷ 2 = 3 remainder 1
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Thus, the binary representation of 30 is 11110.
This method works because each division by 2 breaks the number into its binary components.

Base 10 to Base 5 Conversion

To convert a base 10 (decimal) number to base 5, repeatedly divide the number by 5, and keep track of the remainders. The converted number is found by reading these remainders from bottom to top.
For instance, converting 69 (base 10) to base 5:

• 69 ÷ 5 = 13 remainder 4
• 13 ÷ 5 = 2 remainder 3
• 2 ÷ 5 = 0 remainder 2

Therefore, 69 (base 10) is equivalent to 234 in base 5.
This systematic division helps break the number down into its base 5 components.

Base 3 to Binary Conversion

Converting from base 3 to binary involves a two-step process. First, convert the base 3 number to base 10, then transform the resulting base 10 number to binary.
For example, to convert 1222 (base 3) to binary:

• Convert to base 10: \(1*(3^3) + 2*(3^2) + 2*(3^1) + 2*(3^0) = 27 + 18 + 6 + 2 = 53\)
• Convert 53 (base 10) to binary:
  • 53 ÷ 2 = 26 remainder 1
  • 26 ÷ 2 = 13 remainder 0
  • 13 ÷ 2 = 6 remainder 1
  • 6 ÷ 2 = 3 remainder 0
  • 3 ÷ 2 = 1 remainder 1
  • 1 ÷ 2 = 0 remainder 1

Thus, 1222 (base 3) becomes 110101 in binary.
This dual-step method ensures accurate conversion by leveraging base 10 as an intermediary.

Base 7 to Base 10 Conversion

Converting from base 7 to base 10 uses the positional value method. Each digit in the base 7 number is multiplied by 7 raised to the power of its positional index, starting from 0.
For example, to convert 1234 (base 7) to base 10:

• 1*(7^3) + 2*(7^2) + 3*(7^1) + 4*(7^0) = 343 + 98 + 21 + 4 = 466

Hence, 1234 in base 7 is equal to 466 in base 10.
This method ensures each digit's proper contribution based on its position in the number.

Base 15 to Base 12 Conversion

To convert a number from base 15 to base 12, first convert it to base 10, then from base 10 to base 12.
Let’s convert EEEE (base 15) to base 12:

• Convert to base 10: E = 14, so \(14*(15^3) + 14*(15^2) + 14*(15^1) + 14*(15^0) = 50624\)
• Convert 50624 (base 10) to base 12:
  • 50624 ÷ 12 = 4218 remainder 8
  • 4218 ÷ 12 = 351 remainder 6
  • 351 ÷ 12 = 29 remainder 3
  • 29 ÷ 12 = 2 remainder 5
  • 2 ÷ 12 = 0 remainder 2

Thus, EEEE (base 15) is 25368 in base 12.
This two-step method ensures precise conversion between these exotic bases.

Base 9 to Hexadecimal Conversion

Converting from base 9 to hexadecimal involves initially converting to base 10, followed by converting the base 10 result to hexadecimal.
For example, converting 678 (base 9) to hexadecimal:

• Convert to base 10: \(6*(9^2) + 7*(9^1) + 8*(9^0) = 486 + 63 + 8 = 557\)
• Convert 557 (base 10) to hexadecimal:
  • 557 ÷ 16 = 34 remainder 13 (D)
  • 34 ÷ 16 = 2 remainder 2
  • 2 ÷ 16 = 0 remainder 2

Therefore, 678 (base 9) converts to 22D in hexadecimal.
This method leverages the intermediate base 10 to facilitate the final conversion to hexadecimal.