Question

How many ones are there in 891 if it is a number in each of the following bases? a. Base 10 b. Base 8 c. Base 12 d. Base 13 e. Base 16

Short answer

One '1' in Base 10; zero '1's in Bases 8, 12, 13, 16.

Step-by-step solution

Understanding Base 10

In base 10, the number 891 is simply the numeral itself since it is already given in base 10. We identify how many 1's are in the number '891'. The numeral '891' contains one '1'.

Conversion to Base 8

To convert 891 to base 8, repeatedly divide 891 by 8 and record the remainders. Dividing gives:\( 891 \div 8 = 111 \text{ remainder } 3 \) \\( 111 \div 8 = 13 \text{ remainder } 7 \) \\( 13 \div 8 = 1 \text{ remainder } 5 \) \\( 1 \div 8 = 0 \text{ remainder } 1 \) \Reading the remainders from the last to first, the base 8 representation of 891 is 1573. It contains no '1's.

Conversion to Base 12

Convert 891 to base 12 by dividing by 12 repeatedly and noting remainders:\( 891 \div 12 = 74 \text{ remainder } 3 \) \\( 74 \div 12 = 6 \text{ remainder } 2 \) \\( 6 \div 12 = 0 \text{ remainder } 6 \) \Thus, the base 12 number is 623, which contains no '1's.

Conversion to Base 13

For base 13 conversion, divide by 13 until the quotient is zero:\( 891 \div 13 = 68 \text{ remainder } 5 \) \\( 68 \div 13 = 5 \text{ remainder } 3 \) \\( 5 \div 13 = 0 \text{ remainder } 5 \) \The base 13 number is 535, which also contains no '1's.

Conversion to Base 16

Convert 891 to base 16 by dividing by 16:\( 891 \div 16 = 55 \text{ remainder } 11 \) \\( 55 \div 16 = 3 \text{ remainder } 7 \) \\( 3 \div 16 = 0 \text{ remainder } 3 \) \The base 16 representation is 37B; since 'B' represents 11, there are no '1's.

Key concepts

Base 10

Base 10, also known as the decimal system, is the most common numeral system used in everyday life. It comprises ten symbols: 0 through 9. Each position in a base 10 number represents a power of 10, with the rightmost position representing 10⁰, the next one 10¹, and so on. In the number 891:

• The 8 is in the hundreds place, representing 8 x 100.
• The 9 is in the tens place, representing 9 x 10.
• The 1 is in the units place, representing 1 x 1.

This is why the numeral is read as "eight hundred ninety-one." In this number, there is one '1'. Base 10 is straightforward because it's our standard counting system.

Base 8

Base 8, or the octal system, uses eight symbols: 0 to 7. It is widely used in computer applications. To convert from base 10 to base 8, you must divide the number by 8 and keep track of the remainders. When converting 891 to base 8:

• First divide 891 by 8, giving a quotient of 111 and a remainder of 3.
• Next, divide 111 by 8, giving a quotient of 13 and a remainder of 7.
• Finally, divide 13 by 8, yielding a quotient of 1 and remainder of 5.
• Lastly, divide 1 by 8, leaving a final remainder of 1.

By reading the remainders backward, 891 in base 8 is 1573. There are no '1's in this conversion.

Base 12

Base 12, or the duodecimal system, might seem unusual but offers advantages in division. It uses digits 0 through 9 and two additional symbols, often 'A' and 'B', to represent 10 and 11, respectively. To convert a number from base 10 to base 12:

• Divide 891 by 12, resulting in a quotient of 74 and remainder of 3.
• Then, divide 74 by 12, giving a quotient of 6 and a remainder of 2.
• Finally, divide 6 by 12, resulting in a quotient of 0 and remainder of 6.

Aligning remainders backward, 891 converts to 623 in base 12, which contains no '1's.

Base 13

Base 13 is less common, and like other bases above 10, it uses additional symbols. Here, 'A' represents ten, 'B' eleven, and 'C' twelve. To convert 891 into base 13 involves:

• Dividing 891 by 13, yielding a quotient of 68 and remainder of 5.
• Then, dividing 68 by 13 to get a quotient of 5 and remainder of 3.
• Lastly, dividing 5 by 13, resulting in a quotient of 0 and remainder of 5.

Reading remainders from last to first provides us the base 13 number 535. This conversion also contains no '1's.

Base 16

Base 16, or hexadecimal, uses symbols 0 to 9 and 'A' to 'F' to represent ten to fifteen. It is particularly popular in computing. Converting 891 from base 10 to base 16:

• First, divide 891 by 16 to get a quotient of 55 and a remainder of 11, represented by 'B'.
• Next, dividing 55 by 16 gives a quotient of 3 and remainder of 7.
• Finally, divide 3 by 16, resulting in a remainder of 3.

The sequence of remainders turned around reads as 37B in base 16, and there are no '1's.