When we talk about numerals in base n, we’re diving into a fascinating world of number systems beyond the familiar decimal (base 10) system. Let’s break it down step by step to understand what it means and how it works.
Understanding Number Systems
Decimal System (Base 10)
The decimal system is the most commonly used number system. It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each position in a number represents a power of 10. For example, in the number 345:
$345 = 3 times 10^2 + 4 times 10^1 + 5 times 10^0$
Binary System (Base 2)
The binary system is used in computers and digital systems. It uses only two digits: 0 and 1. Each position represents a power of 2. For example, in the binary number 1011:
$1011_2 = 1 times 2^3 + 0 times 2^2 + 1 times 2^1 + 1 times 2^0 = 8 + 0 + 2 + 1 = 11$
Other Bases
Numerals in base n follow the same principle but use n digits. For instance, in base 5, we use the digits 0, 1, 2, 3, and 4. Each position in a number represents a power of 5. For example, in the base 5 number 243:
$243_5 = 2 times 5^2 + 4 times 5^1 + 3 times 5^0 = 2 times 25 + 4 times 5 + 3 = 50 + 20 + 3 = 73$
Converting Between Bases
Converting from Base n to Decimal
To convert a number from base n to decimal, you multiply each digit by the corresponding power of n and sum the results. For example, let’s convert the base 7 number 321 to decimal:
$321_7 = 3 times 7^2 + 2 times 7^1 + 1 times 7^0 = 3 times 49 + 2 times 7 + 1 = 147 + 14 + 1 = 162$
Converting from Decimal to Base n
To convert a decimal number to base n, you repeatedly divide the number by n and record the remainders. These remainders, read in reverse, give you the number in base n. Let’s convert the decimal number 162 to base 7:
- $162 text{ divided by } 7 text{ gives quotient } 23 text{ and remainder } 1$
- $23 text{ divided by } 7 text{ gives quotient } 3 text{ and remainder } 2$
- $3 text{ divided by } 7 text{ gives quotient } 0 text{ and remainder } 3$
Reading the remainders from bottom to top, we get $321_7$
Practical Applications
Computing
The binary system (base 2) is fundamental in computing. Each bit in a computer’s memory is a binary digit, representing either 0 or 1. This simplicity allows for efficient data processing and storage.
Timekeeping
The hexadecimal system (base 16) is often used in computing for its compact representation of binary data. It uses sixteen digits: 0-9 and A-F. For example, the binary number 1111 1111 can be represented as FF in hexadecimal.
Everyday Use
While we mostly use the decimal system in everyday life, understanding other bases can be useful. For example, the sexagesimal system (base 60) is used in timekeeping (60 seconds in a minute, 60 minutes in an hour).
Examples and Exercises
Example 1: Converting Base 8 to Decimal
Convert the base 8 number 157 to decimal:
$157_8 = 1 times 8^2 + 5 times 8^1 + 7 times 8^0 = 1 times 64 + 5 times 8 + 7 = 64 + 40 + 7 = 111$
Example 2: Converting Decimal to Base 3
Convert the decimal number 45 to base 3:
- $45 text{ divided by } 3 text{ gives quotient } 15 text{ and remainder } 0$
- $15 text{ divided by } 3 text{ gives quotient } 5 text{ and remainder } 0$
- $5 text{ divided by } 3 text{ gives quotient } 1 text{ and remainder } 2$
- $1 text{ divided by } 3 text{ gives quotient } 0 text{ and remainder } 1$
Reading the remainders from bottom to top, we get $1200_3$
Exercise 1
Convert the base 6 number 254 to decimal.
Exercise 2
Convert the decimal number 78 to base 4.
Conclusion
Understanding numerals in base n opens up a world of different number systems, each with its unique applications and benefits. From computing to timekeeping, these systems play a crucial role in our daily lives and technological advancements.
3. Wikipedia – Positional Notation