Number Bases: Conversion and Operations

Introduction to Number Bases and Conversion

A number base (also called a radix) is the number of unique digits, including zero, used to represent numbers in a given numbering system. The number system we use every day, called the decimal (base ten) system, uses ten digits: 0 through 9. However, other number bases exist and are widely used, especially in computing. This topic focuses specifically on converting numbers between different bases, building on a basic understanding of what number bases are.

Common Number Bases

Several number bases are commonly studied in mathematics and computing. Base 2 (binary) uses only two digits, 0 and 1, and is the fundamental language of computers and digital electronics. Base 5 (quinary) uses five digits, 0 to 4. Base 8 (octal) uses eight digits, 0 to 7. Base 10 (decimal) uses ten digits, 0 to 9, and is the system used in everyday counting. Base 16 (hexadecimal) uses sixteen symbols, 0 to 9 and then A to F (representing 10 to 15), and is commonly used in computing for representing colours and memory addresses.

Notation for Number Bases

To avoid confusion about which base a number is written in, a small subscript number is placed after the number to indicate its base. For example, 1011₂ means the number 1011 in base 2 (binary), while 1011₁₀ means the number one thousand and eleven in base 10 (decimal). This notation is essential whenever numbers from different bases might be confused with one another.

Understanding Place Value in Different Bases

Just as in the decimal system, each digit's position in a number represents a power of the base. In base ten, positions from right to left represent 10⁰, 10¹, 10², 10³, and so on (units, tens, hundreds, thousands). Similarly, in base two, positions from right to left represent 2⁰, 2¹, 2², 2³, and so on. Understanding this place value system based on powers of the base is the key to converting numbers between bases.

Converting from Any Base to Base Ten

To convert a number from another base into base ten (decimal), multiply each digit by the base raised to the power corresponding to its position (starting from 0 on the right), then add all the results together. For example, to convert 1101₂ to base ten: (1×2³) + (1×2²) + (0×2¹) + (1×2⁰) = 8 + 4 + 0 + 1 = 13. So 1101₂ = 13₁₀. As another example, to convert 234₅ to base ten: (2×5²) + (3×5¹) + (4×5⁰) = 50 + 15 + 4 = 69. So 234₅ = 69₁₀.

Converting from Base Ten to Another Base

To convert a base ten number into another base, repeatedly divide the number by the target base, recording the remainder at each step, until the quotient becomes zero. The digits of the new base representation are then the remainders read in reverse order (from the last remainder to the first). For example, to convert 45₁₀ to base 2: 45 ÷ 2 = 22 remainder 1; 22 ÷ 2 = 11 remainder 0; 11 ÷ 2 = 5 remainder 1; 5 ÷ 2 = 2 remainder 1; 2 ÷ 2 = 1 remainder 0; 1 ÷ 2 = 0 remainder 1. Reading the remainders from bottom to top gives 101101₂.

Converting Between Two Non-Decimal Bases

To convert a number directly between two bases that are neither base ten, the easiest and most reliable method is to use base ten as an intermediate step: first convert the original number to base ten, then convert that base ten value to the target base. For example, to convert 122₃ to base 5: first convert 122₃ to base ten: (1×3²) + (2×3¹) + (2×3⁰) = 9 + 6 + 2 = 17; then convert 17₁₀ to base 5: 17 ÷ 5 = 3 remainder 2; 3 ÷ 5 = 0 remainder 3; reading remainders gives 32₅. So 122₃ = 32₅.

Addition of Numbers in Other Bases

Addition in a non-decimal base follows the same principles as base ten addition, except that carrying occurs whenever a column sum reaches or exceeds the value of the base, rather than 10. For example, when adding in base 2, any column sum of 2 or more results in a carry to the next column (since 2 = 10₂). For example, 111₂ + 101₂: adding from the right, 1+1=10₂ (write 0, carry 1); 1+0+1(carry)=10₂ (write 0, carry 1); 1+1+1(carry)=11₂ (write 11). The result is 1100₂.

Subtraction of Numbers in Other Bases

Subtraction in another base follows the same borrowing principle as base ten subtraction, except that when borrowing is needed, one unit borrowed from the next column is worth the value of the base, not 10. For example, in base 5 subtraction, borrowing one from the next column adds 5 (not 10) to the current column before subtracting.

Why Binary (Base 2) Is Important

Binary is especially important because digital computers and electronic devices represent all data using only two states: "on" and "off," which correspond directly to the digits 1 and 0. Every piece of information a computer processes — numbers, text, images, and sound — is ultimately stored and processed as combinations of binary digits (called bits). Understanding binary numbers and how to convert them to and from decimal is therefore foundational knowledge in computer science and digital technology.

Practical Applications of Number Bases

Beyond computing, different number bases appear in various real-world contexts: time is measured using a base 60 system for seconds and minutes, and a base 24 system for hours in a day; angles are measured in degrees using base 60 for minutes and seconds of arc; hexadecimal (base 16) is used in computing to represent colours in web design (such as #FF5733) and memory addresses; and historically, some ancient civilisations used different number bases, such as the Babylonians who used base 60.

Common Mistakes in Base Conversion

Common errors when converting between number bases include forgetting that place values are powers of the target base rather than powers of ten, misreading remainders in the wrong order when converting from base ten to another base, using digits that are not valid in a given base (for example, using the digit 5 in a base 5 system, where only digits 0 to 4 are allowed), and making arithmetic errors when carrying or borrowing in bases other than ten.

Summary

Number bases describe different systems for representing numbers using varying quantities of digits, with base ten (decimal) being the most familiar and base two (binary) being fundamental to computing. Converting from another base to base ten involves multiplying each digit by the base raised to its positional power and summing the results, while converting from base ten to another base involves repeated division by the target base and reading the remainders in reverse. Addition and subtraction in other bases follow the same principles as in base ten, but with carrying and borrowing based on the value of the base rather than ten.

Building Confidence with Base Conversion

Mastering base conversion takes patience and careful attention to place value, since the habits built up over years of working only in base ten can make other bases feel unfamiliar at first. Working through many practice conversions, checking answers by converting back to the original base, and relating binary numbers to real digital examples (such as how a byte of 8 bits can represent values from 0 to 255) all help build genuine understanding rather than simple memorisation of the conversion steps.

Worked Examples

Example 1: Convert 1010₂ to base ten. (1×2³) + (0×2²) + (1×2¹) + (0×2⁰) = 8 + 0 + 2 + 0 = 10₁₀.

Example 2: Convert 58₁₀ to base 2. 58÷2=29 r0; 29÷2=14 r1; 14÷2=7 r0; 7÷2=3 r1; 3÷2=1 r1; 1÷2=0 r1. Reading remainders bottom to top: 111010₂.

Example 3: Convert 342₅ to base ten. (3×5²) + (4×5¹) + (2×5⁰) = 75 + 20 + 2 = 97₁₀.

Example 4: Add 110₂ and 101₂. From the right: 0+1=1; 1+0=1; 1+1=10₂ (write 0, carry 1), then the carry gives a leading 1. Answer: 1011₂.

Example 5: Convert 27₁₀ to base 5. 27÷5=5 r2; 5÷5=1 r0; 1÷5=0 r1. Reading remainders bottom to top: 102₅.

Student Exercise

Solve the following problems, showing all your working:

1. Convert 11011₂ to base ten.
2. Convert 39₁₀ to base 2.
3. Convert 213₄ to base ten.
4. Add 1011₂ and 110₂.
5. Subtract 101₂ from 111₂.
6. Convert 46₁₀ to base 5.
7. Convert 132₅ to base ten.
8. Convert 25₁₀ to base 2.
9. Convert 101011₂ to base ten.
10. Convert 76₁₀ to base 8.