Number Bases


2.1  Number Bases

Number systems which consisting of digits from \(0\) to \(9\).

The number systems are made up of numbers with various bases. 

  Extra information   
  • Digits are the symbols used or combined to form a number in the number system. 
  • \(0,1,2, 3,4,5,6,7,8,9\) are the ten digits used in the the decimal number system.
  • Each base has digits from \(0\) to a digit which is less than its base.
Place values that involved in number bases 
  • Each base has place values according to each respective base . 
  • The place values of a base are the repeated multiplication of that base. 
  • For example, \(a\) is the base, then its place value will start with \(a^0, a^1, a^2, a^3, .... a^n\).
  Example questions   
  Find the place value of each digit in \(542_6\)  


Number in base  is \(6\) \(5\) \(4\) \(2\)
Place value  \(6^2\) \(6^1\) \(6^0\)
Value of a particular digit in a number in various bases. 
  The value of a particular digit in a number is the multiplication of a digit and the place value that represents the digit.   
  State the value of \(5\) in \(34\underline52_6\)  

Solution: The base is \(6\), thus 

Number  \(3\) \(4\) \(5\) \(2\)
Place value \(6^3\) \(6^2\) \(6^1\) \(6^0\)
Digit value      \(5 \times 6^1= 30\)  
Numerical value of a number in various bases 
  The numerical value of a number in various bases can be determined by calculating the sum of the digit values of the number   
  Determine the numerical value of \(6452_7\)  

Solution: The base is \(7\), thus 

Number  \(6\) \(4\) \(5\) \(2\)
Place value  \(7^3\) \(7^2\) \(7^1\) \(7^0\)
Digit value  \(6 \times 7^3 =2058\) \(4 \times 7^2= 196\) \(5 \times 7_1= 35\) \(2 \times 7^0 =2\)
Number value \(2058 +196+35+2 = 2291_\text{10}\)
Conversion numbers from one base to another base using various methods
A numbers can be converted to other bases by using various methods, such as division using place value and the division using base value. 
These processes involve converting 
(a)   a number in base ten to another base 
(b)   a number in a certain base ten and then to another base 
(c)   a number in base two directly to base eight 
(d)   a number in base eight directly to base two
A. Conversion of number in base ten into another base 
  • A number in base ten can be converted to another base by dividing the number using the place value or the base value required. 
  Convert \(567_\text{10}\) to a number in base five  


\(\begin{array}{c} 5\\5\\5\\5\\\phantom{-} \end{array} \begin{array}{|c} \quad567\quad\\ \hline \quad113\quad\\ \hline \quad22\quad\\ \hline \quad4\quad\\ \hline \quad0\quad\\ \hline \end{array} \begin{array}{c} \quad{\text{remainder}}\\ \longrightarrow2\\ \longrightarrow3\\ \longrightarrow2\\ \longrightarrow4\\ \end{array}\)

\(\therefore 567_\text{10} =4232_5\)

B. Conversion of a number in a certain base to another base
A number in base \(p\) can be converted to base ten and then to base \(q\)
  Convert \(251_6\) to a number in base eight   


Step 1: Convert the number in base ten 

Number in base \(6\) \(2\) \(5\) \(1\)
Place value  \(6^2\) \(6^1\) \(6^0\)
Value of number in base \(10\) \(2 \times 6^2 = 72\) \(5 \times 6^1 = 30\) \(1 \times 6^0= 1\)

\(\therefore 72 +30+1 = 103_\text{10}\)


Step 2: Convert the number in base ten to base eight 

\(\begin{array}{c} 8\\8\\8\\\phantom{-} \end{array} \begin{array}{|c} \quad103\quad\\ \hline \quad12\quad\\ \hline \quad1\quad\\ \hline \quad0\quad\\ \hline \end{array} \begin{array}{c} \quad{\text{remainder}}\\ \longrightarrow7\\ \longrightarrow4\\ \longrightarrow1\end{array}\)

\(\therefore251_6 = 147_8\)

C. Conversion a number in base two into base eight 
(i)   Separate each of three digits of a number in base two from the right to the left 
(ii)   Determine the sum of the digit values for the combined three digits in base two 
(iii)   Combine the number in base eight 
  Convert \(111101_2\) in base eight   


Number in base two  \(1\) \(1\) \(1\) \(1\) \(0\) \(1\)
Place value  \(2^2\) \(2^1\) \(2^0\) \(2^2\) \(2^1\) \(2^0\)
Digit value  \(4\) \(2\) \(1\) \(4\) \(0\) \(1\)
Base eight  \(4 +2 +1 = 7\) \(4+0+1= 5\)
\(111101_2 = 75_8\)
Subtraction numbers in various bases 
There are two methods which can be perform such as 
(i)   Vertical form 
(ii)   Conversion of base 
Vertical form method 
1.   Substract the given digits starting from the right to the left 

The difference is written in the answer space. 

The difference is always less than the given base and its value is equal to base ten 

3.   This process is repeated until all the digits in the numbers are substracted

Calculate \(375_6 - 150_6 \)


\(\begin{array}{r} 375_6 \\ - 150_6\\ \hline 225_6 \\\hline \end{array}\)

Conversion of base 
1.    Convert a number to base ten and perform substraction
2.    Convert the answer in base ten to the required base