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Definition |
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Number systems which consisting of digits from \(0\) to \(9\).
The number systems are made up of numbers with various bases.
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Extra information |
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- 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.
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- Each base has digits from \(0\) to a digit which is less than its base.
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| Place values that involved in number bases |
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Explanation |
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- 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 \(a^0, a^1, a^2, a^3, .... a^n\).
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Example questions |
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Find the place value of each digit in \(542_6\) |
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Answers:
| Number in base is \(6\) |
\(5\) |
\(4\) |
\(2\) |
| Place value |
\(6^2\) |
\(6^1\) |
\(6^0\) |
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| Value of a particular digit in a number in various bases. |
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Definition |
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The value of a particular digit in a number is the multiplication of a digit and the place value that represents the digit. |
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Example |
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State the value of \(5\) in \(34\underline52_6\) |
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Solution: The base is \(6\), thus
| Number |
\(3\) |
\(4\) |
\(5\) |
\(2\) |
| Place value |
\(6^3\) |
\(6^2\) |
\(6^1\) |
\(6^0\) |
| Digit value |
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\(5 \times 6^1= 30\) |
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| Numerical value of a number in various bases |
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Definition |
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The numerical value of a number in various bases can be determined by calculating the sum of the digit values of the number |
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Example |
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Determine the numerical value of \(6452_7\) |
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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 |
\(2058 +196+35+2 = 2291_\text{10}\) |
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| Conversion numbers from one base to another base using various methods |
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| A numbers can be converted to other bases by using various methods, such as division using place value and the division using base value. |
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| These processes involve converting |
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a number in base ten to another base |
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a number in a certain base ten and then to another base |
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| (c) |
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a number in base two directly to base eight |
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a number in base eight directly to base two |
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| A. Conversion of number in base ten into another base |
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- A number in base ten can be converted to another base by dividing the number using the place value or the base value required.
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Example |
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Convert \(567_\text{10}\) to a number in base five |
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Solution:
\(\begin{array}{c} 5\\5\\5\\5\\\phantom{-} \end{array} \begin{array}{|c} \quad567\quad\\ \hline \quad113\quad\\ \hline \quad22\quad\\ \hline \quad4\quad\\ \hline \quad0\quad\\ \hline \end{array} \begin{array}{c} \quad{\text{remainder}}\\ \longrightarrow2\\ \longrightarrow3\\ \longrightarrow2\\ \longrightarrow4\\ \end{array}\)
\(\therefore 567_\text{10} =4232_5\)
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| B. Conversion of a number in a certain base to another base |
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| A number in base \(p\) can be converted to base ten and then to base \(q\) |
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Example |
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Convert \(251_6\) to a number in base eight |
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Solution:
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 \(10\) |
\(2 \times 6^2 = 72\) |
\(5 \times 6^1 = 30\) |
\(1 \times 6^0= 1\) |
\(\therefore 72 +30+1 = 103_\text{10}\)
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Step 2: Convert the number in base ten to base eight
\(\begin{array}{c} 8\\8\\8\\\phantom{-} \end{array} \begin{array}{|c} \quad103\quad\\ \hline \quad12\quad\\ \hline \quad1\quad\\ \hline \quad0\quad\\ \hline \end{array} \begin{array}{c} \quad{\text{remainder}}\\ \longrightarrow7\\ \longrightarrow4\\ \longrightarrow1\end{array}\)
\(\therefore251_6 = 147_8\)
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| C. Conversion a number in base two into base eight |
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| Rules |
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Separate each of three digits of a number in base two from the right to the left |
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Determine the sum of the digit values for the combined three digits in base two |
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Combine the number in base eight |
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Example |
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Convert \(111101_2\) in base eight |
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Solution:
| 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 |
\(4 +2 +1 = 7\) |
\(4+0+1= 5\) |
| \(111101_2 = 75_8\) |
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| Subtraction numbers in various bases |
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| There are two methods which can be perform such as |
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Vertical form |
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Conversion of base |
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| Vertical form method |
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Substract the given digits starting from the right to the left |
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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
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| 3. |
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This process is repeated until all the digits in the numbers are substracted |
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Example |
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Calculate \(375_6 - 150_6 \)
Solution:
\(\begin{array}{r} 375_6 \\ - 150_6\\ \hline 225_6 \\\hline \end{array}\)
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| Conversion of base |
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Convert a number to base ten and perform substraction |
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Convert the answer in base ten to the required base |
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