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| Number Bases |
- 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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- 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.
- The table below shows the digits used in base two up to base ten:
| Number Base |
Digit |
| Base \(2\) |
\(0,1\) |
| Base \(3\) |
\(0,1,2\) |
| Base \(4\) |
\(0,1,2,3\) |
| Base \(5\) |
\(0,1,2,3,4\) |
| Base \(6\) |
\(0,1,2,3,4,5\) |
| Base \(7\) |
\(0,1,2,3,4,5,6\) |
| Base \(8\) |
\(0,1,2,3,4,5,6,7\) |
| Base \(9\) |
\(0,1,2,3,4,5,6,7,8\) |
| Base \(10\) |
\(0,1,2,3,4,5,6,7,8,9\) |
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| Place Values 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 \(a^0, a^1, a^2, a^3, .... a^n\).
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Find the place value of each digit in \(542_6\).
Solution:
| 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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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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State the value of \(34\underline52_6\).
Solution:
| 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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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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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 |
\(2058 +196+35+2 = 2291_\text{10}\) |
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| Convert 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 number in base ten to another base
- a number in a certain base ten and then to another base
- a number in base two directly to base eight
- a number in base eight directly to base two
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| 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.
Example: Convert \(567_\text{10}\) to a number in base five.
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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| 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 \(q\).
Example: Convert \(251_6\) to a number in base eight.
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 \(10\) |
\(2 \times 6^2 = 72\) |
\(5 \times 6^1 = 30\) |
\(1 \times 6^0= 1\) |
Step 2: Convert the number in base ten to base eight
\(\begin{array}{c} 8\\8\\8\\\phantom{-} \end{array} \begin{array}{|c} \quad103\quad\\ \hline \quad12\quad\\ \hline \quad1\quad\\ \hline \quad0\quad\\ \hline \end{array} \begin{array}{c} \quad{\text{remainder}}\\ \longrightarrow7\\ \longrightarrow4\\ \longrightarrow1\end{array}\)
\(\therefore251_6 = 147_8\) |
| Conversion a Number in Base Two into Base Eight |
- Separate each of three digits of a number in base two from the right to the left.
- Determine the sum of the digit values for the combined three digits in base two.
- Combine the number in base eight.
Example: Convert \(111101_2\) in base eight.
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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| Perform Calculations Involving Addition and Subtraction of Numbers in Various Bases |
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Additon |
Subtraction |
| Vertical Form |
- Add the given digits starting from the right to the left.
- The sum of the digits in base ten is converted to the given base to be written in the answer space.
- This process is repeated until all the digits in the number are added up.
Example:
Calculate \(673_8 + 175_8\).
\(\begin{array}{cccccc}^{\color{red}}&&^{\color{red}1}&^{\color{red}1} \\ && 6 & 7 & 3\,_8 \\ +&& 1 & 7 & 5\,_8 \\ \hline & 1& \color{red}0 & \color{blue}7 & \color{orange}8\,\color{black}_8 \\ \hline \end{array}\\\qquad\qquad\qquad\;\;\;\color{orange}\uparrow\\\qquad\qquad\qquad\;\;\;\small\color{orange}3+5=8_{10}=10_8\\\color{blue}{\qquad\qquad\qquad\uparrow}\\\qquad\qquad\qquad\small{1+7+7=15_{10}=17_8}\\\color{red}\qquad\qquad\;\;\uparrow\\\qquad\qquad\;\;\small{1+6+1=8_{10}=10_8}\)
Answer:
Perform addition as usual and convert the values in base ten to base eight.
\(673_8 + 175_8=1070_8\)
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- Subtract 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.
- This process is repeated until all the digits in the number are substracted.
Example:
Calculate \(6241_7 - 613_7\).
\(\begin{array}{cccccc}^{\color{red}}&^{\color{red}5}&^{\color{red}7}&^{\color{red}3}&^{\color{red}7} \\ &{\not}6& 2 & {\not}4 & 1\;_7 \\ -&& 6 & 1 & 3 \,_7\\ \hline & 5& \color{red}3 & \color{blue}2 & \color{orange}5 \,\color{blacl}_7\\ \hline \end{array}\\\qquad\qquad\qquad\;\;\;\color{orange}\uparrow\\\qquad\qquad\qquad\;\;\;\small\color{orange}7+1-3=5_{7}\\\color{blue}{\qquad\qquad\qquad\uparrow}\\\qquad\qquad\qquad\small{3-1=2_7}\\\color{red}\qquad\qquad\;\;\uparrow\\\qquad\qquad\;\;\small{7+2-6=3_7}\)
Answer:
\(6241_7 - 613_7=5325_7\)
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| Conversion of Base |
- Convert a number to base ten and perform addition.
- Convert the answer in base ten to the required base.
Example:
Calculate \(673_8 + 175_8\).
\(6\,7\,3_8\rightarrow\;\;\;4\,4\,3_{10}\\1\,7\,5_8\rightarrow+\underline{1\,2\,5_{10}}\\ \qquad\qquad\;\;\underline{\,5\,6\,8_{10}}\)
\(\begin{array}{c}\\8\\8\\8\\8\\\phantom{-} \end{array} \begin{array}{|c} \quad568\quad\\ \hline \quad71\quad\\ \hline \quad8\quad\\ \hline \quad1\quad\\ \hline\quad0\quad \end{array} \begin{array}{c} \quad \\ -0 \\-7\\ -0\\-1 \end{array}\)\(\Bigg\uparrow\)
Answer:
\(673_8 + 175_8=1070_8\)
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- Convert a number to base ten and perform substraction.
- Convert the answer in base ten to the required base.
Example:
Calculate \(6241_7 - 613_7\).
\(6\,2\,4\,1_7\rightarrow\;2\,1\,8\,5_{10}\\6\,1\,3_8\rightarrow-\underline{\;\;\;3\,0\,4_{10}}\\ \qquad\qquad\;\;\underline{1\,8\,8\,1_{10}}\)
\(\begin{array}{c}\\7\\7\\7\\7\\\phantom{-} \end{array} \begin{array}{|c} \quad1881\quad\\ \hline \quad268\quad\\ \hline \quad38\quad\\ \hline \quad5\quad\\ \hline\quad0\quad \end{array} \begin{array}{c} \quad \\ -5 \\-2\\ -3\\-5 \end{array}\)\(\Bigg\uparrow\)
Answer:
\(6241_7 - 613_7=5325_7\)
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