## Number Bases

 2.1 Number Bases

Number Bases
 Definition
• 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.
• 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 $$10$$ $$0,1,2,3,4,5,6,7,8,9$$

Place Values Involved in Number Bases
 Explanation
• 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

Find the place value of each digit in $$542_6$$.

Solution:

 Number in Base is $$6$$ $$5$$ $$4$$ $$2$$ Place Value $$6^2$$ $$6^1$$ $$6^0$$

Value of a Particular Digit in a Number in Various Bases
 Definition
The value of a particular digit in a number is the multiplication of a digit and the place value that represents the digit.
 Example

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 $$5 \times 6^1= 30$$

Numerical Value of a Number in Various Bases
 Definition
The numerical value of a number in various bases can be determined by calculating the sum of the digit values of the number.
 Example

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 $$2058 +196+35+2 = 2291_\text{10}$$

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

$$\therefore 567_\text{10} =4232_5$$

 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
1. Separate each of three digits of a number in base two from the right to the left.
2. Determine the sum of the digit values for the combined three digits in base two.
3. 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 $$4 +2 +1 = 7$$ $$4+0+1= 5$$ $$111101_2 = 75_8$$
Perform Calculations Involving Addition and Subtraction of Numbers in Various Bases
 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. $$673_8 + 175_8=1070_8$$ 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: $$6241_7 - 613_7=5325_7$$ 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: $$673_8 + 175_8=1070_8$$ 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: $$6241_7 - 613_7=5325_7$$

## Number Bases

 2.1 Number Bases

Number Bases
 Definition
• 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.
• 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 $$10$$ $$0,1,2,3,4,5,6,7,8,9$$

Place Values Involved in Number Bases
 Explanation
• 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

Find the place value of each digit in $$542_6$$.

Solution:

 Number in Base is $$6$$ $$5$$ $$4$$ $$2$$ Place Value $$6^2$$ $$6^1$$ $$6^0$$

Value of a Particular Digit in a Number in Various Bases
 Definition
The value of a particular digit in a number is the multiplication of a digit and the place value that represents the digit.
 Example

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 $$5 \times 6^1= 30$$

Numerical Value of a Number in Various Bases
 Definition
The numerical value of a number in various bases can be determined by calculating the sum of the digit values of the number.
 Example

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 $$2058 +196+35+2 = 2291_\text{10}$$

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

$$\therefore 567_\text{10} =4232_5$$

 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
1. Separate each of three digits of a number in base two from the right to the left.
2. Determine the sum of the digit values for the combined three digits in base two.
3. 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 $$4 +2 +1 = 7$$ $$4+0+1= 5$$ $$111101_2 = 75_8$$
Perform Calculations Involving Addition and Subtraction of Numbers in Various Bases
 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. $$673_8 + 175_8=1070_8$$ 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: $$6241_7 - 613_7=5325_7$$ 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: $$673_8 + 175_8=1070_8$$ 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: $$6241_7 - 613_7=5325_7$$