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• Expressing Binary Numbers

Question 1

\(111_{(2)} + 1110_{(2)}\)}

\[ 0 1 1 1\] \[ 1 1 1 0\]

Question 2 :

\(11010_{(2)} + 1100_{(2)}\)}

\[ 11010\] \[ 01100\]

---

\[ 010011\] \[ 111110\]

Tutorial Questions

• Addition of Binary Numbers Example 1
• Addition of Binary Numbers Example 2
• Addition of Binary Numbers Example 3

Other Questions

• Binary Addition

• Binary Addition

• Binary Addition

• Binary Subtraction: using Borrowing Method

• Binary Numbers Adding Two Binary Numbers (Example) HD

• Binary Subtraction using “Two’s Complement”

Binary Numbers Multiplication

• Left Shifting Technique for Binary Multiplication

• Binary Subtraction: using Twos Complement

• Binary Multiplication: using Left Shifting

• Relations -Example

• Binary Multiplication ( Binary Numbers Tutorial Sheet)

Introduction to Hexadecimal Numbers

Hexadecimal (or simply “hex”) is a base-16 number system. It uses 16 distinct symbols: the digits 0-9 and the letters A-F, where A stands for 10, B for 11, C for 12, D for 13, E for 14, and F for 15. Hexadecimal is often used in computing and digital electronics because it can represent large binary numbers more compactly.

Why Use Hexadecimal?

• Compact Representation: Hexadecimal representation is more compact compared to binary, reducing the length of numbers.
• Ease of Conversion: It’s easy to convert between binary and hexadecimal. Each hex digit corresponds to four binary digits (bits).
• Human-Friendly: Hexadecimal is more readable and manageable for humans compared to long strings of binary digits.

Hexadecimal Number System

• Base: 16
• Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Converting Between Binary and Hexadecimal

To convert a binary number to hexadecimal, group the binary digits into sets of four (starting from the right). Each group of four binary digits corresponds to a single hexadecimal digit.

Converting Between Decimal and Hexadecimal

To convert a decimal number to hexadecimal, repeatedly divide the number by 16 and keep track of the remainders. The hexadecimal number is obtained by reading the remainders in reverse order.

Example 3: Convert Decimal to Hexadecimal Decimal number: 254 1. Divide by 16: \(254 \div 16 = 15\) remainder \(14\) (E) 2. Divide the quotient by 16: \(15 \div 16 = 0\) remainder \(15\) (F) 3. So, 254 in decimal is FE in hexadecimal.

Example 4: Convert Hexadecimal to Decimal Hexadecimal number: 1A3 1. Convert each hex digit to decimal and multiply by the corresponding power of 16: - 1 = 1 \(\times 16^2 = 256\) - A = 10 \(\times 16^1 = 160\) - 3 = 3 \(\times 16^0 = 3\) 2. Sum the results: \(256 + 160 + 3 = 419\) 3. So, 1A3 in hexadecimal is 419 in decimal.

Practical Uses of Hexadecimal

• Memory Addresses: Hexadecimal is often used to represent memory addresses in programming and debugging.
• Color Codes: In web design, colors are often specified using hexadecimal values (e.g., #FFFFFF for white).
• File Formats: Hexadecimal is used in various file formats and protocols for compact data representation.

Exercises

1. Convert the binary number 110101011 to hexadecimal.
2. Convert the hexadecimal number 3C9 to binary.
3. Convert the decimal number 456 to hexadecimal.
4. Convert the hexadecimal number 2F to decimal.

Gray Code

• The reflected binary code (RBC), also known just as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).

• For example, the representation of the decimal value “1” in binary would normally be “001” and “2” would be “010”.

• In Gray code, these values are represented as “001” and “011”. That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.

• Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems.

Step 1: Understand Binary Representation

First, list all the 3-bit binary numbers in standard order:

000, 001, 010, 011, 100, 101, 110, 111

Step 2: Generate Gray Code

To convert these binary numbers to Gray code, follow the rule where the first bit remains the same, and each subsequent bit is obtained by XOR’ing the previous bit with the current bit.

Let’s go through each binary number:

1. 000 (Binary)

• First bit remains the same: 0

• Second bit: \(0 \oplus 0 = 0\)

• Third bit: \(0 \oplus 0 = 0\)

• Gray code: 000

2. 001 (Binary)

• First bit remains the same: 0

• Second bit: \(0 \oplus 0 = 0\)

• Third bit: \(0 \oplus 1 = 1\)

• Gray code: 001

3. 010 (Binary)

• First bit remains the same: 0

• Second bit: \(0 \oplus 1 = 1\)

• Third bit: \(1 \oplus 0 = 1\)

• Gray code: 011

4. 011 (Binary)

• First bit remains the same: 0

• Second bit: \(0 \oplus 1 = 1\)

• Third bit: \(1 \oplus 1 = 0\)

• Gray code: 010

5. 100 (Binary)

• First bit remains the same: 1

• Second bit: \(1 \oplus 0 = 1\)

• Third bit: \(1 \oplus 0 = 1\)

• Gray code: 110

6. 101 (Binary)

• First bit remains the same: 1

• Second bit: \(1 \oplus 0 = 1\)

• Third bit: \(1 \oplus 1 = 0\)

• Gray code: 111

7. 110 (Binary)

• First bit remains the same: 1

• Second bit: \(1 \oplus 1 = 0\)

• Third bit: \(0 \oplus 1 = 1\)

• Gray code: 101

8. 111 (Binary)

• First bit remains the same: 1

• Second bit: \(1 \oplus 1 = 0\)

• Third bit: \(0 \oplus 1 = 1\)

• Gray code: 100

Step 3: List the Gray Code Sequence

Here is the 3-bit Gray code sequence:

000, 001, 011, 010, 110, 111, 101, 100

Summary

By following the rule of keeping the first bit the same and XOR’ing subsequent bits, we have successfully converted 3-bit binary numbers into Gray code. This sequence ensures that only one bit changes between consecutive values.

Question 1

Part (a) How many Symbols are used in each System

1. Decimal (Base 10)
2. Binary (Base 2)
3. Hexadecimal (Base 16)

Question 2

State which of the following sets the following numbers belong to.

[1)] \(18\) [2)] \(8.2347\ldots\) [3)] \(\pi\) [4)] \(1.33333\ldots\) [5)] \(17/4\) [6)] \(4.25\) [7)] \(\sqrt{\pi}\) [8)] \(\sqrt{25}\)

The possible answers are

[a] Natural number : \(\mb{N} \subseteq \mb{Z } \subseteq \mb{Q} \subseteq \mb{R}\) [b] Integer : $ $ [c] Rational Number : $ $ [d] Real Number \(\mb{R}\)

\end{enumerate}

%

\begin{framed}

\(\mb{N}\) : natural numbers (or positive integers) \(\{1,2,3,\ldots\}\)

\(\mb{Z}\) : integers \(\{-3,-2,-1,0,1,2,3,\ldots\}\)

(The letter \(\mb{Z}\) comes from the word which means ``numbers” in German.)

\(\mb{Q}\) : rational numbers \(\mb{R}\) : real numbers

\(\mb{N} \subseteq \mb{Z } \subseteq \mb{Q} \subseteq \mb{R}\)

(All natural numbers are integers. All integers are rational numbers. All rational numbers are real numbers.)

Question 1

Perform the following binary additions.

1. \((110101)_{2}\) + \((1010111)_{2}\)
2. \((1010101)_{2}\) + \((101010)_{2}\)
3. \((11001010)_{2}\) + \((10110101)_{2}\)
4. \((1011001)_{2}\) + \((111010)_{2}\)

Question 2

Perform the following binary subtractions.

1. \((110101)_{2}\) - \((1010111)_{2}\)
2. \((1010101)_{2}\) - \((101010)_{2}\)
3. \((11001010)_{2}\) - \((10110101)_{2}\)
4. \((1011001)_{2}\) - \((111010)_{2}\)

Question 1

Perform the following binary multiplications.

1. \((1001)_{2}\times( 1000)_{2}\) % 9 by 8
2. \((101)_{2}\times(1101)_{2}\) % 5 by 11
3. \((111)_{2}\times(1111)_{2}\) % 7 by 15
4. \((10000)_{2}\times(11001)_{2}\)%16 by 25

Perform the following binary multiplications. %\begin{multicols}{2} % %a. \((1001000)_{2} \div ( 1000)_{2}\) %b. \((101101)_{2} \div (1001)_{2}\) %c. \((1001011000)_{2} \div (101000)_{2}\) %d. \((1100000)_{2} \div (10000)_{2}\) % %

Which of the following binary numbers is the result of this binary division: \((10)_{2} \times ( 1101)_{2}\). % % (2) / (13) \begin{multicols}{2}

1. \((11010)_{2}\) %26
2. \((11100)_{2}\) %28
3. \((10101)_{2}\) %21
4. \((11011)_2\) %27

Which of the following binary numbers is the result of this binary division: \((101010)_{2} \times( 111 )_{2}\). % (4) / (6) \begin{multicols}{2}

1. \((11000)_{2}\) %24
2. \((11001)_{2}\) %25
3. \((10101)_{2}\) %21
4. \((11011)_2\) %27

Which of the following binary numbers is the result of this binary division: \((1001110)_{2}\times ( 1101 )_{2}\). % (9) / (3) \begin{multicols}{2}

1. \((11000)_{2}\) %24
2. \((11001)_{2}\) %25
3. \((10101)_{2}\) %21
4. \((11011)_2\) %27

Perform the following binary divisions. %\begin{multicols}{2} % %a. \((1001000)_{2} \div ( 1000)_{2}\) %b. \((101101)_{2} \div (1001)_{2}\) %c. \((1001011000)_{2} \div (101000)_{2}\) %d. \((1100000)_{2} \div (10000)_{2}\) % %

\begin{enumerate}

Which of the following binary numbers is the result of this binary division: \((111001)_{2} \div ( 10011)_{2}\). % (57) / (19) \begin{multicols}{2}

1. \((10)_2\) %2
2. \((11)_{2}\) %3
3. \((100)_{2}\) %4
4. \((101)_{2}\) %5

Which of the following binary numbers is the result of this binary division: \((101010)_{2} \div ( 111 )_{2}\). % (42) / (7) \begin{multicols}{2}

1. \((11)_2\) %3
2. \((100)_{2}\) %4
3. \((101)_{2}\) %5
4. \((110)_{2}\) %6

Which of the following binary numbers is the result of this binary division: \((1001110)_{2} \div ( 1101 )_{2}\). % (78) / (13) \begin{multicols}{2}

1. \((100)_{2}\) %4
2. \((110)_{2}\) %6
3. \((111)_{2}\) %7
4. \((1001)_2\) %9

Question 1

Answer the following questions about the hexadecimal number systems

1. How many characters are used in the hexadecimal system?
2. What is highest hexadecimal number that can be written with two characters?
3. What is the equivalent number in decimal form?
4. What is the next highest hexadecimal number?

Question 2

Which of the following are not valid hexadecimal numbers?

1. \(73\)
2. \(A5G\)
3. \(11011\)
4. $EEF $

Question 3

Express the following decimal numbers as a hexadecimal number.

1. \((73)_{10}\)
2. \((15)_{10}\)
3. \((22)_{10}\)
4. \((121)_{10}\)

Question 4

Compute the following hexadecimal calculations.

1. \(5D2+A30\)
2. \(702+ABA\)
3. \(101+111\)
4. \(210+2A1\)

Question 1

Express the following decimal numbers as binary numbers.

1. \((73)_{10}\)

2. \((15)_{10}\)

3. \((22)_{10}\)

Question 2

All three answers are among the following options.

a.\((10110)_{2}\) %22

b.\((1111)_{2}\) %15

3. \((1001001)_{2}\) %73

4. \((1000010)_{2}\) %64

Question 3

Express the following binary numbers as decimal numbers

1. \(101\)

2. \(1101\)

3. \(11011\)

4. \(100101\)

Question 4

Express the following binary numbers as decimal numbers.

a.\((101010)_{2}\)

b.\((10101)_{2}\)

c.\((111010)_{2}\)

4. \((11010)_{2}\)

#### Question 5

Express the following binary numbers as decimal numbers.

a.\((110.10101)_{2}\)

b.\((101.0111)_{2}\)

3. \((111.01)_{2}\)

4. \((110.1101)_{2}\)

#### Question 6

Express the following decimal numbers as binary numbers

1. 6
2. 15
3. 37
4. 77

Express the following decimal numbers as binary numbers.

a.\((27.4375)_{10}\)% b.\((5.625)_{10}\) c. \((13.125)_{10}\) d. \((11.1875)_{10}\)

Perform the following binary additions.

a.\((110101)_{2}\) + \((1010111)_{2}\) b.\((1010101)_{2}\) + \((101010)_{2}\) c. \((11001010)_{2}\) + \((10110101)_{2}\) d. \((1011001)_{2}\) + \((111010)_{2}\)

Part B: Number Systems - Binary Arithmetic

Perform the following binary subtractions.

1. \((110101)_{2}\) - \((1010111)_{2}\)
2. \((1010101)_{2}\) - \((101010)_{2}\)
3. \((11001010)_{2}\) - \((10110101)_{2}\)
4. \((1011001)_{2}\) - \((111010)_{2}\)

Question 2 Perform the following binary multiplications.

1. \((1001)_{2}\times( 1000)_{2}\) % 9 by 8
2. \((101)_{2}\times(1101)_{2}\) % 5 by 11
3. \((111)_{2}\times(1111)_{2}\) % 7 by 15
4. \((10000)_{2}\times(11001)_{2}\)%16 by 25

%http://www.csgnetwork.com/binaddsubcalc.html (See section 1.1.3 of the text)

Question 1

Which of the following binary numbers is the result of this binary multiplication: \((10)_{2} \times ( 1101)_{2}\). % % (2) / (13)

1. \((11010)_{2}\) %26
2. \((11100)_{2}\) %28
3. \((10101)_{2}\) %21
4. \((11011)_2\) %27

Question 3

Which of the following binary numbers is the result of this binary multiplication: \((1001110)_{2}\times ( 1101 )_{2}\). % (9) / (3)

1. \((11000)_{2}\) %24
2. \((11001)_{2}\) %25
3. \((10101)_{2}\) %21
4. \((11011)_2\) %27

Question 5

Which of the following binary numbers is the result of this binary division: \((111001)_{2} \div ( 10011)_{2}\). % (57) / (19)

1. \((10)_2\) %2
2. \((11)_{2}\) %3
3. \((100)_{2}\) %4
4. \((101)_{2}\) %5

Question 6

Which of the following binary numbers is the result of this binary division: \((101010)_{2} \div ( 111 )_{2}\). % (42) / (7)

1. \((11)_2\) %3
2. \((100)_{2}\) %4
3. \((101)_{2}\) %5
4. \((110)_{2}\) %6

Question 7

Which of the following binary numbers is the result of this binary division: \((1001110)_{2} \div ( 1101 )_{2}\). % (78) / (13)

1. \((100)_{2}\) %4
2. \((110)_{2}\) %6
3. \((111)_{2}\) %7
4. \((1001)_2\) %9

Question 3

Which of the following binary numbers is the result of this binary division: \((1001110)_{2}\times ( 1101 )_{2}\). % (9) /(3)

a.\((11000)_{2}\) %24 b.\((11001)_{2}\) %25 c. \((10101)_{2}\) %21 d. \((11011)_2\) %27

Question 2

Which of the following binary numbers is the result of this binary division: \((101010)_{2} \times( 111 )_{2}\).

% (4) /(6)

a.\((11000)_{2}\) %24 b.\((11001)_{2}\) %25 c. \((10101)_{2}\) %21 d. \((11011)_2\) %27

Question 3

Which of the following binary numbers is the result of this binary division: \((1001110)_{2}\times ( 1101 )_{2}\). % (9) /(3)

a.\((11000)_{2}\) %24 b.\((11001)_{2}\) %25 c. \((10101)_{2}\) %21 d. \((11011)_2\) %27

Which of the following binary numbers is the result of this binary division: \((1001110)_{2}\times ( 1101 )_{2}\). % (9) /(3)

a.\((11000)_{2}\) %24 b.\((11001)_{2}\) %25 c. \((10101)_{2}\) %21 d. \((11011)_2\) %27

Question 7

Perform the following binary divisions.

a.\((1001000)_{2} \div ( 1000)_{2}\) b.\((101101)_{2} \div (1001)_{2}\) c. \((1001011000)_{2} \div (101000)_{2}\) d. \((1100000)_{2} \div (10000)_{2}\)

Question 1 : Binary numbers

Express the following binary numbers as decimal numbers

( i) \(101\) ( ii) \(1101\) ( iii) \(11011\) ( iv) \(100101\)

( b) Express the following decimal numbers as binary numbers

( i) 6 ( ii) 15 ( iii) 37 ( iv) 77

Question 2

A number is expressed in base 5 as \((234)_5\). What is it as decimal number? Suppose you multiply \((234)_5\) by 5. what would be the answer in base 5.

Question 3

Perform the following binary additions

1. \(1011 + 1111\)

2. \(10101 + 10011\)

3. \(1010 + 11010\)

4. \(101010 + 10101 + 101\)

Question 4

Perform the binary additions

• \((10111)_2 +(111010)_2\)

• \((1101)_2 + (1011)_2 + (1111)_2\)

Question 5}

Perform the binary subtractions using both the bit-borrowing method and the two’s complement method.

• \((1001)_2 -(111)_2\)

• \((110000)_2 -(10111)_2\)

Question 6}

Perform the binary multiplications

• \((1101)_2 \times (101)_2\)

• \((1101)_2 \times (1101)_2\)

Question 7

( a) What is highest Hexadecimal number that can be written with two characters, and what is it’s equivalent in decimal form? What is the next highest hexadecimal number?

% \[FF = 255\] % Remark ( b) Which of the following are not valid hexadecimal numbers?

( i) A5G ( ii) 73 ( iii) EEF ( iv) 101

Question 8 : Binary Substraction}

( i) 110 - 10 ( ii) 101 - 11
( iii) 1001 - 11 ( iv) 10001 - 100 ( v) 101001 - 1101 ( vi) 11010101-1101

Question 9

( a) Suppose 2341 is a base-5 number Compute the equivalent in each of the following forms:

( i) decimal number ( ii) hexadecimal number ( iii) binary number

( b) Perform the following binary additions

( i) \(1011+ 1111\) ( ii) \(10101 + 10011\) ( iii) \(1010 + 11010\)

Question 1

Perform the following binary multiplications.

1. \((1001000)_{2} \div ( 1000)_{2}\)
2. \((101101)_{2} \div (1001)_{2}\)
3. \((1001011000)_{2} \div (101000)_{2}\)
4. \((1100000)_{2} \div (10000)_{2}\)

Question 2

Perform the following binary divisions.

1. \((1001000)_{2} \div ( 1000)_{2}\)
2. \((101101)_{2} \div (1001)_{2}\)
3. \((1001011000)_{2} \div (101000)_{2}\)
4. \((1100000)_{2} \div (10000)_{2}\)

Question 3

Which of the following binary numbers is the result of this binary division: \((111001)_{2} \div ( 10011)_{2}\). (57) / (19)

1. \((10)_2\) 2
2. \((11)_{2}\) 3
3. \((100)_{2}\) 4
4. \((101)_{2}\) 5

Question 4

Which of the following binary numbers is the result of this binary division: \((101010)_{2} \div ( 111 )_{2}\). (42) / (7)

1. \((11)_2\) 3
2. \((100)_{2}\) 4
3. \((101)_{2}\) 5
4. \((110)_{2}\) 6

Question 5

Which of the following binary numbers is the result of this binary division: \((1001110)_{2} \div ( 1101 )_{2}\). (78) / (13)

1. \((100)_{2}\) 4
2. \((110)_{2}\) 6
3. \((111)_{2}\) 7
4. \((1001)_2\) 9

Question 7

Perform the following binary multiplications.

1. \((1001000)_{2} \div ( 1000)_{2}\)

2. \((101101)_{2} \div (1001)_{2}\)

3. \((1001011000)_{2} \div (101000)_{2}\)

4. \((1100000)_{2} \div (10000)_{2}\)

Question 9

Perform the following binary division exercises.

1. \((1001000)_{2} \div ( 1000)_{2}\)

2. \((101101)_{2} \div (1001)_{2}\)

3. \((1001011000)_{2} \div (101000)_{2}\)

4. \((1100000)_{2} \div (10000)_{2}\)

Question 10

Calculate working in hexadecimal

\((BBB)_{16} + (A56)_{16}\)

\((BBB)_{16} - (A56)_{16}\)

Question 11}

Write the hex number \((EC4)_{16}\) in binary. Write the binary number \((11110110101|)_2\) in hex.

Question 12}

Express the decimal number 753 in binary , base 5 and hexadecimal.

Question 13}

Express 42900 as a product of its prime factors, using index notation for repeated factors.

Question 14}

Expresse the recuring decimals

( i) \(0.727272\ldots\) ( ii) \(0.126126126....\) ( iii) \(0.7545454545...\)

as rational numbers in its simplest form.

Question 15}

Given that \(\pi\) is an irrational number, can you say whether \(\frac{\pi}{2}\) is rational or irrational. or is it impossible to tell?

• % #### Question 16

1. Given x is the irrational positive number \(\sqrt{2}\), express \(x^8\) in binary notation

2. From part (i), is \(x^8\) a rational number?

Question 17

1. 5/7 lies between 0.714 and 0.715.
2. \(\sqrt(2)\) is at least 1.41.
3. \(\sqrt(3)\) 9s at least 1.732 and at most 1.7322.

Question 18

1. Write down the numbers 0.0000526 in floating point form.

2. How is the number 1 expressed in floating point form.

Question 19

1. Deduce that every composite integer \(n\) has a prime factor such that \(p \leq \sqrt{n}\).

2. Decide whether 899 is a prime.

Question 20

1. What would be the maximum numbber of digits that a decimal fraction with denominator 13 could have in a recurring block in theory?

2. Can you predict which other fractions with denominator 13 will have the same digits as 1/13 in their recurring block?

Question 1

Perform the following binary subtractions.

1. 110 - 10
2. 101 - 11
3. 1001 - 11
4. 10001 - 100
5. 101001 - 1101
6. 11010101-1101

Question 4

Perform the binary subtractions using both the bit-borrowing method and the two’s complement method.

1. \((1001)_2 -(111)_2\)
2. \((110000)_2 -(10111)_2\)

Question 1

Express the following decimal numbers as binary numbers.

1. \((73)_{10}\)
2. \((15)_{10}\)
3. \((22)_{10}\)

All three answers are among the following options.

1. \((10110)_{2}\) %22
2. \((1111)_{2}\) %15
3. \((1001001)_{2}\) %73
4. \((1000010)_{2}\) %64

Question 2

Express the following binary numbers as decimal numbers.

1. \((101010)_{2}\)
2. \((10101)_{2}\)
3. \((111010)_{2}\)
4. \((11010)_{2}\)

Question 3

Express the following binary numbers as decimal numbers.

1. \((110.10101)_{2}\)
2. \((101.0111)_{2}\)
3. \((111.01)_{2}\)
4. \((110.1101)_{2}\)

Express the following decimal numbers as binary numbers.

\((73)_{10}\) \((15)_{10}\)

\((22)_{10}\)

All three answers are among the following options.

1. \((10110)_{2}\) %22
2. \((1111)_{2}\) %15
3. \((1001001)_{2}\) %73
4. \((1000010)_{2}\) %64

Express the following binary numbers as decimal numbers.

1. \((101010)_{2}\)
2. \((10101)_{2}\)
3. \((111010)_{2}\)
4. \((11010)_{2}\)

Express the following binary numbers as decimal numbers.

1. \((110.10101)_{2}\)
2. \((101.0111)_{2}\)
3. \((111.01)_{2}\)
4. \((110.1101)_{2}\)

Express the following decimal numbers as binary numbers.

1. \((27.4375)_{10}\) %
2. \((5.625)_{10}\)
3. \((13.125)_{10}\)
4. \((11.1875)_{10}\)

Perform the following binary multiplications. % % %(a) \((1001000)_{2} \div ( 1000)_{2}\) %(b) \((101101)_{2} \div (1001)_{2}\) %(c) \((1001011000)_{2} \div (101000)_{2}\) %(d) \((1100000)_{2} \div (10000)_{2}\) % %

%

Perform the following binary divisions.

%(a) \((1001000)_{2} \div ( 1000)_{2}\) %(b) \((101101)_{2} \div (1001)_{2}\) %(c) \((1001011000)_{2} \div (101000)_{2}\) %(d) \((1100000)_{2} \div (10000)_{2}\)

%http://www.csgnetwork.com/binaddsubcalc.html (See section 1.1.3 of the text)

Perform the following binary additions.

1. \((110101)_{2}\) + \((1010111)_{2}\)
2. \((1010101)_{2}\) + \((101010)_{2}\)
3. \((11001010)_{2}\) + \((10110101)_{2}\)
4. \((1011001)_{2}\) + \((111010)_{2}\)

Perform the following binary subtractions.

1. \((110101)_{2}\) - \((1010111)_{2}\)
2. \((1010101)_{2}\) - \((101010)_{2}\)
3. \((11001010)_{2}\) - \((10110101)_{2}\)
4. \((1011001)_{2}\) - \((111010)_{2}\)

Perform the following binary multiplications.

1. \((1001)_{2}\times( 1000)_{2}\) % 9 by 8
2. \((101)_{2}\times(1101)_{2}\) % 5 by 11
3. \((111)_{2}\times(1111)_{2}\) % 7 by 15
4. \((10000)_{2}\times(11001)_{2}\)%16 by 25

Perform the following binary multiplications. % % %a. \((1001000)_{2} \div ( 1000)_{2}\) %b. \((101101)_{2} \div (1001)_{2}\) %c. \((1001011000)_{2} \div (101000)_{2}\) %d. \((1100000)_{2} \div (10000)_{2}\) % %

Which of the following binary numbers is the result of this binary division: \((10)_{2} \times ( 1101)_{2}\). % % (2) / (13)

1. \((11010)_{2}\) %26
2. \((11100)_{2}\) %28
3. \((10101)_{2}\) %21
4. \((11011)_2\) %27

Which of the following binary numbers is the result of this binary division: \((101010)_{2} \times( 111 )_{2}\). % (4) / (6)

1. \((11000)_{2}\) %24
2. \((11001)_{2}\) %25
3. \((10101)_{2}\) %21
4. \((11011)_2\) %27

Which of the following binary numbers is the result of this binary division: \((1001110)_{2}\times ( 1101 )_{2}\). % (9) / (3)

1. \((11000)_{2}\) %24
2. \((11001)_{2}\) %25
3. \((10101)_{2}\) %21
4. \((11011)_2\) %27

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Perform the following binary divisions. % % %a. \((1001000)_{2} \div ( 1000)_{2}\) %b. \((101101)_{2} \div (1001)_{2}\) %c. \((1001011000)_{2} \div (101000)_{2}\) %d. \((1100000)_{2} \div (10000)_{2}\) % %

Which of the following binary numbers is the result of this binary division: \((111001)_{2} \div ( 10011)_{2}\). % (57) / (19)

1. \((10)_2\) %2
2. \((11)_{2}\) %3
3. \((100)_{2}\) %4
4. \((101)_{2}\) %5

Which of the following binary numbers is the result of this binary division: \((101010)_{2} \div ( 111 )_{2}\). % (42) / (7)

1. \((11)_2\) %3
2. \((100)_{2}\) %4
3. \((101)_{2}\) %5
4. \((110)_{2}\) %6

Which of the following binary numbers is the result of this binary division: \((1001110)_{2} \div ( 1101 )_{2}\). % (78) / (13)

1. \((100)_{2}\) %4
2. \((110)_{2}\) %6
3. \((111)_{2}\) %7
4. \((1001)_2\) %9

#### *{Question 1}

* When is the positive integer \(p\) said to be a prime number? * Express the following hexadecimal number as a decimal number, and as a binary number: \[(A32.8)_{16}\]

* Convert the following decimal number into base 2, showing all your working: \((253)_{10}\). * Covert the decimal integer \((407)_{10}\) to binary notation.

* Showing your working, express the following number \[ 1.024024024024\ldots\] as a ration number in its simplest form. * Compute the following \(101101_2 + 1101_2\)

Hexadecimal Numbers - Worksheet 1 {data-navmenu=“Exercises”} ==================================

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