# Lecture 2 Bits, Bytes & Number systems

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Lecture 2 Bits, Bytes & Number systems

Representation of Numbers

Different ways to say “how many”…

Human: decimal number system

Radix-10 or base-10 Base-10 means that a digit can have one of ten possible values

• 0 through 9.

Computer: binary number system

Radix-2 or base-2 Why binary? Each digit can have one of two values

• 0 or 1

Bits and Bytes

A binary digit is a single numeral in a binary number.

Each 1 and 0 in the number below is a binary digit: 1 0 0 1 0 1 0 1

The term “binary digit” is commonly called a “bit.”

Eight bits grouped together is called a “byte.”

Relationship between Decimal & Binary

Background:

Number systems are positional

There are 10 symbols that represent decimal quantities

Multi-digit numbers are interpreted as in the following example

79310

= 7 x 100 + 9 x 10 + 3

= 7 x 102 + 9 x 101 + 3 x 100

Remember that the position index starts from 0.

Each place value in a decimal number is a power of 10.

We can get a general form of this

ABCbase A x (base)2 + B x (base)1 + C x (base) 0

Indicate positions

Relationship between Decimal & Binary

Binary numbers are represented using the digits 0 and 1. Multi-digit numbers are interpreted as in the following example 101112

= 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20 = 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 1 x 1

Each place value in a binary number is a power of 2.

Converting binary numbers to decimal

Step 1: Starting with the 1’s place, write the binary place value over each digit in the binary number being converted.

Step 2: Add up all of the place values that have a “1” in them.

Interpret the binary number 101012 in decimal

TRY: Interpret the binary number 010101102 in decimal

Converting decimal numbers to binary

We have now learnt how to convert from binary to decimal

Using positional representation

But how about decimal to binary

Repeated division method

• Simply keep dividing it by 2 and record the remainder • Repeat above step as many times as necessary until you get a

quotient that can’ t be divided by 2 • Remainders give the binary digits, starting from the last

remainder

Let’s look at some examples…

Converting decimal numbers to binary

Let’s convert decimal 23 to binary.

Step 1: 23/2 = 11 remainder 1 Step 2: 11/2 = 5 remainder 1 Step 3: 5/2 = 2 remainder 1 Step 4: 2/2 = 1 remainder 0

The last quotient “1” cannot be divided by 2 any more. So the process ends. The final binary number is read from the very end including the last quotient: 1 0 1 1 1

Try: Convert decimal 73, 96, 127, 128 to binary.

Hexadecimal

Computers use binary number system because of the electric voltage (high or low voltage)

Very difficult to express for large number representation

Hexadecimal to rescue

Hexadecimal system is interface between human brain and computer brain

4 bits from binary are read together and represented using a single digit

• Such 4-bits are known as nibble • This gives a total of 16 different options

The hexadecimal number system is a Base-16 number system: There are 16 symbols that represent quantities:

Represented by the symbols 0-9 and A-F where the letters represent values: A=10, B=11, C=12, D=13, E=14, and F=15

Numbering systems

Decimal

0 1 2 3 4 5 6 7

Hexadecimal

0 1 2 3 4 5 6 7

Binary

0000 0001 0010 0011 0100 0101 0110 0111

Decimal

8 9 10 11 12 13 14 15

Hexadecimal

8 9 A B C D E F

Binary

1000 1001 1010 1011 1100 1101 1110 1111

Representation of Numbers

Different ways to say “how many”…

Human: decimal number system

Radix-10 or base-10 Base-10 means that a digit can have one of ten possible values

• 0 through 9.

Computer: binary number system

Radix-2 or base-2 Why binary? Each digit can have one of two values

• 0 or 1

Bits and Bytes

A binary digit is a single numeral in a binary number.

Each 1 and 0 in the number below is a binary digit: 1 0 0 1 0 1 0 1

The term “binary digit” is commonly called a “bit.”

Eight bits grouped together is called a “byte.”

Relationship between Decimal & Binary

Background:

Number systems are positional

There are 10 symbols that represent decimal quantities

Multi-digit numbers are interpreted as in the following example

79310

= 7 x 100 + 9 x 10 + 3

= 7 x 102 + 9 x 101 + 3 x 100

Remember that the position index starts from 0.

Each place value in a decimal number is a power of 10.

We can get a general form of this

ABCbase A x (base)2 + B x (base)1 + C x (base) 0

Indicate positions

Relationship between Decimal & Binary

Binary numbers are represented using the digits 0 and 1. Multi-digit numbers are interpreted as in the following example 101112

= 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20 = 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 1 x 1

Each place value in a binary number is a power of 2.

Converting binary numbers to decimal

Step 1: Starting with the 1’s place, write the binary place value over each digit in the binary number being converted.

Step 2: Add up all of the place values that have a “1” in them.

Interpret the binary number 101012 in decimal

TRY: Interpret the binary number 010101102 in decimal

Converting decimal numbers to binary

We have now learnt how to convert from binary to decimal

Using positional representation

But how about decimal to binary

Repeated division method

• Simply keep dividing it by 2 and record the remainder • Repeat above step as many times as necessary until you get a

quotient that can’ t be divided by 2 • Remainders give the binary digits, starting from the last

remainder

Let’s look at some examples…

Converting decimal numbers to binary

Let’s convert decimal 23 to binary.

Step 1: 23/2 = 11 remainder 1 Step 2: 11/2 = 5 remainder 1 Step 3: 5/2 = 2 remainder 1 Step 4: 2/2 = 1 remainder 0

The last quotient “1” cannot be divided by 2 any more. So the process ends. The final binary number is read from the very end including the last quotient: 1 0 1 1 1

Try: Convert decimal 73, 96, 127, 128 to binary.

Hexadecimal

Computers use binary number system because of the electric voltage (high or low voltage)

Very difficult to express for large number representation

Hexadecimal to rescue

Hexadecimal system is interface between human brain and computer brain

4 bits from binary are read together and represented using a single digit

• Such 4-bits are known as nibble • This gives a total of 16 different options

The hexadecimal number system is a Base-16 number system: There are 16 symbols that represent quantities:

Represented by the symbols 0-9 and A-F where the letters represent values: A=10, B=11, C=12, D=13, E=14, and F=15

Numbering systems

Decimal

0 1 2 3 4 5 6 7

Hexadecimal

0 1 2 3 4 5 6 7

Binary

0000 0001 0010 0011 0100 0101 0110 0111

Decimal

8 9 10 11 12 13 14 15

Hexadecimal

8 9 A B C D E F

Binary

1000 1001 1010 1011 1100 1101 1110 1111

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