Memory Size

Memory Size Table

Unit	Size (Bytes)
Bit (b)	1 bit or 1/8 byte
Nibble	4 bits
Byte (B)	8 bits
Kilobyte (KB)	1024 bytes
Megabyte (MB)	1024 KB
Gigabyte (GB)	1024 MB
Terabyte (TB)	1024 GB
Petabyte (PB)	1024 TB
Exabyte (EB)	1024 PB
Zettabyte (ZB)	1024 EB
Yottabyte (YB)	1024 ZB

Memory size used in x86-64 architecture

Storage (Unit)	Size(bits)	Size(bytes)
Byte	8-bits	1 byte
Word	16-bits	2 bytes
Double-word	32-bits	4 bytes
Quardword	64-bits	8 bytes
Double quardword	128-bits	16 bytes

Memory size in C/C++

C/C++ Declaration	Storage (Unit)	Size (bits)	Size (bytes)
char	Byte	8 bits	1 byte
short	Word	16 bits	2 bytes
int	Double-word	32 bits	4 bytes
unsigned int	Double-word	32 bits	4 bytes
long1	Quadword	64 bits	8 bytes
long long	Quadword	64 bits	8 bytes
int *2	Quadword	64 bits	8 bytes
char *	Quadword	64 bits	8 bytes
float	Double-word	32 bits	4 bytes
double	Quadword	64 bits	8 bytes

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Addition of Bits

Example:
You’re adding:

  00110111   (which is 55 in decimal)
+ 00000001   (which is 1 in decimal)
------------
= 00111000   (which is 56 in decimal)

Let’s Add Bit by Bit (Right to Left):

Bit position:  7 6 5 4 3 2 1 0
               0 0 1 1 0 1 1 1   ← 00110111
           +   0 0 0 0 0 0 0 1   ← 00000001
           -------------------
               0 0 1 1 1 0 0 0   ← 00111000

Explanation of the rightmost bits:

• 1 + 1 = 10 → you write down 0, carry over 1
• 1 + 0 + carry 1 = 10 → write down 0, carry 1
• 1 + 0 + carry 1 = 10 → write down 0, carry 1
• 0 + 0 + carry 1 = 1 → write 1, carry 0
• Remaining bits add up normally.

---

7-bit integer encoding

It is also known as Base-127 Varint encoding

Chunk Position	Continuation Bit (MSB)
First (LSB side)	1 if more follow
Last (MSB side)	Always 0

Value Range	Bytes Used	MSB of Each Byte
0 – 127	1 byte	0xxxxxxx (MSB = 0)
128 – 16383	2 bytes	First: 1xxxxxxx, Second: 0xxxxxxx
>16383	3+ bytes	All but last: 1xxxxxxx, Last: 0xxxxxxx

1. Convert 300 to binary.
2. Break into 7-bit chunks.
3. Add continuation bits

Example: Encoding integer 300

Step 1: Convert 300 to binary.

• 00000001 00101100 → 16-bit
• =0b100101100

Step 2: Break into 7-bit chunks.

• 00 0000010 0101100
• $1^{st}$ chunk = 0101100
• $2^{nd}$ chunk = 0000010

Step 3: Add continuation bits
Converting chunks to 8-bit and then adding 1 to MSB, to the chunk who follows more bytes and 0 to MSB, to the chunk who follows least bytes.

$1^{st}$ chunk = 0101100
= 00101100
$2^{nd}$ chunk = 0000010
= 00000010

Adding 1 to $1^{st}$ chunks MSB and 0 to $2^{nd}$ chunks MSB.

$1^{st}$ chunk = 00101100 + 1
= 10101100
= 0xAC → Hexadecimal
$2^{nd}$ chunk = 00000010 + 0
= 00000010
= 0x02 → Hexadecimal

Result:
Encoded as bytes:

[0xAC, 0x02]

Or in binary:

10101100 00000010

Conversion

Integer : $1,2,6,55,58,106,etc.$
Decimal : $0.5,1.75,12.345,0.001,etc.$
Binary : $10101,10001,10100011,1001,etc.$
Hexadecimal : $0x1A,0x2F,0xFF,0x100,etc.$

Integer to Binary

To convert a decimal number to binary:

1. Divide the number by 2.
2. Record the remainder.
3. Repeat the process with the quotient until you reach 0.
4. The binary representation is the remainders read in reverse order.

Example: Convert 13 to binary.

Division Step	Quotient	Remainder
13 / 2	6	1
6 / 2	3	0
3 / 2	1	1
1 / 2	0	1

Binary of 13: 1101

Binary to Integer

To convert a binary number to decimal:

1. Multiply each bit by 2 raised to its position (from right, starting at 0).
2. Sum the results.

Example: Convert 1101 to decimal.

$$1\times 2^3+1\times 2^2+0\times 2^1+1\times 2^0$$ $$8+4+0+1=13$$

Decimal of 1101: 13

Integer to Hexadecimal

Hex Table (0–15):

Decimal	Hex
0	0
1	1
…	…
10	A
11	B
12	C
13	D
14	E
15	F

To convert a decimal to hexadecimal:

1. Divide the decimal number by 16 till Quotient < 16.
2. Convert Quotient and Reminders values to hex digits.
3. Put them together.

Example: Convert 65 to Hexadecimal.

Step 1:

• 65 ÷ 16 = 4 remainder 1

This gives:

• Quotient = 4
• Remainder = 1

Step 2:

• Quotient: 4 → hex digit is 4
• Remainder: 1 → hex digit is 1

Step 3:

• So, 65 in decimal = 0x41 in hex

Exercise:

1. Convert 10 to hexadecimal. → Answer

2. Convert 15 to hexadecimal. → Answer

3. Convert 31 to hexadecimal. → Answer

4. Convert 64 to hexadecimal. → Answer

5. Convert 127 to hexadecimal. → Answer

6. Convert 255 to hexadecimal. → Answer

7. Convert 1023 to hexadecimal. → Answer

8. Convert 4096 to hexadecimal. → Answer

9. Convert 12345 to hexadecimal. → Answer

10. Convert 65535 to hexadecimal. → Answer

Hexadecimal to Integer

To convert a hexadecimal number to decimal, follow this simple method:

1. Write the hexadecimal number.
2. Replace each hex digit with its decimal equivalent.
3. Multiply each by 16 raised to its positional power.
4. Add all the products together.

Take each digit, multiply it by 16 raised to the power of its position (counting from right to left, starting at 0), and sum the results.

Example: Convert 2F to decimal

= (2 × 16¹) + (F × 16⁰)
= (2 × 16) + (15 × 1)
= 32 + 15
= 47

Example: Convert 3A7 to decimal

= (3 × 16²) + (A × 16¹) + (7 × 16⁰)
= (3 × 256) + (10 × 16) + (7 × 1)
= 768 + 160 + 7
= 935

Exercise:

1. Convert hexadecimal A to decimal.

2. Convert hexadecimal 1F to decimal.

3. Convert hexadecimal 3C to decimal.

4. Convert hexadecimal 7E to decimal.

5. Convert hexadecimal FF to decimal.

6. Convert hexadecimal 100 to decimal.

7. Convert hexadecimal 1A3 to decimal.

8. Convert hexadecimal 2F7 to decimal.

9. Convert hexadecimal 3E8 to decimal.

10. Convert hexadecimal FFFF to decimal.

Decimal to Binary

To convert a decimal integer to binary:

1. Divide the number by 2.
2. Record the remainder (it will be 0 or 1).
3. Divide the quotient again by 2.
4. Repeat until the quotient is 0.
5. The binary number is the remainders read from bottom to top.

Example: Convert 10.625

Divide the numbers into two part.
$10+0.625=10.625$

Now, find binary for 10 and 0.625.

10 = 1010 Refer: Integer to Binary Conversion

0.625 × 2 = 1.25 → 1
0.25  × 2 = 0.5  → 0
0.5   × 2 = 1.0  → 1

Binary of 0.625 = .101

So, 10.625 (decimal) = 1010.101 (binary)

Example: Convert 7.77

Divide the numbers into two part.
$7+0.77=7.77$

Now, find binary for 7 and 0.77.

7 = 0111 Refer: Integer to Binary Conversion

Binary to Decimal

To convert a binary number to decimal:

1. Separate the integer and fractional parts (if any).
2. For the integer part, multiply each bit by $2^n$, where n is the position from right to left, starting at 0.
3. For the fractional part, multiply each bit by $2^{-n}$, where n starts at 1 and increases to the right.
4. Add all the values together.

Example: Convert 1010.101 to Decimal

Divide the binary into two parts.
1010 + 0.101 = 1010.101

Now, convert both parts to decimal.

Integer Part: 1010

(1 × 2³) + (0 × 2²) + (1 × 2¹) + (0 × 2⁰)  
= 8 + 0 + 2 + 0 = 10

Fractional Part: .101

(1 × 2⁻¹) + (0 × 2⁻²) + (1 × 2⁻³)  
= 0.5 + 0 + 0.125 = 0.625

So, 1010.101 (binary) = 10.625 (decimal)

Example: Convert 111.110001 to Decimal

Divide the binary into two parts.
111 + 0.110001 = 111.110001

Now, convert both parts to decimal.

Integer Part: 111

(1 × 2²) + (1 × 2¹) + (1 × 2⁰)  
= 4 + 2 + 1 = 7

Fractional Part: .110001

(1 × 2⁻¹) + (1 × 2⁻²) + (0 × 2⁻³) + (0 × 2⁻⁴) + (0 × 2⁻⁵) + (1 × 2⁻⁶)  
= 0.5 + 0.25 + 0 + 0 + 0 + 0.015625  
= 0.765625

So, 111.110001 (binary) = 7.765625 (decimal)

Footnotes

1. The long type declaration is compiler dependent. Type shown is for gcc and g++ compilers. ↩

2. int * means the address of an integer. ↩