Number System Conversion

Number systems

• Binary number system (Radix-2): 0 and 1
• Octal number system (Radix-8): 0 to 7
• Decimal number system (Radix-10): 0 to 9
• Hexadecimal (Hex) number system (Radix-16): 0 to 9, A to F

Decimal to other base

1. Divide the decimal number by the value of new base
2. Get the remainder from step 1 as the rightmost (least significant) digit of the new base
3. Divide the quotient of the previous by the new base
4. Get the remainder from step 3 as the next digit Repeat step 3 and 4 until the quotient becomes zero

Example

Convert from decimal to binary: Decimal number: 29${}_{10}$

Step	Operation	Result	Remainder
Step 1	29 / 2	14	1
Step 2	14 / 2	7	0
Step 3	7 / 2	3	1
Step 4	3 / 2	1	1
Step 5	1 / 2	0	1

The resulting binary is 11101${}_2$

Other base to decimal

1. Determine the column (positional) value of each digit
2. Multiply the column value by the digits in the corresponding columns
3. Sum the products calculated in step 2

Example

Convert the binary number to decimal: Binary number: 11101${}_2$

Step	Decimal Number
Step 1	((1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20))
Step 2	16 + 8 + 4 + 0 + 1
Step 3	29

The resulting decimal is 29${}_{10}$

Other base to non-decimal

1. Convert the original number to decimal number
2. Convert the decimal number obtained to new base number

Shortcut - Binary to octal

1. Split binary digits into group of three (start from the right)
2. Convert each group of three binary digits to one octal digit

Example

Convert the binary number to octal: Binary number: 10101${}_2$

Step	Binary Number	Octal Number
Step 1	10101${}_2$	010 101
Step 2	10101${}_2$	2${}_8$ 5${}_8$
Step 3	10101${}_2$	25${}_8$

The resulting octal is 25${}_8$

Shortcut - Octal to binary

1. Convert each octal digit to three binary number (you may treat octal as binary here)
2. Combine all resulting binary groups into a single binary number

Example

Convert the octal to binary: Octal number: 25${}_8$

Step	Octal Number	Binary Number
Step 1	25${}_8$	2${}_{10}$ 5${}_{10}$
Step 2	25${}_8$	010${}_2$ 101${}_2$
Step 3	25${}_8$	010101${}_2$

The resulting binary is 010101${}_2$

Shortcut - Binary to hex

1. Split binary digits into groups of four (start from the right)
2. Convert each group to one hex symbol (you may treat them as decimal)

Example

Convert the binary to hexadecimal: Binary number: 10101${}_2$

Step	Binary Number	Hexadecimal Number
Step 1	10101${}_2$	0001 0101
Step 2	10101${}_2$	1${}_{10}$ 5${}_{10}$
Step 3	10101${}_2$	15${}_{16}$

The resulting hex is 15${}_{16}$

Shortcut - Hex to binary

1. Convert each hex symbol to a four-digit binary (you may treat hex as decimal here)
2. Combine all resulting binary groups into a single binary digit

Example

Convert the hex to binary: Hexadecimal number: 15${}_{16}$

Step	Hexadecimal Number	Binary Number
Step 1	15${}_{16}$	1${}_{10}$ 5${}_{10}$
Step 2	15${}_{16}$	0001${}_2$ 0101${}_2$
Step 3	15${}_{16}$	00010101${}_2$

The resulting binary is 00010101${}_2$

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Operating_system INFO1112 Data_representation Number_system_conversion Binary Hex Hexadecimal