# Convert between basic number systems

This post is lesson 7 of 42 in the subject Computer Hardware

This article will help you know how to convert between the number systems. The first thing to pay attention to is how to convert from base 10 to any base system like 2, 8, 16 and vice versa. In addition, how to convert between the base systems 2, 8, 16 will also be mentioned.

## 1. Convert base b to base 10

• A(b) = A’(10)
• A’(10) = anbn + an-1bn-1 +…+ a0b0 + a-1b-1 +…+ a-mb-m
• Example:

(6 5  4  3  2  1  0  -1 -2  -3  -4) is the exponent

1 1  0  1  0  0  1 . 1  0  1  1(2)

= 1.26 + 1.25 + 0.24 + 1.23 + 0.22 + 0.21 + 1.20 + 1.2-1 + + 0.2-2 + 1.2-3 + 1.2-4

= 64 + 32 + 8 + 1 + 0.5 + 0.125 + 0.0625

= 105.6875(10)

## 2. Convert base 10 to base b

Whole-number part conversion: Take the whole-number part of N(10) and divide by b to get quotient and remainder. Then take quotient divide by b to get quotient and remainder, repeating this operation until the quotient is 0. The result of the conversion M(b) is the remainders in the division written out in reverse order.

Decimal part conversion: Take the decimal part (after the comma) of N(10) and multiply by b to get whole-number part and decimal part. Then take decimal part multiply by b to get whole-number part and decimal part, repeating this operation until the decimal part is 0. The result of the conversion number M(b) is the integer part numbers in the multiplication, are written in order of multiplication.

## 3. How to convert between base systems 2, 8, 16

### 3.1. Convert from base 2 to base 8

Rule: A sequence of 3 consecutive bits from right to left in base 2 will correspond to a number in base 8.

Example: 111111111110(2) = ?(8)

Gather 3 bits from right to left (111)(111)(111)(110) to get (7) (7) (7) (6)

Result: 111111111110(2) = 7776(8)

### 3.2. Convert from base 2 to base 16

Quy tắc: A sequence of 4 consecutive bits from right to left in base 2 will correspond to a number in base 16.

Example: 11111111111111111110(2) = ?(16)

Gather 4 bits from right to left (1111)(1111)(1111)(1111)(1110) to get (15) (15) (15) (15) (14)

Result: 11111111111111111110(2) = FFFFE(16)

### 3.3. Convert from base 8 to base 2

Rule: A digit in base 8 will correspond to a sequence of 3 consecutive bits in base 2.

Example: 27(8) = ? (2)

Convert each digit (2) (7) into 3 consecutive bit (010) (111)

Result: 27(8) = 010111(2)

### 3.4. Convert from base 16 to base 2

Rule: A digit in base 16 will correspond to a sequence of 4 consecutive bits in base 2.

Example: 27(16) = ? (2)

Convert each digit (2) (7) into 4 consecutive bit (0010) (0111)

Result: 27(16) = 00100111(2)

### 3.5. Convert from base 8 to base 16 and vice versa

The most simple way is to change base 8 to base 2 and then change to base 16.

Example: 27(8) = ? (16)

Convert each digit (2) (7) into 3 consecutive bit (010) (111). We get binary sequence 010111(2). Convert to base 16 by gathering 4 bits from right to left (0001)(0111) to get (1)(7).

Result: 27(8) = 00010111(2) = 17(16)

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