Conversion to Decimal

We can use following formula:

$$N_{(base\ r)} = \sum_{i=0}^{n-1} d_i r^i$$

where,
N is the number with Radix r
n is the total number of digits in number N
d_i is the value of digit at position i

Let"s try with converting Decimal to Decimal :) :

Here,
N = 2024
r = 10
d₀ = 4
d₁ = 2
d₂ = 0
d₃ = 2

So,
$$2024_{(base\ 10)} = 4 * 10^0 + 2 * 10^1 + 0 * 10^2 + 2 * 10^3$$
$$= 4 * 1 + 2 * 10 + 0 * 100 + 2 * 1000$$
$$= 4 + 20 + 0 + 2000 = 2024_{(base\ 10)}$$

Trying with binary,

Here,
N = 1101
r = 2
d₀ = 1
d₁ = 0
d₂ = 1
d₃ = 1

So,
$$1101_{(base\ 2)} = 1 * 2^0 + 0 * 2^1 + 1 * 2^2 + 1 * 2^3$$
$$= 1 * 1 + 0 * 2 + 1 * 4 + 1 * 8$$
$$= 1 + 0 + 4 + 8 = 13_{(base\ 10)}$$

For binary, another way to convert is to add the value of 2 to the power of the position where the value is 1.

1	1	0	1
d3	d2	d1	d0
23	22	-	20
8	4	-	1

adding all of them gives us 13. We ignored d₁ because the value at that position is 0.

Let"s try with converting Hexadecimal to Decimal :

Here,
N = 6D1A
r = 16
d₀ = A
d₁ = 1
d₂ = D
d₃ = 6

So,
$$6D1A_{(base\ 16)} = A * 16^0 + 1 * 16^1 + D * 16^2 + 6 * 16^3$$
Converting A and D to their decimal equivalent:
$$= 10 * 16^0 + 1 * 16^1 + 13 * 16^2 + 6 * 16^3$$
$$= 10 * 1 + 1 * 16 + 13 * 256 + 6 * 4096$$
$$= 10 + 16 + 3328 + 24576 = 27930_{(base\ 10)}$$

Converting Decimal to other number system

• Take the decimal number D
• Divide by radix R
• Note down the remainder, r
• Divide the quotient q, with radix R again and again note down the remainder
• Repeat this until $q<R$
• Write down the remainders bottom-up from least significant to most significant digit

Example, converting 19153 to Hexadecimal:

Number	Division	Quotient	Remainder
19153	$19153/16 = 1197$, r=1	1197	1
1197	$1197/16 = 74$, r=13	74	13
74	$74/16 = 4$, r=10	4	10
4	$4/16 = 0$, r=4	0	4

Stopped as r < R, remainder is less than radix

Writing the remainders from bottom to top, from last to first:

4 : 10 : 13 : 1 = 4AD1_{base 16}

Conversion from Hexadecimal to Binary

Simply convert the individual digits to their binary equivalent, also add extra 0s so that the binary equivalent number becomes 4 digit one.

4	A	D	1
0100	1010	1101	0001

and combining them, we get
(4AD1)₁₆ = (0100101011010001)₂

We can remove the first zero, so it becomes (100101011010001)₂

If we want to convert binary to hexadecimal, do the reverse, but first break the binary number to group of 4 digits starting from the least significant bit, add the number of 0s in the group where the number of digits in less than 4 to complete it, and then write the binary equivalent to it.

Taking the above example, to convert (100101011010001)₂ to hexadecimal:

0100	1010	1101	0001
4	A	D	1

and combining them, we get
(0100101011010001)₂ = (4AD1)₁₆

Here, we broke the bits into groups of 4 from the least significant bit(right to left), and added a 0 to make 100 4-digit one, 0100, and then converted those groups to hexadecimal.