Decimal

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DECIMAL

This short tutorial explains how to convert Decimal between different bases, including binary.

Converting decimal to binary (or another base):

Converting a base 10 number to binary (base 2) is very straightforward. You repeatedly divide the number by 2, each time recording the remainder. At the end of this line up all of the remainders and reverse their order. For example:

CONVERTING DECIMAL TO BINARY

Converting 20 (dec) to binary - step 1

20/2 = 10 (+ remainder 0)
10/2 = 5 (+ remainder 0)
5/2 = 2 (+ remainder 1)
2/2 = 1 (+ remainder 0)
1/2 = 0 (+ remainder 1)

Once the answer (without remainder) equals 0, stop. The remainders from each division were: 0,0,1,0,1, reverse these digits to get the result. 20dec = 10100bin

By replacing 2 with a different number, one can convert to that base. Here's an example showing 14dec being converted to base 8 (octal):

CONVERTING DECIMAL TO OCTAL

Converting 14dec to octal

14/8 = 1 (6 remainder)
1/8 = 0 (1 remainder)

The remainders were 6,1, reversed this is 16

14dec = 16oct

So what about hexadecimal? The process is the same, but with one extra thing to keep in mind: hexadecimal is base 16, and hence makes use of 6 extra symbols (ABCDEF). A = 10, and F, 15. 16 is written as 10. When converting to any base above 10 you must keep in mind that a remainder above equal to or greater than 10 has to be written using the correct symbol.

CONVERTING DECIMAL TO HEXADECIMAL

Converting 1446dec to hexadecimal

1446/16 = 90 (6 remainder)
90/16 = 5 (10 remainder)
5/16 = 0 (5 remainder)

The remainders were 6,10,5.

10 should be written as A. Reversed, this equals 5A6

1446dec = 5A6hex

CONVERTING BINARY TO DECIMAL

Converting binary (or another base) to decimal:

A decimal number can be represented as 100s, 10s and units, so 234 is 2 hundreds, 30 tens and 4. Each digit in a number has a place value and a face value. The face value is a number's actual value, while the place value is a digit's value in the base to which it belongs, and depends on its position in the number. These are illustrated in the next example:

Place values:

Number: 1589

Number: 1 5 8 9
Face value: 1 5 8 9
Place value: 1000 100 10 1

Knowing this, it is possible to convert between bases easily. The next example converts 11011bin to decimal:

Step 1: find the place value and face value of each digit:

Digit: 1 1 0 1 1
Face value: 1 1 0 1 1
Place value: 16 8 4 2 1

Step 2: Multiply each digit's face value by its place value and add all together:

(1*16)+(1*8)+(0*4)+(1*2)+(1*1)
16+8+2+1 = 27

11011bin = 27dec

It may seem a bit unnecessary to bother with face values for binary conversion, however with higher bases you can see why these are important. The next example shows how to convert 4C9hex to decimal:

Digit : 4 C 9
Face value : 4 12 9 (Reminder: C means 12 - A=10,B=11,C=12,D=13,E=14,F=15)
Place value: 256 16 1

(256*4)+(16*12)+9
1024+192+9 = 1225

4C9h = 1225d

CONVERTING DECIMAL TO HEXADECIMAL

Steps:

Divide the decimal number by 16. Treat the division as an integer division.
Write down the remainder (in hexadecimal).
Divide the result again by 16. Treat the division as an integer division.
Repeat step 2 and 3 until result is 0.
The hex value is the digit sequence of the remainders from the last to first.

Note: a remainder in this topic refers to the left over value after performing an integer division.

Example 1
Convert the number 1128 DECIMAL to HEXADECIMAL

NOTES	DIVISION	RESULT	REMAINDER (in HEXADECIMAL)
Start by dividing the number by 16. In this case, 1128 divided by 16 is 70.5. So the integer division result is 70 (throw out anything after the decimal point). The remainder is (70.5 - 70) multiplied with 16; or (0.5 times 16), which is 8.	1128 / 16	70	8
Then, divide the result again by 16 (the number 70 on the DIVISION column comes from the previous RESULT). In this case, 70/16=4.375. So the integer division result is 4 (throw out anything after the decimal point) The remainder is (0.375 multiplied with 16, which is 6.	70 / 16	4	6
Repeat. Note here that 4/16=0.25. So the integer division result is 0. The remainder is (0.25-0) multiplied with 16, which is 4.	4 / 16	0	4
Stop because the result is already 0 (0 divided by 16 will always be 0)
Well, here is the answer. These numbers come from the REMAINDER column values (read from bottom to top)			468

Side note: You can get the remainder of a division using the Modulus or % operator. Ie: 1128%16=8.

Example 2
Convert the number 256 DECIMAL to HEXADECIMAL

DIVISION	RESULT	REMAINDER (in HEX)
256 / 16	16	0
16 / 16	1	0
1 / 16	0	1

ANSWER		100

Example 3
Convert the number 921 DECIMAL to HEXADECIMAL

DIVISION	RESULT	REMAINDER (in HEX)
921 / 16	57	9
57 / 16	3	9
3 / 16	0	3

ANSWER		399

Example 4
Convert the number 188 DECIMAL to HEXADECIMAL

DIVISION	RESULT	REMAINDER (in HEX)
188 / 16	11	C (12 decimal)
11 / 16	0	B (11 decimal)

ANSWER		BC

Note that here, the answer would not be 1112, but BC. Remember to write down the remainder in hex, not decimal.

Example 5
Convert the number 590 DECIMAL to HEXADECIMAL

DIVISION	RESULT	REMAINDER (HEX)
590 / 16	36	E (14 decimal)
36 / 16	2	4 (4 decimal)
2 / 16	0	2 (2 decimal)

ANSWER		24E