Numbers in Different Bases

 

People use base 10 to count. Many people aren't even aware that other number bases exist. However, you can count in any base you want. To understand other bases it helps to first understand base ten.

In base 10, we start at 0 and count till we reach 9. Then we carry one group of tens to the tens place and start the ones place over at 0. We keep doing this until we reach 9 tens and 9 ones (99). When we add one, we drop the ones place back to zero, but this means the tens place is full, so we carry one the "ten tens" (hundreds) place and drop the tens place back to zero. Base ten numbers can be interpreted like this.

234= 2 X 10² + 3 X 10¹ + 4 X 10⁰ = 200 + 30 + 4

Remember that 10⁰=1.

When you count in different bases, you start at 0 and go to one less than the base (we went to 9 before carrying). When you add one more, you carry and drop the preceding place back to 0. You can get the place value by raising the base to the appropriate power. Work your way from right to left and start with zero. A couple of illustrations may help.

To count in base four, you would count 0, 1, 2, 3, 10 (1 four and 0 ones), 11 (1 four and 1 one), 12, 13 (1 four and 3 ones), 20 (2 fours and 0 ones).

You can also think of it this way. In base four you would read this number like this:

123= 1 X 4² + 2 X 4¹ + 3 X 4⁰ = 16 + 8 + 3

That means 123 in base 4 equals 27 in base 10.

Sometimes the base is larger than 10. Since we don't have special symbols to stand for the bigger numbers, we use letters. In hexadecimal(base 16)  the number twenty three (23) is

23_base16 = 23₁₆ =  2x16¹ + 3 x 16⁰

Here is a look at some base 16 numbers

Base10	Base 16		To convert from base 16 to base 10	(*remember any base to the 0 power = 1)
1	1		1st: Write the number in expanded form:	23base16 = 2316 =  2x161 + 3 x 160 OR                     =     2x161 + 3 x 1
2	2
3	3		2nd: Do the multiplications	=  2x161 + 3 x 160               =      32  +  3
4	4
5	5		3rd: Add the values together - that is your base 10 value	= 32  + 3               = 35
6	6
7	7		Now You Try:   Change these base 16 numbers to base 10
8	8		1) 1c16 =
9	9		2) 2116 =
10	a		3) 10016 =
11	b		4) 8016 =
12	c		5) ad 16 =
13	d
14	e		To convert from base 10 to base 16		(*remember any base to the 0 power = 1)
15	f		1st: Take the number in base 10 and divide by 16  (keep remainder)		198 / 16 = 12   r 5
16	10
17	11
18	12
19	13		2nd Adjust numbers to "a…f " if needed		12   r5 -> c r5 because 12 = c
20	14
21	15		3rd Tack remainder into the ones place.		c r5 -> c5   198 = c516
22	16
23	17
24	18
25	19		Now You Try:   Change these base 10 numbers to base 16 (the first ones you can check on the list to see if you are right.) Show your work.
26	1a
27	1b
28	1c		1)  20   = _____16
29	1d
30	1e		2) 36     = _____16
31	1f
32	20		3) 40    = _____16
33	21
34	22		4) 80 =   ______16
35	23
36	24		5) 200 = ______ 16
37	25