From binary to hexadecimal now, eh? Well, converting hexadecimal to decimal is quite different from converting binary to hexadecimal, yet is a bit similar.

First thing first, you need to memorize this:

1=1
2=2
3=3
4=4
5=5
6=6
7=7
8=8
9=9
A=10
B=11
c=12
D=13
E=14
F=15

This is the first fifteen numbers in hexadecimal. These will be important to know for converting hexadecimal into decimal and vice versa.

First: Hexadecimal into decimal

This is quite similar to binary in converting it into decimal, but instead of base-2, we're dealing with base-16. Therefore, we would have x(16^n). If you read the earlier one about conversion of binary, this formula is very reminiscent of the one found there. For newcomers:

x-the number that the hexadecimal stands for.

n: the distance from the last bit...

Similar to binary, hexadecimal also has a least significant bit of sorts, which is the last number or letter to be in the sequence. So for example:

FA0

The 0 would be the last bit. Now, remember that list at the beginning? This is where it comes into play. See, A will stand for 10 in the x in the equation above, since it is 1 from the least bit, therefore:

10*(16^1)=160

Then 15*(16^2)=3840

Add the two together: 3840+160=4000

Simple?

Now for decimal to hexadecimal

This is where things get a bit tricky. Instead of doing the number-16^(x), we are instead dividing by 16 and multiplying the decimal places by sixteen to gain the hexadecimal value. For example back to 4000 since we used it


4000/16, which equals 250. Since this technically equal 250.0, we would times sixteen by the .0, giving us zero, which would be the least significant bit. Then we divide 250 by 16, which would equal 15.625. Now, we times 0.625 by sixteen, giving us 10. Now ten on our chart equals A. Finally, we divide by 15, not 15.625 since we already used the .625 part and we get 0.9375 times this by 16 and you will get fifteen. (Yes, I realize the redundancy as I was typing this explaining it.) In any case, fifteen will be symbolize by F. So if we work from down to up from this sentence, we have FA0, which was what we begin with in the first place!

Don't you just love circular processes?

Any questions?

Now, since I talked about how to convert binary to decimal and vice versa, let's talk about adding in binary.

Okay so let's say we have 1001111+111. Now, you can either convert the numbers to decimal, but let's say we didn't have time or the test requires you to keep everything in binary. So what are we going to do?

Well as stated before in the converting lesson, binary uses 1 and 0's and no other numbers. Therefore, if one adds binary together and get's something like 2, one should carry it over, just how one carries over the one when adding 15+6.

1001111 +
111

The least bit would add to 2. Now, since binary is composed of two numbers, subtract two from the number to get the sum, which would be zero.
Carry over the one. The next bit will add to three. Again, subtract two and you'll get one. Carry over another one and continue the pattern.

1001111+111=1010110

Note: if you want to check, 111 should equal 7, 1001111 equals 79, 1010110 should equal 86.

1001111+111=

Learning binary will probably be a very important skill for high school students to learn, especially if said high school student is thinking about taking a physics I.B. test or any science test in general. With that said, it is very hard to learn and perhaps even harder to explain, but I will try my best.

Converting: Binary to Decimal

So as one knows, binary numbers consists of 1's and 0's of varying lengths. The last bit in the sequence is known as the least significant bit.

10010001 ---Least significant bit is one

The last significant bit is, unlike what the name might suggest, very important for it serves as the placeholder of sorts for the number, with numbers stretching to the left of the bit increasing in value and decreasing in value the farther right from the least significant figure. Now, you may just be wondering what do I mean by increasing and decreasing in value. Well this is the formula that will be key converting binary into decimal.

n(2^(x))

Where

n represents the bit (whether the number is one or zero)

x represents the bit's position from the least significant bit.

So for example, 10010001, the least significant bit would be 1(2^0) or 1, since 2^0=1. Note: It is not one, it is zero for...Well an analogy is that you can't say you're away from the house if you just move to another room in the house. Next, you do the rest of them, so 10010001 1(2^4)=16 and 10010001 1(2^7)=128. Finally add the results together. 16+128+1= 145. Confusing at first, but it becomes easier over time.

Converting Decimal to Binary

Need to convert to decimal to binary, eh? Well, that is hard as well. But, if you reverse engineer the process above, it becomes easier. For example, let's say you have a number like ten. First, ask yourself "What number can be expressed in the form 2^x, can be subtracted from ten, and is the highest without making it negative. Or...

10-2^?=/= 0> -n

The number would be eight or 2^3. If we look at the above process, that would mean that the number is 3 places from the least significant bit or 1??B, where b represents the least significant bit. Also note, that since binary uses only 1 and 0 and 0 serves as an un-active bit, one would be used. In any case, one would use 10-8 and that would equal two. Since two is only 2^1, that would mean that is one spot from the least significant bit, meaning that it would be 1?1B. Now, you(well some of you) are probably wondering 'We ran out of numbers! We can't figure out the least significant bit! And we didn't even figure out one of the earlier bits.' Well, think of it as this way: Since we didn't figure out those bits during the reverse engineering process, that would mean that they are 'un-active.' Thus, you would put zero for the question mark and the bit. Or...

1010

And that is all there is to it. Hard to understand, probably. Especially with my way of explaining things. But, if you practice this technique, it will become easier.

Bonus: How to convert binary to fractions

Okay so, I mentioned earlier that when one moves to the right from the least significant bit, the more negative in value. So let's say we have 0.0101(decimal point is to help differentiate the least significant bit). In a similar way with converting binary to decimal, find out the position of the number from the least significant number. So for the first one, it is two places from the least significant bit. Now as aforementioned, the position number would be negative. Therefore,

(2^(-2))*1=1/4.

So 0.0101 in binary would be 1/4 + 1(2^(-4))= 5/16

... So yeah...

---

By: so64 | On: Mon Jun 08, 09 1:35 PM

2 Comments | Post A Comment