Today, we got lesson about Binary Number system. We also learn how to calculate the binary number system…^_^ This is what I can simplified from what I learn and little bit instruction to calculate.
Let's look at base-two, or binary, numbers. How would you write, for instance, 1210 ("twelve, base ten") as a binary number? You would have to convert to base-two columns, the analogue of base-ten columns. In base ten, you have columns or "places" for 100 = 1, 101 = 10, 102 = 100, 103 = 1000, and so forth. Similarly in base two, you have columns or "places" for 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24= 16, and so forth.
The first column in base-two math is the units column. But only "0" or "1" can go in the units column. When you get to "two", you find that there is no single solitary digit that stands for "two" in base-two math. Instead, you put a "1" in the twos column and a "0" in the units column, indicating "1 two and 0ones". The base-ten "two" (210) is written in binary as 102.
A "three" in base two is actually "1 two and 1 one", so it is written as 112. "Four" is actually two-times-two, so we zero out the twos column and the units column, and put a "1" in the fours column; 410 is written in binary form as 1002. Here is a listing of the first few numbers:
| decimal (base 10) | binary (base 2) | |
| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 | 0 ones1 one1 two and zero ones1 two and 1 one1 four, 0 twos, and 0 ones1 four, 0 twos, and 1 one1 four, 1 two, and 0 ones1 four, 1 two, and 1 one1 eight, 0 fours, 0 twos, and 0 ones1 eight, 0 fours, 0 twos, and 1 one1 eight, 0 fours, 1 two, and 0 ones1 eight, 0 fours, 1 two, and 1 one1 eight, 1 four, 0 twos, and 0 ones1 eight, 1 four, 0 twos, and 1 one1 eight, 1 four, 1 two, and 0 ones1 eight, 1 four, 1 two, and 1 one1 sixteen, 0 eights, 0 fours, 0 twos, and 0 ones |
Converting between binary and decimal numbers is fairly simple, as long as you remember that each digit in the binary number represents a power of two.
- Convert 1011001012 to the corresponding base-ten number.
I will list the digits in order, and count them off from the RIGHT, starting with zero:
| digits: | 1 0 1 1 0 0 1 0 1 |
| numbering: | 8 7 6 5 4 3 2 1 0 |
The first row above (labelled "digits") contains the digits from the binary number; the second row (labelled " numbering") contains the power of 2 (the base) corresponding to each digits. I will use this listing to convert each digit to the power of two that it represents:
1×28 + 0×27 + 1×26 + 1×25 + 0×24 + 0×23 + 1×22 + 0×21 + 1×20 = 1×256 + 0×128 + 1×64 + 1×32 + 0×16 + 0×8 + 1×4 + 0×2 + 1×1 = 256 + 64 + 32 + 4 + 1 = 357 Copyright © Elizabeth Stapel 2001-2011 All Rights Reserved
Then 1011001012 converts to 35710.
Converting decimal numbers to binaries is nearly as simple: just divide by 2.
- Convert 35710 to the corresponding binary number.
To do this conversion, I need to divide repeatedly by 2, keeping track of the remainders as I go. Watch below:
As you can see, after dividing repeatedly by 2, I ended up with these remainders:
These remainders tell me what the binary number is. I read the numbers from around the outside of the division, starting on top and wrapping my way around and down the right-hand side. As you can see:
35710 converts to 1011001012.
- This method of conversion will work for converting to any non-decimal base. Just don't forget to include that first digit on the top, before the list of remainders. If you're interested, an explanation of why this method works is available here.
You can convert from base-ten (decimal) to any other base. When you study this topic in class, you will probably be expected to convert numbers to various other bases.
