In mathematics there exists a large complex set of number systems that have different bases and have distinctive characteristics among them. One of these is the binary system, which works with two numbers, “0” and “1”.
This system is very important for computer science, since this is where the term arises bit. Through electronic impulses and bitsSoftware executes actions on a computer.
In the following paragraphs you will find the most complete information about this base 2 system. We will show you the necessary steps to convert binary numbers to hexadecimal, octal, and decimal..
What is the binary system and what is it used for in computer science and computing?
Binary numbers are a mathematical system with base 2, meaning the allowed values are “0” and “1”. Therefore, there is a significant difference with the decimal system that we frequently use in various aspects of our lives. In computing, the numbers zero and one are called binary digits and are translated as “binary digit” or commonly known as "bit".
So, what can be said... “bit 0” and the “bit 1” to represent electronic pulses, generally using 0 as a disconnection and 1 as a connection or positive signal. In this way, Programs installed on a computer can represent computer instructions, read texts, or represent data..
This is achieved through the work of microprocessors in detecting the presence or absence of electrical pulse signals. Processors have different transfer rates depending on their manufacturing process and They can group up to 8 bits into a single signalThis set is what is called “byte” and it is the unit of measurement used in computing.
History and origin: When was the binary system discovered and what was its impact?
The first appearance of binary system appears through the mathematician Pingala, 3 centuries before the birth of ChristHowever, others claim that there is a work in the Chinese oracle book. I Chin, whose creation dates back to 1200 BC and in which certain patterns are established with binary numbers. With this same prayer book, an arrangement was made for a decimal sequence that included the numbers 0 to 63.
This work was carried out by the philosopher Shao Yong and performed it in the 11th century AD. But it was not until 1605 that the Chancellor of England, the philosopher and mathematician Francis BaconHe adopted a system in which he established that letters could be represented in binary numbers.
This led to his publishing, 70 years later The Spanish monk Juan Caramuel gave a more realistic description of this base-2 systemAlready in the 19th century, what would become a paradigm shift of the binary number systemA detailed logic was established for how these numbers should be treated. This was done by the British George Boole, which later became known as Boolean algebra.
In the late 1890s, the United States needed to conduct a census of its population, so it hired Herman Hollerith to carry out this workThe German used a binary system by means of punched cards based on Joseph Marie Jacquard, which implemented binary control for its looms. It is at this moment that binary numbers enter the world of computing.
From that moment on, information began to be processed automatically without human intervention. Years later, the work of the electrical engineer appeared. Claude Shannon and the scientist George Stibitz. These were the first to establish the standards for the use of the binary system in computing.
Binary vs decimal vs octal vs hexadecimal: What are the differences between them?
binary system It has a base of 2, meaning the numbers used are the 0 and the 1. On the other hand the decimal system its base is 10 And it's the system we use every day. Whereas the system octagonal The base is 8 and the numbers are used 0, 1, 2, 3, 4, 5, 6, 7. Finally, the system Hexadecimal has a base of 16 and is represented by Arabic numerals and letters of the alphabet, which are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
The binary system accepts negative or inverse integers.0 being the reference point for the incorporation of the sign, but these numbers are represented in a particular way. Whereas in the decimal system Whole numbers and fractions are accepted and can have positive and negative signs represented with a – in front of each number prior to zero.
Regarding the octal system It accepts fractions and positive and negative numbers. Hexadecimal considers any number less than 7 to be negative.Therefore, everything F If it appears as the first digit, it means it is negative and should not be calculated.
System Conversion: How is data migrated from one system to another?
To convert numbers from one system to another, you will need to perform a few steps, which we will show you below:
Binary and decimal
To convert a binary number You must start counting the digits from the right, with the first one being "0".This way, if you have the number 11011 you will count from the last "1" like the number 0. Therefore, your account will look like this:
- 1(4) 1(3) 0(2) 1(1) 1(0)> 4, 3, 2, 1, 0
Next, you will have to do a sum of powers of base 2, where the exponents are the numbers 4, 3, 2, 1, 0. To each addend you must multiply it by binary numbers. That is, by 1, 1, 0, 1, 1.
Thus, the conversion would yield:
- 24 + 1.23 + 0.22 + 1.21 + 1.10 =
- 16 + 8 + 0 + 2 + 0 =
- 26
That is, 11011 In binary system, it is equal to the number 26 in a decimal system.
In case you want to convert from decimal to binary you will have to take the decimal number and divide it in twoThen you'll have to apply that result again. divide into 2 and so on until it ends.
If The rest you got was 1You will need to take this into account, since you will be taking all the quotients and expressing them in reverse order to form the binary number.
For example, if you have the number 100, you will have to do the following:
- 100 / 2 = 50 the remainder is 0
- 50 / 2 = 25 the remainder is 0
- 25 / 2 = 12 the remainder is 1
- 12 / 2 = 6 the remainder is 0
- 6 / 2 = 3 the remainder is 0
- 3 / 2 = 1 the rest is 1
Therefore, the number binary that is formed 1100100This means that the number 100 in a The decimal system is equal to the binary number that has been obtained..
Binary and hexadecimal
To convert a To convert a binary number to a hexadecimal number, you must group the digits in groups of 4.If you have fewer digits, you will have to complete with leading zeros.
Then you will have to put the numbers in each digit. 8, 4, 2, 1Next, you'll need to multiply those numbers by each digit and add them together. For example, if you have a binary number 1010 you must do the following: 8x1 + 4x0 + 2x1 + 1x0 = 10 And since 10 is greater than 9, the corresponding number is A.
Therefore, the binary number 1010 represents the number A in the hexadecimal systemIf you have larger digits, you should follow the same steps, but for each group of four digits.
In other words, if you had the number 1100101101, you should do the following:
- Separate into four digits 11 0010 1101.
- Complete the first group with 0s until you have a group of 4. That is, you will have 0011 0010 1101
Perform the previous steps by multiplying and adding each group:
- 8 x 0 + 4 x 0 + 2 x 1 + 1 x 1 = 3
- 8 x 0 + 4 x 0 + 2 x 1 + 1 x 0 = 2
- 8 x 1 + 4 x 1 + 2 x 0 + 1 x 1 = 13 > greater than 9, therefore the number D corresponds
This means that the binary number 1100101101 is equal to 32D in the hexadecimal system.
In case you want Convert a hexadecimal number to binary You will have to directly replace the digit with its corresponding one binary number.
To do that, you will need to take into account the following relationship:
- 0 = 0000
- 1 = 0001
- 2 = 0010
- 3 = 0011
- 4 = 0100
- 5 = 0101
- 6 = 0110
- 7 = 0111
- 8 = 1000
- 9 = 1001
- A = 1010
- B = 1011
- C = 1100
- D = 1101
- E = 1110
- F = 1111
If we do it with the previous case 32D, We will have to replace each digit.
Hence:
- 3 = 0011
- 2 = 0010
- D = 1101
This is how the binary number is formed 001100101101, or in other words 1100101101That is, the number that was formed is the same one we used to convert it in the step of binary to hexadecimal.
Binary and octagonal
As with the conversion to a hexadecimal system You will need to group the numbers, but in this case in groups of three digitsYou will also need to fill in with 0s on the left when they do not reach a group of 3 digits.
In addition, you should keep in mind the following replacement table, with the first digits being binary and the second their corresponding octal digits:
- 0 = 0
- 001 = 1
- 010 = 2
- 011 = 3
- 100 = 4
- 101 = 5
- 110 = 6
- 111 = 7
What you should do now is group and then replace the numbers by means of the tableFor example, if you have the number 1011101.
You will need to follow this process:
- Do the groups of 3 digitsThat is, you will have 1 011 101.
- Now Complete the first group with 0., you will get a result of 001 011 101.
Replace according to the list that we showed you before.
You'll end up with:
- 001 = 1
- 011 = 3
- 101 = 5
Therefore, the binary number 10111012 corresponds to an octagonal system to the number 1358. In case you want convert a number of octagonal to binary system You will have to follow the path ConverseThat is, you will have to replace each digit you have from the octal system in groups of three binary digits. You will need Keep in mind the list we mentioned earlier..
So, if you have a number 24578 and you want to convert it to binaryYou'll have to separate it into 2, 4, 5, 7.
Next, you will need to follow the replacements for each number:
- = 2 010
- = 4 100
- = 5 101
- = 7 111
Thus it is concluded that the number 24578 = 101001011112
