In mathematics there is a large complex set of number systems that have different bases and have distinctive characteristics between them. One of these is the binary system, which works with two numbers, “0” and “1”.
This system is very important for computingsince from this arises the term bit. Through electronic impulses and bitssoftware performs 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 you must take to convert binary numbers into hexadecimal, octal and decimal.
What is the binary system and what is it used for in computer science and computing?
Binary numbers are a base 2 mathematical system, that is, the admitted values are “0” and “1”. So there is a big difference with the decimal system that we use frequently in different aspects of our lives. In computing, the number zero and one are called binary digits and it is translated as “binary digit” or commonly known as “bit”.
From what can be said “bit 0” and the “bit 1” to represent electronic pulses, usually using 0 as a disconnect and 1 as a connect or positive signal. In this way, programs installed on a computer can represent computer instructions, read text or represent data.
This can be done through the work carried out by the microprocessors to detect the presence or absence of the signal of the electrical pulses. The processors have a different transfer rate according to their manufacture and can pack up to 8 bits into a single signal. This set is called “byte” y is the unit of measurement used in computing.
History and origin When was the binary system discovered and what is its impact?
The first appearance of binary system appear by means of the mathematician Pingala, 3 centuries before the birth of Christ. But, others claim that there is a work in the Chinese oracular book I Chin whose creation dates from the year 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 did it in the eleventh century AD. But it was not until the year 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, 70 years later, publishing the Spanish monk Juan Caramuel a more realistic description of this base 2 system. Already in the 19th century, there was produced what would be a paradigm break of the binary number system. A detailed logic was established for how these numbers should be treated. This was done by the British George Boolewhich later became known as the Boolean algebra.
In the late 1890s, the United States needed to take a census of its population, so it hired Herman Hollerith to carry out this work. The German used a binary system by means of perforated cards based on Joseph Marie Jacquardwhich carried out a binary control for its looms. It is at this moment where binary numbers enter computing.
From that moment the information began to be processed automatically without the help of the human being. Years later, works of the electrical engineer appeared Claude Shannon and the scientist George Stibitz. In these, the standards for the use of the binary system in computing began to be established.
Binary system vs decimal vs octagonal vs hexadecimal How are they different?
the binary system has its base 2, that is, the numbers used are the 0 and 1. On the other hand the decimal system its base is 10 and it is the system we use daily. while the system octagonal the base is 8 and the numbers are used 0, 1, 2, 3, 4, 5, 6, 7. Lastly, the system hexadecimal has its base of 16 and is represented by the Arabic numbers 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 integersbeing 0 the point of reference for the incorporation of the sign, but these numbers are represented in a particular way. While in the decimal system whole numbers and fractions are accepted and can have positive and negative signs represented by a – in front of each number before zero.
Regarding the octal system accepts fractions and positive and negative numbers. Hexadecimal considers any number less than 7 to be negative.so everything F that it is found as the first digit means that it is negative and should not be calculated.
Conversion between systems How do you migrate data from one to another?
To convert numbers from one system to another you will have to perform some steps, which we will show you below:
binary and decimal
To convert a binary number You should start counting the digits starting from the right and the first being “0”. This way if you have the number 11011 you will count from the last “1” like the number 0. So your account will look like this:
- 1(4) 1(3) (two) 1(1) 1() > 4, 3, 2, 1, 0
Next, you will have to make a sum of power of base 2being the exponents the numbers 4, 3, 2, 1, 0. For each addend you must multiply it for binary numbers. That is, by 1, 1, 0, 1, 1.
In this way, the conversion would give:
- 24 + 1.23 + 0.22 + 1.21 + 1.1 =
- 16 + 8 + 0 + 2 + 0 =
- 26
I mean, 11011 in binary system 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 by two. Then you will have to return to that result split in 2 and so on until finished.
If he rest you stayed in 1you should take this into account, since you will take all the quotients and expose them inversely to form the binary number.
For example, if you have the number 100 you will have to do the following:
- 100 / 2 = 50 the rest is
- 50 / 2 = 25 the rest is
- 25 / 2 = 12 the rest is 1
- 12 / 2 = 6 the rest is
- 6 / 2 = 3 the rest is
- 3 / 2 = 1 the rest is 1
Therefore, the number binary what is formed is 1100100. This means that the number 100 in a decimal system is equal to the binary number that has been obtained.
binary and hexadecimal
To convert a binary number to a number of the hexadecimal system you must group by groups of 4 digits. In case you have less digits you will have to complete with leading zeros.
Then you will have to put in each digit the numbers 8, 4, 2, 1. You will then need to multiply those numbers by each digit and add them up. For example, if you have a binary number 1010 you will need to do the following: 8 x 1 + 4 x 0 + 2 x 1 + 1 x 0 = 10 and since 10 is greater than 9 corresponds to the number A.
Therefore, the binary number 1010 represents the number A in the hexadecimal system. If you had larger digits you should do the same steps, but for each group of four digits.
That is, if you had the number 1100101101 you should do the following:
- Separate into four digits 11 0010 1101.
- Complete the first group with 0 until you can have the group of 4. That is, you will have 0011 0010 1101.
Perform the above steps by multiplying and adding each group:
- 8 x + 4 x + two x 1 + 1 x 1 = 3
- 8 x + 4 x + two x 1 + 1 x = 2
- 8 x 1 + 4 x 1 + two x + 1 x 1 = 13 > greater than 9, so the number D corresponds
This implies that the binary number 1100101101 is equal in hexadecimal to 32D.
In case you want convert a hexadecimal number to binary you will have to directly replace the digit with its corresponding binary number.
For that you will have 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.
So that:
- 3 = 0011
- 2 = 0010
- D = 1101
In this way the binary number is formed 001100101101, or what is the same 1100101101. That is, the number that was formed is the same one that we took to convert it in the step of binary to hexadecimal.
binary and octagonal
As is the case with the conversion to a hexadecimal system you will have to group the numbers, but in this case in groups of three digits. You will also have to fill in with 0s to the left when they do not reach a group of 3 digits.
In addition, you must keep in mind the following replacement table, with the first binary digits and the second corresponding to an octal system:
- 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 through the tableFor example, if you have the number 1011101.
You will have to follow this process:
- do the groups of 3 digitsthat is, you will have 1 011 101 left.
- Now, complete with 0 the first groupyou will have as a result 001 011 101.
Replace according to list that we showed you before.
You will have:
- 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 reverse. That is, you will have to replace each digit you have of the octal system by groups of three binary digits. you shall keep in mind the list that we mentioned before.
So if you have a number 24578 and you want to convert it to binaryyou will 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
In this way it is concluded that the number 24578 = 101001011112
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