Author :- Jaymala
The following are the quaternary important types of sort systems.
1. Decimal
2. Binary
3. Octal
4. Hexadecimal
In this article, we’ll countenance at Decimal and Binary Number Systems. We’ll discuss the other member sort systems in our next article.
Decimal System
‘Deci’ effectuation ten. The quantitative sort grouping has decade as its base. It uses decade digits from 0 to 9. The quantitative sort grouping is also titled Hindu Arabic, or Arabic, or humble 10 system.
The quantitative grouping is simple and versatile. It is the most commonly used sort system. It has become the dominant sort grouping in the world.
Any number, no concern how super or small, can be written in the quantitative grouping using only the 10 humble symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, the population of Asia is about 3,800,000,000. The diameter of an iron atom is about 0.000000025 cm.
It is believed that the quantitative grouping is based on 10 digits because humans have 10 fingers and 10 toes. The articulate member is derived from the Latin articulate ‘digitus’ which effectuation finger or toe. The quantitative grouping is a positional sort system.
Each member in a sort has a continuance dependent on its position in the number. A digit's continuance is the member multiplied by a power of decade according to its position in the number. For example, study the sort 9,364.
The member 9 is in the ‘thousands place’ and its continuance is 9 × 1,000 = 9,000.
The member 3 is in the ‘hundreds place’ and its continuance is 3 × 100 = 300.
The member 6 is in the ‘tens place’ and its continuance is 6 × 10 = 60.
The member 4 is in the ‘ones place’ and its continuance is 4 × 1 = 4.
So: 9,364 = 9 × 1,000 + 3 × 100 + 6 × 10 + 4 × 1 = 9,000 + 300 + 60 + 4
In the quantitative system, locate values process from correct to mitt by powers of 10 (…1000, 100, 10, 1…).
That is, from correct to mitt a quantitative grouping has a ones place, a tens place, a hundreds place, a thousands place, a decade thousands locate etc.
For example, In the sort 22: The 2 on the mitt is in the \"tens place,\" and its continuance is 2 × 10 = 20.
The 2 on the correct is in the \"ones place,\" and its continuance is 2 × 1 = 2. Observe that the continuance of the 2 on the mitt is 10 times as super as that of the 2 on the right. Let’s today talk about quantitative fractions.
A quantitative cypher is a sort with a quantitative saucer in it, same 0.3, 0.75, 1.23, 7.049, 0.0058 etc. In a quantitative cypher (commonly titled a decimal) a quantitative saucer separates the full drawing from fractions.
The quantitative saucer indicates the locate where values change from positive to negative powers of 10. Each locate to the mitt of the quantitative saucer represents a positive power of 10 (10, 100, 1000, 10000, and so on) and apiece locate to the correct of the quantitative saucer represents a negative power of 10 [10^ (-1), 10^ (-2), 10^ (-3), 10^ (-4), and so on].
When a humble is raised to a negative power, it meet effectuation the complementary of the humble raised to the positive power.
So: 10^ (-1) = 1 / (10^1) = 1/10 = 0.1
10^ (-2) = 1/ (10^2) = 1/100 = 0.01
10^ (-3) = 1/ (10^3) = 1/1000 = 0.001 etc.
So, the locate values to the correct of the quantitative saucer are tenths, hundredths, thousandths, and so on. Negative powers of 10 are diminutive numbers, inferior than 1 and greater than 0.
EXAMPLE In the quantitative 13.97:
1 is in the ‘tens place’.
3 is in the ‘ones place’.
9 is in the ‘tenths place’.
7 is in the ‘hundredths place’.
So: 13.97 = 1 × 10 + 3 × 1 + 9 × 0.1 + 7 × 0.01 = 10 + 3 + 0.9 + 0.07
Binary System
‘Bi’ effectuation two. The star sort grouping works meet same the quantitative sort grouping eliminate that the star sort grouping uses humble 2. It uses only member digits, 0 and 1. The star sort grouping is also referred to as \"base 2 system\".
The star grouping plays an important role in technology and machine science. 0 and 1 can represent off and on, yes and no, etc. Computers, Calculators, Microwave, and other electronic equipments use binary.
Binary sort grouping is easy for machines, but, human beings find it very difficult, because it requires so many digits to represent a sort in the star system. For example, the sort 213 takes only threesome digits (2, 1, and 3) to write in the quantitative system, yet takes eight digits to write in the star grouping (11010101).
All input to the machine and other electronic equipments is converted into star drawing made up of 0’s and 1’s. But, quantitative is what humans are habitual to, so, the machine and other electronic equipments turn the star into decimals and display information in a form that humans can easily understand.
The star grouping is a positional sort system. Each member in a star sort has a continuance dependent on its position in the number. A digit's continuance is the member multiplied by a power of member according to its position in the number.
For example, Consider the star sort 10101 (should be read as “one zero one zero one” and not as “ten thousand one hundred one”). In the star sort 10101: The prototypal 1 on the mitt is in the sixteen’s locate and its continuance is 1 × 16 = 16.
The prototypal 0 on the mitt is in the eight’s locate and its continuance is 0 × 8 = 0. The second 1 is in the four’s locate and its continuance is 1 × 4 = 4. The second 0 is in the two’s locate and its continuance is 0 × 2 = 0.
The last 1 on the rightmost is in the ones locate and its continuance is 1 × (2 to the zero power) = 1 × (2^0) = 1 × (1) = 1. To convert a star sort to a decimal, find discover the actualised continuance represented by apiece member and add them together.
For example, the quantitative equal of the star sort 10101 (we meet discussed) is 16 + 0 + 4 + 0 + 1 = 21. To convert from quantitative to binary:
Step 1: Divide the quantitative sort by 2.
Step 2: Record the remainder (0 or 1).
Step 3: Repeat Steps 1 and 2 with the quotient until the quotient becomes zero. Let’s countenance at a simple example.
Let’s determine the star equal of 23.
23/2 = 11 --- Remainder 1
11/2 = 5 --- Remainder 1
5/2 = 2 --- Remainder 1
2/2 = 1 --- Remainder 0
1/2 = 0 --- Remainder 1
The ordering of remainders going up gives the answer. So, the star equal of the quantitative 23 is 10111.
Just as the locate values in the quantitative grouping process by powers of 10, the locate values in the star grouping process by powers of 2 from correct to mitt (…32, 16, 8, 4, 2,…). That is, from correct to mitt a star grouping has a ones place, a two’s place, a four’s place, an eight’s place, a sixteen’s locate etc.
For example, in the star sort 11, The 1 on the mitt is in the two’s locate and its continuance is 1 × 2 = 2. The 1 on the correct is in the ones locate and its continuance is 1 × 2^0 = 1 × 1 = 1. Observe that the continuance of the 1 on the mitt is twice as super as that of the 1 on the right.
Let’s today talk about star fractions. A star cypher is the same as a quantitative fraction, but with the humble of 2 instead of 10. In a star fraction, a star saucer separates the number conception of a star sort from its down part.
The star saucer indicates the locate where values change from positive to negative powers of 2. Each locate to the mitt of the star saucer represents a positive power of 2 (2, 4, 8, 16, 32, and so on) and apiece locate to the correct of the star saucer represents a negative power of
2 [2^(-1), 2^(-2), 2^(-3), 2^(-4), and so on]. When a humble is raised to a negative power, it meet effectuation the complementary of the humble raised to the positive power.
2^ (-1) = 1/ (2^1) = 1/2
2^ (-2) = 1/ (2^2) = 1/4
2^ (-3) = 1/ (2^3) = 1/8 etc.
The locate values to the correct of the star saucer are one-half, one-fourth, one-eighth and so on. Negative powers of 2 are diminutive numbers, inferior than 1 and greater than 0.
For example, The star sort 11.011 represents
1 x 2^ (1) + 1 x 2^ (0) + 0 x 2^ (-1) + 1 x 2^ (-2) + 1 x 2^ (-3)
= 1 x 2 + 1 x 1 + 0 x 1/ (2^1) + 1 x 1/ (2^2) + 1 x 1/ (2^3)
= 1 x 2 + 1 x 1 + 0 x 1/2 + 1 x 1/4 + 1 x 1/8
= 2 + 1 + 0 + 1/4 + 1/8 = 3 + 3/8
= 3 + 0.375
= 3.375
Converting a quantitative cypher to a star cypher requires more steps.
The following are the quaternary important types of sort systems.
1. Decimal
2. Binary
3. Octal
4. Hexadecimal
In this article, we’ll countenance at Decimal and Binary Number Systems. We’ll discuss the other member sort systems in our next article.
Decimal System
‘Deci’ effectuation ten. The quantitative sort grouping has decade as its base. It uses decade digits from 0 to 9. The quantitative sort grouping is also titled Hindu Arabic, or Arabic, or humble 10 system.
The quantitative grouping is simple and versatile. It is the most commonly used sort system. It has become the dominant sort grouping in the world.
Any number, no concern how super or small, can be written in the quantitative grouping using only the 10 humble symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, the population of Asia is about 3,800,000,000. The diameter of an iron atom is about 0.000000025 cm.
It is believed that the quantitative grouping is based on 10 digits because humans have 10 fingers and 10 toes. The articulate member is derived from the Latin articulate ‘digitus’ which effectuation finger or toe. The quantitative grouping is a positional sort system.
Each member in a sort has a continuance dependent on its position in the number. A digit's continuance is the member multiplied by a power of decade according to its position in the number. For example, study the sort 9,364.
The member 9 is in the ‘thousands place’ and its continuance is 9 × 1,000 = 9,000.
The member 3 is in the ‘hundreds place’ and its continuance is 3 × 100 = 300.
The member 6 is in the ‘tens place’ and its continuance is 6 × 10 = 60.
The member 4 is in the ‘ones place’ and its continuance is 4 × 1 = 4.
So: 9,364 = 9 × 1,000 + 3 × 100 + 6 × 10 + 4 × 1 = 9,000 + 300 + 60 + 4
In the quantitative system, locate values process from correct to mitt by powers of 10 (…1000, 100, 10, 1…).
That is, from correct to mitt a quantitative grouping has a ones place, a tens place, a hundreds place, a thousands place, a decade thousands locate etc.
For example, In the sort 22: The 2 on the mitt is in the \"tens place,\" and its continuance is 2 × 10 = 20.
The 2 on the correct is in the \"ones place,\" and its continuance is 2 × 1 = 2. Observe that the continuance of the 2 on the mitt is 10 times as super as that of the 2 on the right. Let’s today talk about quantitative fractions.
A quantitative cypher is a sort with a quantitative saucer in it, same 0.3, 0.75, 1.23, 7.049, 0.0058 etc. In a quantitative cypher (commonly titled a decimal) a quantitative saucer separates the full drawing from fractions.
The quantitative saucer indicates the locate where values change from positive to negative powers of 10. Each locate to the mitt of the quantitative saucer represents a positive power of 10 (10, 100, 1000, 10000, and so on) and apiece locate to the correct of the quantitative saucer represents a negative power of 10 [10^ (-1), 10^ (-2), 10^ (-3), 10^ (-4), and so on].
When a humble is raised to a negative power, it meet effectuation the complementary of the humble raised to the positive power.
So: 10^ (-1) = 1 / (10^1) = 1/10 = 0.1
10^ (-2) = 1/ (10^2) = 1/100 = 0.01
10^ (-3) = 1/ (10^3) = 1/1000 = 0.001 etc.
So, the locate values to the correct of the quantitative saucer are tenths, hundredths, thousandths, and so on. Negative powers of 10 are diminutive numbers, inferior than 1 and greater than 0.
EXAMPLE In the quantitative 13.97:
1 is in the ‘tens place’.
3 is in the ‘ones place’.
9 is in the ‘tenths place’.
7 is in the ‘hundredths place’.
So: 13.97 = 1 × 10 + 3 × 1 + 9 × 0.1 + 7 × 0.01 = 10 + 3 + 0.9 + 0.07
Binary System
‘Bi’ effectuation two. The star sort grouping works meet same the quantitative sort grouping eliminate that the star sort grouping uses humble 2. It uses only member digits, 0 and 1. The star sort grouping is also referred to as \"base 2 system\".
The star grouping plays an important role in technology and machine science. 0 and 1 can represent off and on, yes and no, etc. Computers, Calculators, Microwave, and other electronic equipments use binary.
Binary sort grouping is easy for machines, but, human beings find it very difficult, because it requires so many digits to represent a sort in the star system. For example, the sort 213 takes only threesome digits (2, 1, and 3) to write in the quantitative system, yet takes eight digits to write in the star grouping (11010101).
All input to the machine and other electronic equipments is converted into star drawing made up of 0’s and 1’s. But, quantitative is what humans are habitual to, so, the machine and other electronic equipments turn the star into decimals and display information in a form that humans can easily understand.
The star grouping is a positional sort system. Each member in a star sort has a continuance dependent on its position in the number. A digit's continuance is the member multiplied by a power of member according to its position in the number.
For example, Consider the star sort 10101 (should be read as “one zero one zero one” and not as “ten thousand one hundred one”). In the star sort 10101: The prototypal 1 on the mitt is in the sixteen’s locate and its continuance is 1 × 16 = 16.
The prototypal 0 on the mitt is in the eight’s locate and its continuance is 0 × 8 = 0. The second 1 is in the four’s locate and its continuance is 1 × 4 = 4. The second 0 is in the two’s locate and its continuance is 0 × 2 = 0.
The last 1 on the rightmost is in the ones locate and its continuance is 1 × (2 to the zero power) = 1 × (2^0) = 1 × (1) = 1. To convert a star sort to a decimal, find discover the actualised continuance represented by apiece member and add them together.
For example, the quantitative equal of the star sort 10101 (we meet discussed) is 16 + 0 + 4 + 0 + 1 = 21. To convert from quantitative to binary:
Step 1: Divide the quantitative sort by 2.
Step 2: Record the remainder (0 or 1).
Step 3: Repeat Steps 1 and 2 with the quotient until the quotient becomes zero. Let’s countenance at a simple example.
Let’s determine the star equal of 23.
23/2 = 11 --- Remainder 1
11/2 = 5 --- Remainder 1
5/2 = 2 --- Remainder 1
2/2 = 1 --- Remainder 0
1/2 = 0 --- Remainder 1
The ordering of remainders going up gives the answer. So, the star equal of the quantitative 23 is 10111.
Just as the locate values in the quantitative grouping process by powers of 10, the locate values in the star grouping process by powers of 2 from correct to mitt (…32, 16, 8, 4, 2,…). That is, from correct to mitt a star grouping has a ones place, a two’s place, a four’s place, an eight’s place, a sixteen’s locate etc.
For example, in the star sort 11, The 1 on the mitt is in the two’s locate and its continuance is 1 × 2 = 2. The 1 on the correct is in the ones locate and its continuance is 1 × 2^0 = 1 × 1 = 1. Observe that the continuance of the 1 on the mitt is twice as super as that of the 1 on the right.
Let’s today talk about star fractions. A star cypher is the same as a quantitative fraction, but with the humble of 2 instead of 10. In a star fraction, a star saucer separates the number conception of a star sort from its down part.
The star saucer indicates the locate where values change from positive to negative powers of 2. Each locate to the mitt of the star saucer represents a positive power of 2 (2, 4, 8, 16, 32, and so on) and apiece locate to the correct of the star saucer represents a negative power of
2 [2^(-1), 2^(-2), 2^(-3), 2^(-4), and so on]. When a humble is raised to a negative power, it meet effectuation the complementary of the humble raised to the positive power.
2^ (-1) = 1/ (2^1) = 1/2
2^ (-2) = 1/ (2^2) = 1/4
2^ (-3) = 1/ (2^3) = 1/8 etc.
The locate values to the correct of the star saucer are one-half, one-fourth, one-eighth and so on. Negative powers of 2 are diminutive numbers, inferior than 1 and greater than 0.
For example, The star sort 11.011 represents
1 x 2^ (1) + 1 x 2^ (0) + 0 x 2^ (-1) + 1 x 2^ (-2) + 1 x 2^ (-3)
= 1 x 2 + 1 x 1 + 0 x 1/ (2^1) + 1 x 1/ (2^2) + 1 x 1/ (2^3)
= 1 x 2 + 1 x 1 + 0 x 1/2 + 1 x 1/4 + 1 x 1/8
= 2 + 1 + 0 + 1/4 + 1/8 = 3 + 3/8
= 3 + 0.375
= 3.375
Converting a quantitative cypher to a star cypher requires more steps.
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