Number System

The Number System includes any of the numerous sets of symbols and the rules for using them to denote numbers, which are used to state how many objects are there in a given set. Thus, the idea of “oneness” can be denoted by the Roman numeral I, by the Greek letter alpha α which was the first letter used as a numeral, by the Hebrew letter aleph which is the first letter used as a numeral, or by the modern number 1, which is nothing but Hindu-Arabic in origin.

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Number System Definition

Number system is a mathematical presentation of numbers of a given set. For further discussion, let us understand number systems.

Most likely, the beginning system of the inscribed symbol in ancient Mesopotamia was a system of symbols for numbers. The present number systems are place-value systems. That is, the value of these numbers depends upon the position or place of the numbers in the representation; for example, the 3 in 30 and 300 represents 3 ten’s and 3 hundred, respectively. In the ancient systems, such as the Egyptian civilization, Roman, Hebrew, and Greek Number systems, did not have a positional characteristic, and it was a very complicated arithmetical calculation. Other systems though, including the Babylonian, one version each of the Chinese and Indian, as well as the Mayan system, did use the principle of place value. 

Number Systems Through the Ages

The development of numbers grew in different directions and has dissimilar versions. Some of the different versions of number systems were developed in Egypt, Babylon, Rome and some other versions are called Hindu-Arabic, Mayans, and the modern American number systems. It is believed that mathematical evolution had begun before the counting systems of numbers started.

Let us discuss the different divisions under the number system.

• Unary Numeral System

The simplest number system of numerals is the unary numeral system, in which each and every natural number is represented by an equivalent number of symbols. They are denoted by /, for instance, then the number seven would be represented by /////// such that we must write seven /.

Tally marks represent one such system which is still in regular use. But, the urinary system can only be used for numbers which are of a smaller size with a small value, although this plays a related role in theoretical computer science. Do you know Elias gamma coding, which is generally used in data compression, denotes arbitrary-sized numbers by using unary to indicate the length of binary numbers?

For example, if / stands for one, * for ten and - for 100, then the number 304 can be compactly represented as --- //// and the number 123 as - ** /// without any need for zero and this is called sign-value notation.

• Binary Numeral System

A binary number is used in the system of mathematics, and the world of science. In this system, the numbers are expressed in the form of a base-2 system or binary numerical system, which uses only two numbers that are 0's (zeros) and 1's (ones). The positional representation is done with a radix of 2 in the base-2 system.

In computers, the main system of numbers is based on the positional system in base 2 which is a binary numeral system, with two digits of binary, 0's and 1's. Positional systems received by grouping binary digits by three, which is the octal numeral system or four which is the hexadecimal numeral system, are commonly used. For very large numbers, bases 232 or 264, you can group the binary digits by 32 or 64, which is the portion of the machine word.

Number system represents a valuable set of numbers that consists of natural numbers, integers, real numbers, irrational numbers, rational numbers and so on.

• Natural Numbers

The natural (or counting) numbers are from 1, 2, 3, 4, 5, 6, 7, 8, 9, etc. Natural numbers are infinite numbers. The set of natural numbers, {1, 2, 3, 4, 5, 6, 7, 8...}, is denoted by N in short form.

whole numbers start from 0 and include the natural numbers.

The addition of any 2 natural numbers is also a natural number (for instance, 4 + 5000 = 5004), and the product of any two natural numbers is also a natural number (4 × 3500 = 14000). The subtraction and division of any two or more natural numbers is not equal to natural numbers, though.

• Integers

Integers are natural numbers consisting of the set of real numbers, their additive inverses and zero.

{..., -9, -8, -7, -6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}

The set of integers is denoted by J or Z in short form. These integer values can be represented on the number line. These numbers can be a whole number or decimal numbers. You can represent -6.5 on a number, 6 on a number line.

The addition, product, and difference between any two integers is also an integer. But this is not true for division.

• Rational Numbers

Rational numbers are figures which can be formulated as a ratio between two integers. For instance, the fractions 13 and −11118 are both rational numbers. Rational numbers include all the integers since any integer is denoted by z.

All decimals which terminate are rational numbers (since 8.27 can be written as 827/100.) Decimals which are a recurrent pattern in nature after some point are also called a rational number.

• Irrational Numbers

An irrational number is a number that cannot be written as a proportion (or fraction). The irrational number does not end or repeats in a decimal form. The ancient Greek mathematician discovered that not all integers are rational numbers; there are equations that cannot be solved using ratios of integers.

The square root of 2 is about 1.414, because 1.4142=1.999396, which is almost equal to 2. You will never get the exact number by squaring the fraction (or terminating decimal numbers). The square root of 2 is an irrational number, meaning its decimal equivalent goes on forever, with no repetitive pattern

The golden ratio is another famous irrational number, a number which has great importance in biology:

1 + 5√2 = 1.61803398874989…

π(pi), the proportion of the perimeter of a circle to its diameter:

π = 3.14159265358979…

and e, the most significant number in calculus:

e = 2.71828182845904…

• Real Numbers

Real numbers are the set of numbers containing all the irrational numbers and all the rational numbers. They consist of “all the numbers” on the number scale. Real numbers are infinite numbers just as there are infinitely many numbers in each of the other sets of numbers. But, the infinity of the real numbers is a bigger infinity.

• Complex Numbers

Complex numbers are the set {a+bi | real numbers are a and b}, where the imaginary unit is i, -1

Complex numbers include the set of real numbers, which includes the set of both rational and irrational numbers. The real numbers, in the complex system, are denoted in the form a + 0i = a. a real number.

This set is always denoted by C in the short form. The set of complex numbers is significant because for any polynomial p(x) with real number coefficients, all the solutions of p(x) = 0 will be in C.

Important Number Systems

At present, there are four types of number system; these are:

• Binary system

• Decimal system

• Octal System

• Hexadecimal system

Number System Conversions

Binary System Conversions

• Binary to Octal

• Binary to Decimal

• Binary to Hexadecimal

Binary to Octal

Example: 1100011

Octal means 8 

Starting from the least significant bit, we will make a group of three bits First pair = 011

Second group = 100

Third group = 001

Now, we know that 011 = 3, 100 = 4, and 001 = 1, so the equivalent octal number system will be (143)8.

Binary to Decimal

Example: 1101001

Starting from the least significant bit from 20, then 21, till the nth most significant bit, i.e., 2ⁿ.

1 = 2⁰ = 1

1 = 2³.1 = 8

0 = 2⁴ .0= 0

1 = 2⁵ = 32

1 = 2⁶ = 64

Now, adding all these:

= 1 + 0 + 0 + 8 + 0 + 32 + 64 = (105)10

Binary to Hexadecimal

Example: 101010001

We know that hexadecimal means 16, so we will start from the least significant bit and make groups of 4 bits:

First pair = 0001 = 1

Second pair = 0101 = 5

Third pair = 0001 = 1

So, the hexadecimal equivalent will be (151)16.

Decimal System Conversions

• Decimal to Binary

• Decimal to Octal

• Decimal to Hexadecimal

Decimal to Binary

Example: (131)10

Divide 131 by 2 and continue till you get the quotient as 0.

1. Divide 131 by 2, we get the quotient as 65 and remainder as 1

2. Divide 65 by 2, we get 32, remainder as 1

3. Divide 32 by 2, we get 16, and remainder as 0

4. Divide 16 by 2,we get 8 and remainder as 0

5. Divide 8 by 2, we get 4, and remainder as 0

6. Divide 4 by 2, then 2 by 2, and finally divide 1 by 2,with the quotient as 0, we get the remainder as 1

Starting from LSB to MSB, write the binary system so formed:

(10000011)2

Decimal to Octal Conversion

Example: (425)10

Divide 425 by 8 till you get the quotient as 0. 

1. Dividing 425 by 8, we get the quotient as 53, and the remainder as 1

2. Divide 53 by 8, we get the quotient as 6, and the remainder as 5

3. Divide 6 by 8 by taking the quotient as 0, we get the remainder as 6 

So, the octal equivalent of (425)10 is (651)8.

Decimal to Hexadecimal

Example: (650)10

1. Divide 650 by 16, we get the quotient as 40 and the remainder as 10 or A

2. Divide 40 by 16, we get the quotient as 2 and the remainder as 8

3. Divide 2 by taking the quotient as 0, we get the remainder as 2

So, the hexadecimal equivalent of (650)10 is (28A)16.

Octal System Conversions

Example: (79)8 to ()10

= 7 Χ 8¹ + 9 Χ 8⁰= 56 + 9 = 65

So, the decimal equivalent of (79))8 is (65)10.

First 20 Binary number system conversions to Decimal, Octal, and Hexadecimal systems:

Number System Chart

Number System in Computer

In computers, the technique of presenting the work is the numbers system. Computers use the decimal number system the most, while the information converted from higher-level to the lower one is in the form of bits, i.e., 0’s and 1’s, which is actually the binary system.

Conclusion

The most commonly used Number system is the decimal positional number system, the decimal referring to the use of 10 numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to construct all the required numbers. This discovery was made by the Indians. There are other two common number systems which are used in computers and computing science. They are the binary system, and these are denoted by 0's and 1's, and the hexadecimal system, which has 16 symbols (We can understand by the term Hex which is 16) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

Fill in the Blanks:

1. The similar ___________of symbols may represent different symbols in various numeral systems (Ans: sequence)

2. ___________________________ developed by the Hindus in India, slowly radiated to other surrounding countries due to their commercial and military activities with India.

(Ans: The numerical system and the concept of zero)

3. The main system of numerals is based on ____________________in base 2 with two digits of binary, 0 and 1. (Ans: the positional system)

State True or False and Correct the Statements:

1. The natural numbers start from 0 to infinite numbers( ) (Ans: false)

2. The complex numbers include the set of real numbers, that is, which includes the set of both rational and irrational numbers. () (Ans: true )

3. The real numbers are the set of numbers containing all of the irrational numbers ( ) (Ans: false)

4. The integers are the natural numbers consisting of the set of real numbers, their additive inverses and zero. ( ) (Ans: true)

5. The quaternions were discovered by William. Rowan Hamilton( ) (Ans: true)

Tick the Correct Answers

1. Unary Numeral System is Denoted by_________________

a. 0,1

b. \

c. ()

d. /

(Ans: d. /)

2. Real Number Includes _________________________

a. Only rational numbers

b. Only irrational numbers

c. Both the rational and irrational numbers

d. None of the above

(Ans: c. both the rational and irrational numbers)

3. The Natural (or counting) Numbers are from_________________

a. 0,1

b. 0, 1, 2, 3 ,4……………..

c. 1, 2, 3, 4………..

d. ….-2, -1, 0 , 1, 2 ,3……..

(Ans: c.1, 2, 3, 4………..)

4. The Complex Numbers are the Set {a + bi | __________________are a and b},

a. The rational numbers

b. The irrational numbers

c. The real numbers

d. The integer numbers

(Ans: c. the real numbers). 

5. Which Number System Cannot be Written as a Proportion?

a. Rational numbers

b. Irrational number

c. Real numbers

d. Integers numbers

(Ans: irrational numbers).