Number System

Introduction to Number System:


Number system, any of 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 a Hindu-Arabic in origin.

A concise treatment of these numeral systems follows. For further discussion, let us understand number systems: Number systems.


Very 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 very complicated arithmetical calculations. 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. 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 was a discovery 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.

Let us discuss the different divisions under the number system.


UNARY NUMERAL SYSTEM


The simplest number system of the 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 still in regular use. But, the unary system can only be used for numbers which are a smaller size such that 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, denoting 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 which 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 with a radix of 2 in the base-2 system.

In computers, the main system of number 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 is the hexadecimal numeral system is 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. 

For example, 10001 represents (1 X 24) + (0 X 23) + (0 X 22) + (0 X 21) + (1 X 20), or 16 + 0 + 0 + 0 + 1, or 17.
Number system represents a valuable set of numbers that consists of natural numbers, integers, real numbers, irrational numbers, rational numbers and goes on.

THE 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...}, it is denoted by N in short form.

The whole number 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 outcome of any two natural numbers is a natural number (4×3500=14000). The subtraction and division of any two or more natural number are not equal to natural numbers, though.

THE INTEGERS:


The integers are the 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 value 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 outcome, and difference between any two integers is also an integer. But this is not true for division

THE RATIONAL NUMBERS:


The rational numbers are those figures which can be formulated as a ratio between two integers. For instance, the fractions 13 and −11118 are both rational numbers. The rational number includes all the integers since any integer is denoted by z can be written as the ratio z1.

All decimals which terminate are rational numbers (since 8.27 can be written as 827100.) Decimals which are a recurrent pattern in nature after some point are also called a rational number:

for example,0.0833333...=112.

THE IRRATIONAL NUMBERS:


An irrational number is those number that which cannot be written as a proportion (or fraction). The irrational number does not end or neither repeats in a decimal form. The ancient Greek mathematician discovered that not all integers are the rational number; there are equations that cannot be solved using ratios of integers.

The first step was to study the equation 2=x2. What number times itself equals to 2?

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 it is decimal equivalent goes on forever, with no repetitious pattern:F

The golden ratio is another famous irrational number, a number which as 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...

THE REAL NUMBERS:


The real numbers are the set of numbers containing all the irrational numbers and all the rational numbers. The real numbers consist of “all the numbers” on the number scale. The real numbers are infinite numbers just as there are infinitely many numbers in each of the other sets of numbers. But, it can be explained that the infinity of the real numbers is a bigger infinity.
The "smaller numbers ", or countable infinity numbers of the integers and rational numbers are sometimes called ℵ0(alef-naught), and the uncountable infinity numbers of the reals are called ℵ1(alef-one).

There are even "bigger" infinities, but one should know to take a set theory class for that specific set.

THE COMPLEX NUMBERS:


The complex numbers are the set {a+bi | the real numbers are a and b}, where the imaginary unit is
i, −1−√-. 
 
The complex numbers include the set of real numbers, that is, 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.

BEYOND…


The mathematicians have used even bigger sets of numbers. The quaternions were discovered by William H. Hamilton in 1845, form a number system with different imaginary figures.

Answer the following questions:


  • 1. Define the term number system?

  • 2. What are the different types of numeral system? Explain with examples?

  • 3. What are the natural numbers?

  • 4. Explain the difference between rational and irrational numbers?

  • 5. what are complex numbers? Give an example?

  • Fill in the blanks:


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

  • 2. Value is the number the numeral represents (Ans: Value)

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

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


  • State whether the following statements are 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 H. 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).