Numeral systems
David Eugene Smith asserts in his book Hindu Arabic Numerals that the most commonly used system of numerals is known as Hindu numerals or Hindu–Arabic numerals. Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero. Till then the simplest numeral system is the unary numerical system,(base 1 like ||| = 3 ). The ancient Egyptian numeral system was of this type, and the Roman numeral system was a modification of this idea.
David Eugene Smith, Louis Charles Karpinski publisher Ginn and Company, 1911
Type of number:
Binary (base 2)
Natural numbers: 1, 2, 3, 4,
Integers: …, -3, -2, -1, 0, 1, 2, 3,
Rational numbers: m/n, where m and n are integers and n¹ 0.
Examples: ½, 5/3, ¼ = 0.25 1/3 = 0.3333…
Irrational numbers,
examples:√2 ≈1.414, Π≈ 22/7, e ≈ 2.718.
Real number = any number
Some Number system:
Octal (base 8)
Decimal (base 10)
Hexadecimal (base 16)
Tutorial:
Write decimal number in exponential format, and vice versa.
Convert binary to decimal and vice versa.
Convert binary to hexadecimal and vice versa.
Definition of number representation system:
“Any notation for the representation of numbers”
To represent the positive integer one hundred and twenty-five as a decimal number, we can write (with the positive sign implied). The subscript 10 denotes the number as a base 10 (decimal) number.
12510 = 1*100 + 2*10 + 5*1 = 1*102 + 2*101 + 5*100
Representing fractions is a simple extension of this idea.
25.4310 = 2*10 + 5*1 + 4*0.1 + 3*0.01 = 2*101 + 5*100 + 4*10-1 + 3*10-2
Many electronic component of computer are in binary format.
Binary representations of positive can be understood in the same way as their decimal counterparts.
The subscript 2 denotes a binary number. Each digit in a binary number is called a bit.
11112 = 2*8 +2*4 + 2*2*1= 23 +22 +21 +20
It is often convenient to handle groups of bits, rather than individually. The most common grouping is 8 bits, which forms a byte A single byte can represent 256 (28) numbers. Memory capacity is usually referred to in bytes. Two bytes is usually called a word, or short word (though word-length depends on the application).
http://www.swarthmore.edu/NatSci/echeeve1/Ref/BinaryMath/NumSys.html#Intro
Type of number
Natural numbers: 1, 2, 3, 4…..
There seems to be no general agreement about whether to include 0 in the set of natural numbers. The set of natural numbers is denoted N
Integers: …, -3, -2, -1, 0, 1, 2, 3,
Are formed by the natural numbers including 0 (0, 1, 2, 3, ..) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). The set of integer is denoted Z or J
Rational numbers: m/n, where m and n are integers and n ≠ 0.
A rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero.
Examples: ½, 5/3, ¼ = 0.25 1/3 = 0.3333…
Rational numbers is denoted Q
Irrational numbers,
examples:√2 ≈1.414, Π ≈ 22/7, e ≈ 2.718 1.618....
An irrational number is any real number which cannot be expressed as a fraction a/b, where a and b are integers, with b non-zero, and is therefore not a rational number.
Real number = any number Real numbers is denoted R.
Binary representation system:
Binary Multiplication division works as in decimal. In binary subtraction when we borrow 1 from the first nonzero digit from the left each 0 became 10 which make 10-1=1
Octal representation system:
Hexadecimal representation system:
