Number system.pdf
- 1. NUMBER SYSTEM A number system is a system of writing for expressing numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation to every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, and division.
- 2. DIFFERENT TYPES OF NUMBER SYSTEM 1. Decimal number system (Base- 10) 2. Binary number system (Base- 2) 3. Octal number system (Base-8) 4. Hexadecimal number system (Base- 16)
- 3. HOW NUMBER SYSTEM USED IN DAILY LIFE 1. Calling a member of a family or a friend using mobile phone. 2. Calculating your daily budget for your food, transportation, and other expenses. 3. Cooking, or anything that involves the idea of proportion and percentage. 4. Weighing fruits, vegetables, meat, chicken, and others in market. 5. Using elevators to go places or floors in the building. 6. Looking at the price of discounted items in a shopping mall. 7. Looking for the number of people who liked your post on Facebook. 8. Switching the channels of your favorite TV shows. 9. Telling time you spent on work or school. 10. Computing the interest you gained on your business.
- 4. DECIMAL NUMBER SYSTEM Decimal number system, also called Hindu- Arabic, or Arabic, number system, in mathematics, positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It also requires a dot (decimal point) to represent decimal fractions. In this scheme, the numerals used in denoting a number take different place values depending upon position. In a base-10 system the number 543.21 represents the sum (5 × 102 ) + (4 × 101 ) + (3 × 100 ) + (2 × 10−1 ) + (1 × 10−2 ). See numerals and numeral systems. This number system, with its associated arithmetic algorithms, has furnished the basis for the development of Western commerce and science since its introduction to the West in the 12th century AD. EXAMPLE Here is the number "forty-five and six-tenths" written as a decimal number: Thedecimal point goes between Ones and Tenths. 45.6 has 4 Tens, 5 Ones and 6 Tenths, like this: Now, let's discover how it all works ...
- 5. BINARY NUMBER SYSTEM Machine language is binary. This meansthat the machinelanguage has binary values or two values, the combinationof which represents the data. Thesetwo statesare “on” state representedby 1 and “off” state, represented by “0”. Let us start with the more fThe binary numbersystem is also a positionalnotation numberingsystem,but in this case, the base is not ten but is instead two. Each digit position in a binary numberrepresentsa power of two. When we write a binary number, each binary digit is multiplied by an appropriatepower of 2 which is based on theirpositionin the number.amiliarnumbersystem,the one where we use the numbers0 to 1
- 6. EXAMPLE….. 101101 = 1 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 = 1 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1 = 32 + 8 + 4 + 1 In the binary number system, there are only two possible values that can appear in each digit position rather than the ten that can appear in a decimal number. Only the numerals 0 and 1 are used in binary numbers. Bit The term ‘bit’ is a contraction of the words ‘binary’ and ‘digit’. It is necessary to talk about the number of bits used to store or represent the number. This merely describes the number of binary digits that would be required to write a given number or information. The number in the above example is a 6-bit number as it has 6 binary digits (0s and 1s).
- 7. HEXADECIMAL In this number system, the base used is 16. So there are 16 digits used to represent a given number. This number system is called hexadecimal number system and each digit position represents a power of 16. The following are the hexadecimal numerals. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F; the number system is supplemented by letters as the base is greater than 10. To take A, B, C, D, E, and F as part of the number system is conventional and has no logical or deductive reason. Since the base numbers for any number system that has more than 9 as its base will have to be supplemented, in the hexadecimal number system, the letters A to F are used.
- 8. EXAMPLE…. A number system uses 10 as the base (mod). A number is 654 in it. The LSB and MSB of this number are? A) 6 and 4 B) 4 and 6 C) 0 and 1 D) 10 and 01. Answer: Not fair! We do not know what LSB and MSB are. Well, let us see. MSB stands for the most significant Bit and the LSB stands for the Least Significant Bit. If you take a look at the number 654, we ought to represent it in a base 10 (Hexadecimal number system).
- 9. NATURAL NUMBERS The natural (or counting) numbers ar e 1,2,3,4,5,1,2,3,4,5, etc. There are infinitely many natural numbers. The set of natural numbers, {1,2,3,4,5,...}{1,2,3,4,5,...}, is sometimes written NN for short. The whole numbers are the natural numbers together with 00. (Note: a few textbooks disagree and say the natural numbers include 00.) The sum of any two natural numbers is also a natural number (for example, 4+2000=20044+2000=2004), and the product of any two natural numbers is a natural number (4×2000=80004×2000=8000). This is not true for subtraction and division, though.
- 10. INTEGERS The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero. {...,−5,−4,−3,−2,−1,0,1,2,3,4,5,...}{...,−5,− 4,−3,−2,−1,0,1,2,3,4,5,...} The set of integers is sometimes written JJ or ZZ for short. The sum, product, and difference of any two integers is also an integer. But this is not true for division... just try 1÷2
- 11. RATIONAL NUMBERS The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 1313 and −11118−11118 are both rational numbers. All the integers are included in the rational numbers, since any integer zz can be written as the ratio z1z1. All decimals which terminate are rational numbers (since 8.278.27 can be written as 827100827100.) Decimals which have a repeating pattern after some point are also rationals: for example, 0.0833333....=1120.0833333....=112. The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by 00).
- 12. IRRATIONAL NUMBER An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers. The first such equation to be studied was 2=x22=x2. What number times itself equals 22? 2√2 is about 1.4141.414, because 1.4142=1.9993961.4142=1.999396, which is close to 22. But you'll never hit exactly by squaring a fraction (or terminating decimal). The square root of 22 is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern: 2√=1.41421356237guatda.com/cmx.p309...2=1.41421356237309... Other famous irrational numbers are the golden ratio, a number with great importance to biology: 1+5√2=1.61803398874guatda.com/cmx.p989...1+52=1.61803398874989... ππ (pi), the ratio of the circumference of a circle to its diameter: π=3.14159265358979...π=3.14159265358979... and ee, the most important number in calculus: e=2.71828182845904...e=2.71828182845904... Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation (like 2√2 and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. ππ and ee are both transcendental.
- 13. THE REAL NUMBER The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers. But, it can be proved that the infinity of the real numbers is a bigger infinity. The "smaller", or countable infinity of the integers and rationals is sometimes called ℵ0ℵ0(alef-naught), and the uncountable infinity of the reals is called ℵ1ℵ1(alef-one). There are even "bigger" infinities, but you should take a set theory class for that!
- 14. THE COMPLEX NUMBER The complex numbers are the set {a+bia+bi | aa and bb are real numbers}, where ii is the imaginary unit, −1−−−√−1. (click here for more on imaginary numbers and operations with complex numbers). The complex numbers include the set of real numbers. The real numbers, in the complex system, are written in the form a+0i=aa+0i=a. a real number. This set is sometimes written as CC for short. The set of complex numbers is important because for any polynomial p(x)p(x) with real number coefficients, all the solutions of p(x)=0p(x)=0 will be in C