Number - Wikipedia. A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers. In common usage, number may refer to a symbol, a word, or a mathematical abstraction. In mathematics, the notion of number has been extended over the centuries to include 0.
Their study or usage is called arithmetic. The same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Today, number systems are considered important special examples of much more general categories such as rings and fields, and the application of the term . The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. Roman numerals remained dominant in Europe until the spread of the superior Arabic numeral system nearing the late 1.
Arabic numeral system remains the most common system for representing numbers in the world today. For systems for expressing numbers, see Numeral system. Numbers can be classified into sets, called number systems, such as the natural numbers and the real numbers.
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Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 1. The mathematical symbol for the set of all natural numbers is N, also written N. The radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 1. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss. Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.
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Integers. Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written . When the set of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of integers, Z also written Z. Here the letter Z comes from German.
Zahl, meaning 'number'. The set of integers forms a ring with the operations addition and multiplication. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as positive integers, and the natural numbers with zero are referred to as non- negative integers.
Rational numbers. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction m/n represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is: 1.
Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example . The symbol for the rational numbers is Q (for quotient), also written Q.
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The symbol for the real numbers is R, also written as R. Real numbers are usually represented by using decimal numerals, in which a decimal point is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one- tenth of the place value of the digit to its left. For example, 1. 23.
A finite decimal representation allows us to represent exactly only the integers and those rational numbers whose denominators have only prime factors which are factors of ten. Thus one half is 0. To represent the rest of the real numbers requires an infinite sequence of digits after the decimal point.
Since it is impossible to write infinitely many digits, real numbers are commonly represented by rounding or truncating this sequence, or by establishing a pattern, such as 0. Thus 1. 23. 4. 56 is an approximation of any real number between 1. Negative real numbers are written with a preceding minus sign: - 1. Every rational number is also a real number. It is not the case, however, that every real number is rational. A real number that is not rational is called irrational. A decimal represents a rational number if and only if it has a finite number of digits or eventually repeats forever, after any initial finite string of digits.
For example, 1/2 = 0. On the other hand, the real number .
Other irrational numbers include. There are infinitely many other ways of representing the number 1, for example 1. Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the least upper bound property. When a real number represents a measurement, there is always a margin of error.
This gives rise to rounding or truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself can grant are removed. The remaining digits are called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0. If the sides of a rectangle are measured as 1.
Since not even the second digit after the decimal place is preserved, the following digits are fully insignificant. Therefore, sensibly, the result is usually rounded to 5. It can be shown that any ordered field, which is also complete, is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field, because they do not include a solution (often addressed as the square root of minus one) to the algebraic equation x.
This set of numbers arose historically from trying to find closed formulas for the roots of cubic and square polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a square root of . The complex numbers consist of all numbers of the forma+bi. Because of this, complex numbers correspond to points on the complex plane, a vector space of two real dimensions.
In the expression a + bi, the real number a is called the real part and b is called the imaginary part. If the real part of a complex number is 0, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or C.
Like the real number system, the complex number system is a field and is complete, but unlike the real numbers, it is not ordered. That is, there is no consistent meaning assignable to saying that i is greater than 1, nor is there any meaning in saying that i is less than 1. In technical terms, the complex numbers lack of a total order that is compatible with field operations. Subclasses of the integers.
The first few prime numbers are 2, 3, 5, 7, and 1. The prime numbers have been widely studied for more than 2.
The study of these questions belong to number theory. An example of a question that is still unanswered is whether every even number is the sum of two primes.
This is called Goldbach's conjecture. A question that has been answered is whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes.
This is called fundamental theorem of arithmetic. A proof appears in Euclid's Elements. Other classes of integers. Example of such sets of integers are Fibonacci numbers and perfect numbers. For more examples, see Integer sequence.
Subclasses of the complex numbers. Real numbers that are not rational numbers are called irrational numbers. Complex numbers which are not algebraic are called transcendental numbers. The algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers. Computable numbers. Equivalent definitions can be given using . The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers.
The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not. The set of computable numbers has the same cardinality as the natural numbers.
Therefore, almost all real numbers are non- computable. However, it is very difficult to produce explicitly a real number that is not computable. Extensions of the concept.
The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties. The set of the p- adic numbers contains the rational numbers, but is not contained in the complex numbers. The elements of an algebraic function field over a finite field and algebraic numbers have many similar properties (see Function field analogy). Therefore, they are often regarded as numbers by number theorists.
The p- adic numbers play an important role in this analogy. Hypercomplex numbers.
They are sometimes called hypercomplex numbers.
The Resurrection of Jesus Christ, Fact or Fiction? The Resurrection of Jesus Christ is either .. Did Jesus Christ actually rise from the dead? Greenleaf, the Royal Professor of Law at Harvard University, was one of the greatest legal minds that ever lived.
He wrote the famous legal volume entitled, A Treatise on the Law of Evidence, considered by many the greatest legal volume ever written. Simon Greenleaf believed the Resurrection of Jesus Christ was a hoax. And he determined, once and for all, to expose the .
After thoroughly examining the evidence for the resurrection — Dr. Greenleaf came to the exact opposite conclusion! He wrote a book entitled, An Examination of the Testimony of the Four Evangelists by the Rules of Evidence Administered in the Courts of Justice. In which he emphatically stated. Greenleaf was so convinced by the overwhelming evidence, he committed his life to Jesus Christ! What changed his mind?
What evidence did Dr. Greenleaf encounter that so drastically turned him around? What facts did he discover that he could not rationally ignore? In order to answer that, let's go back to that scene nearly 2,0.
That eery night in the garden of Gethsemene, as a blood- thirsty mob came and arrested Jesus Christ. He was brought before Pilate, and there in plain view, Jesus Christ was spit upon, cursed, he was laughed at, whipped, beaten. He was branded a blasphemer. And He was sentenced to die by the most extreme, shameful form of capital punishment ever devised — crucifixion.
Galatians 3: 1. 3 reads, . He died as a criminal. He died in open shame and disgrace. In the eyes of the world, Jesus Christ died a total disgrace! And there stood His apostles — in stunned unbelief. Here was their hope.
Here were 1. 1 men that had forsaken all and followed this man. They believed in this man for 3 years. Can you imagine what was going on in their minds? Didn't He heal the sick? Didn't we see Him walk on water? Didn't we see Him raise people from the dead? Didn't we see Him still the raging storm?
And to end like this? THAT NIGHT, THERE STOOD 1. DEFEATED AND CONFUSED MEN! They had given up everything for naught. And then to die like He did.
A complete disgrace! And Mark 1. 4: 5. Jesus was dead! He was crucified as a blasphemer! One of the women saw Peter and said, . The Bible says . Less than two months later — they went back into Jerusalem boldly preaching, at the threat of death, that — JESUS CHRIST WAS ALIVE! Listen to Peter in Acts 2,. THIS JESUS HATH GOD RAISED UP, WHEREOF WE ALL ARE WITNESSES.
But Peter and John answered and said unto them, Whether it be right in the sight of God to hearken unto you more than unto God, judge ye. What had these men seen? What had these men heard? What was it that beatings, threatenings, jail, torture and even death could not silence? In Acts 5, they put in jail and beaten — and verse 4.
Two major problems: One, If the body of Jesus Christ was available, you'd better believe, the Roman authorities would have paraded His body for all to see. Christianity was spreading like a wild fire! And nothing could have quenched that flame quicker than the body of Jesus Christ!
An article, suggesting Jesus faked His death and resurrection, appeared in the prestigious British Journal of the Royal College of Physicians. The article makes the assertion, . They were there! They died a horrible and martyrs death!
John's College, Nottingham, writes concerning the apostles, . You could imprison them, flog them, kill them, but you could not make them deny their conviction that on the third day he rose again. Greenleaf wrote, . And to exterminate Christianity totally, became his passion!
The Bible describes him. One day traveling the dusty road to Damascus, Saul burning with vengeance, with documents authorizing him to capture all the followers of Jesus.
But on the road to Damascus something happened to Saul of Tarsus. And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest. For Saul of Tarsus, among the greatest enemies of Jesus Christ, that ever lived — became the Apostle Paul — THE GREATEST CHRISTIAN THAT EVER LIVED! What happened to this man? What could so transform this man, that he completely dedicated his life to spreading the gospel of Jesus Christ? And was later beheaded at Rome in 6.
A. D. There is only one possible explanation —THE RESURRECTION OF JESUS CHRIST! Gilbert West and Lord Lyttleton, two Oxford students, believed Christianity was a . Lyttleton resolved to disprove the conversion of Saul of Tarsus, and West would refute the resurrection of Jesus Christ. They figured a careful, rational, examination of the evidence would easily disprove the Christian faith. But after examining the evidence — they both separately came to the opposite conclusion! Lyttleton concluded that Saul of Tarsus did, in fact, convert to the Apostle Paul through Jesus Christ.
And Gilbert West concluded that the Resurrection of Jesus Christ was among the best established facts in all of history! West went on to write a book entitled, . Professor Thomas Arnold, former chair of history at Oxford, and author of the famous volumes, History of Rome, was skillfully educated in the study of historical facts.
Professor Arnold, stated, . And friend that's a very serious thing to do. If Jesus Christ actually rose from the dead — what He said about ! And Jesus Christ warned of a place called hell — over and over! He mentioned hell eight times more than heaven! In Luke 1. 6: 2. 3, Jesus described a man that had died, and Jesus said, . If not, Friend you could be — one year, one month, one hour, one minute, one heartbeat — away from eternity in a lake of fire!
Revelation 2. 0: 1. Peter 3: 9 says, . Revelation 1: 5 says, . Unto him that loved us, and washed us from our sins in his own blood.! For all have sinned, and come short of the glory of God.
Unto him that loved us, and washed us from our sins in his own blood. Lord Jesus, I know that I am a sinner, and unless you save me I am lost forever. I thank you for dying for me at Calvary. I come to you now, the best way I know how, and ask you to save me. I now receive you as my Savior. In Jesus Christ name, Amen.