Real numbers can be divided into rational numbers and irrational numbers.
The difference between irrational numbers and rational numbers:
1. When both rational and irrational numbers are written as decimals, rational numbers can be written as finite decimals and infinite cyclic decimals.
For example, 4 = 4.0, 4/5 = 0.8, 1/3 = 0.33333 ... and irrational numbers can only be written as infinite acyclic decimals.
For example, √ 2 =1.414213562. ..............................................................................................................................
2. All rational numbers can be written as the ratio of two integers; And irrational numbers can't. Accordingly, it is suggested that irrational numbers should be labeled as "unreasonable", and rational numbers should be renamed as "comparative numbers" and irrational numbers as "non-comparative numbers". After all, irrational numbers are not unreasonable, but people didn't know much about them at first.
Using the main differences between rational numbers and irrational numbers, it can be proved that √2 is irrational.
Proof: Suppose √2 is not an irrational number, but a rational number.
Since √2 is a rational number, it must be written as the ratio of two integers:
√2=p/q
Since P and Q have no common factor that can be reduced, p/q can be regarded as a reduced fraction, that is, the simplest form of fraction.
Square √ 2 = both sides of p/q.
Get 2 = (p 2)/(q 2)
That is 2 (q 2) = p 2.
Because 2q^2 is even, P must be even. Let p=2m.
From 2 (q 2) = 4 (m 2)
Q 2 = 2m 2。
Similarly, q must be an even number, let q=2n.
Since both P and Q are even numbers, they must have a common factor of 2, which contradicts the previous assumption that p/q is a fractional number. This contradiction is caused by the assumption that √2 is a rational number. So √2 is an irrational number.
Origin:
Pythagoras (about 885 BC to 400 BC) was very clever from an early age. Once he was walking in the street with firewood on his back. An elder saw that his method of bundling firewood was different from others, and said, "This child is a genius in mathematics and will become a scholar in the future." Hearing this, he left firewood, crossed the Mediterranean and went to tellez Gate to study. Pythagoras is very clever. Under Taylor's guidance, many math problems were solved by him. Among them, he proved that the sum of the internal angles of a triangle is equal to 180 degrees; It can be calculated that if you want to use ceramic tiles to pave the floor, only the bricks of three regular polygons, namely regular triangle, regular quadrangle and regular hexagon, can just pave the floor. It is also proved that there are only five regular polyhedrons in the world, namely, regular 4, 6, 8, 12 and icosahedron. He also discovered odd numbers, even numbers, triangular numbers, quadrilateral numbers, perfect numbers, friendship numbers, and even Pythagoras numbers. But his greatest achievement was the discovery of Pythagorean theorem (Pythagorean theorem) named after him later, that is, the sum of the areas of the squares of the two right-angled sides of a right-angled triangle is equal to the area of the square of the hypotenuse. It is said that Pythagoras invented this method when he saw craftsmen paving the floor with square bricks in the temple and often had to calculate the area.
After Pythagoras skillfully used mathematical knowledge, he felt that he could not be satisfied with solving problems, so he tried to expand from the field of mathematics to the field of philosophy and explain the world from the perspective of numbers. After some hard training, he put forward the view that "everything is counted". The elements of numbers are the elements of all things, and the world is made up of numbers. Everything in the world can't be expressed by numbers, and numbers themselves are the order of the world. Pythagoras also established the Youth Brotherhood around him. About 200 years after his death, his disciples developed this theory and formed a powerful Pythagorean school.
One day, members of the school had just finished an academic seminar and were taking a cruise to enjoy the scenery to dispel the fatigue of the day. On this day, it was a sunny day, and the sea breeze blew gently, causing layers of waves. Everyone is very happy. A bearded scholar looked at the vast sea and said excitedly, "Mr Pythagoras' theory is not bad at all. "You see the waves are layered, with peaks and valleys, just like odd and even numbers. The world is the order of numbers. " "Yes, yes." Then a big man who was rowing came in and said, "Let's talk about this boat and the sea. Measuring sea water by ship is sure to get an accurate figure. Everything can be represented by numbers. "
"I don't think so." At this moment, a scholar at the stern suddenly asked a question. He said quietly, "What if it's not an integer in the end?"
"That's a decimal." "What if the decimal is not divisible?"
"Impossible, everything in the world can be directly and accurately expressed by numbers."
At this time, the scholar said calmly in a tone that he didn't want to argue any more: "Not everything in the world can be expressed by the numbers we know now. Take the right triangle, which Mr Pythagoras studied most, as an example. If it is an isosceles right-angled triangle, you can't accurately measure the hypotenuse with a right-angled side. "
The scholar who raised this question is hippasus, a clever, diligent and independent mathematician in Pythagoras School. If it weren't for today's debate, I wouldn't want to express my new views. Hearing this, the big man who paddled stopped and shouted, "No way, Mr. Wang's theory can be applied everywhere." Hippasus blinked his clever eyes and held out his hands, comparing the two escapes to an isosceles right triangle.
"If the straight side is 3, what is the hypotenuse?"
"4。"
"More accurate?"
"4.2。"
"More accurate?"
"4.24。"
"More accurate?"
The big man flushed and couldn't answer for a moment. Hippasus said, "You can't count 10 or 20 digits in the future, which is the most accurate. I have calculated many times that one side and the other side of any isosceles right triangle cannot be represented by an accurate number. " It was like a bolt from the blue, and the whole ship immediately burst into a roar: "How dare you violate Mr. Pythagoras' theory and destroy the creed of our school!" "Dare to believe that numbers are the world!" Hippasus was very calm at this moment. He said, "This is a new discovery. Even Mr Pythagoras will reward me when he is alive. You can check it at any time. " But people didn't listen to his explanation and shouted angrily, "rebellion!" Mr.' s unscrupulous disciple. " "Kill him! Batch him to death! " Beard rushed up and punched him in the chest. Hippasus protested: "You ignore science, you are so unreasonable!" "The creed of the guardian is always justified. "At this time, the big man also rushed over and picked him up at once:" Give you a highest reward! " He said, and threw hippasus into the sea. The blue sea soon flooded his body and never came out again. At this time, several white clouds were floating in the sky, and several waterfowl passed by the sea. After a storm, the Mediterranean sea seems to be quiet again.
In this way, a very talented mathematician was destroyed by a scholar of slave autocracy. But it does make people see the ideological value of hippasus. After this incident, the Pythagorean school members really discovered that not only the right side of an isosceles right triangle can not measure the hypotenuse, but also the diameter of a circle can not measure the circumference. That number is 3.141592653589726 ... and it will never be accurate. Slowly, they feel regret, regret the unreasonable action to kill hippasus. They gradually understand that intuition is not absolutely reliable, and some things must be proved by science; They understand that in the past, in addition to rational numbers such as the number "0" and natural numbers, there were some infinite acyclic decimals. This is really a newly discovered number-it should be called "irrational number". This name reflects the original appearance of mathematics, but it also truly records the arrogance of Pythagoras school.
The mathematical crisis caused by irrational numbers continued until19th century. 1872, the German mathematician Dai Dejin started from the requirement of continuity, defined irrational numbers through the division of rational numbers, and established the theory of real numbers on a strict scientific basis, thus ending the era when irrational numbers were regarded as "irrational numbers" and the first great crisis in the history of mathematics that lasted for more than two thousand years.
Rational number (rational number):
Infinitely cyclic decimals and numbers with infinite roots are called irrational numbers.
Integers and fractions are collectively called rational numbers.
Mathematically, a rational number is the ratio of two integers, usually written as a/b, where b is not zero. Fraction is a common expression of rational number, while integer is a fraction with denominator of 1, which is also a rational number.
Include integers and fractions, and can also be expressed as finite decimals or infinite cyclic decimals.
This definition applies to decimal and other decimal (such as binary) numbers. Infinitely cyclic decimals and numbers with infinite roots are called irrational numbers.
Integers and fractions are collectively called rational numbers.
Mathematically, a rational number is the ratio of two integers, usually written as a/b, where b is not zero. Fraction is a common expression of rational number, while integer is a fraction with denominator of 1, which is also a rational number.
Mathematically, a rational number is the ratio of an integer a to a nonzero integer b, which is usually written as a/b, so it is also called a fraction. The Greek name is λ ο γ ο? The original meaning is "rational number", but the Chinese translation is not appropriate, and it has gradually become "rational number". Real numbers that are not rational numbers are called irrational numbers.
The set of all rational numbers is expressed as q, and the fractional part of rational numbers is finite or cyclic.
reference data
For example, 3, -98. 1 1, 5.272 and 7/22 are all rational numbers.
Rational numbers can also be divided into positive rational numbers, negative rational numbers and 0.
All rational numbers form a set, that is, rational number set, which is represented by bold letter Q, while some modern math books are represented by hollow letter Q.
The rational number set is a subset of the real number set. For related contents, see number system expansion.
A set of rational numbers is a field, that is, four operations can be performed in it (except that 0 is a divisor). For these operations, the following algorithms hold (a, b, c, etc. Represents any rational number):
① the commutative law of addition A+B = B+A;
② the associative law of addition A+(B+C) = (A+B)+C;
(3) There is a number 0, so that 0+a = a+0 = a;
(4) For any rational number A, there is an addition inverse element, which is denoted as -a, so that a+(-a) = (-a)+a = 0;
⑤ The commutative law of multiplication AB = BA
⑥ Multiplicative associative law A (BC) = (AB) C;
⑦ Distribution law A (B+C) = AB+AC;
⑧ Multiplication has a unit 1≠0, so that for any rational number A,1a = a1= a;
Pet-name ruby For rational number A which is not 0, there is a multiplication inverse 1/a, so a (1/a) = (1/a) A =1.
⑩0a=0
In addition, rational number is an ordered domain, that is, there is an ordered relation ≤
Rational number is also an Archimedes field, that is, rational numbers a and b, a≥0, B >；; 0, we can find a natural number n, which makes nb >；; Answer: It is not difficult to infer that there is no maximum rational number.
The name rational number is worth mentioning. The name "rational number" is puzzling, and rational numbers are no more "reasonable" than other numbers. In fact, this seems to be a mistake in translation. The word rational number comes from the west and is rational in English. Rational usually means "rational". China translated western scientific works in modern times into "rational numbers" according to Japanese translation methods. However, this word comes from ancient Greece, and its English root is ratio, which means ratio (the root here is English and the Greek meaning is the same). So the meaning of this word is also very clear, that is, the "ratio" of integers. In contrast, "irrational number" is a number that cannot be accurately expressed as the ratio of two integers, but it is not unreasonable.
Rational number addition and subtraction mixed operation
1. The significance of unifying addition and subtraction of rational numbers into addition;
For subtraction in addition and subtraction mixed operation, we can convert subtraction into addition according to the rational number subtraction rule, thus unifying the mixed operation into addition operation. The unified formula is the sum of several positive numbers or negative numbers. We call this formula algebraic sum.
2. The method and steps of rational number addition and subtraction mixed operation:
(1) The subtraction in rational number mixing operation is converted into addition by using the subtraction rule.
(2) Use the law of addition, additive commutative law and the law of addition combination to perform simple operations.
The concepts of absolute value and reciprocal in rational number range have the same meaning in real number range.