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Rational number is the collective name of integer (positive integer, 0, negative integer) and fraction, and it is the set of integer and fraction.

Integers can also be regarded as fractions with a denominator of one. Real numbers that are not rational numbers are called irrational numbers, that is, the fractional part of irrational numbers is infinite cyclic numbers. It is one of the important contents in the field of number and algebra, and it is widely used in real life. It is the basis for continuing to learn real numbers, algebraic expressions, equations, inequalities, rectangular coordinate systems, functions, statistics and other mathematical contents and related disciplines.

The set of rational numbers can be represented by the capital black orthographic symbol Q, but Q does not represent rational numbers. The set of rational numbers and rational numbers are two different concepts. The set of rational numbers is the set of all rational numbers, and rational numbers are all elements in the set of rational numbers.

Chinese name

rational number

Foreign name

rational number

definition

Integer and fraction

Show time

About 580 BC to 500 BC

Scope of ownership

real number

quick

navigate by water/air

Basic arithmetic, mixed arithmetic, related problems

brief introduction

Named source

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. [ 1]

Understanding of rational numbers

Rational numbers are integers (positive integer, 0, negative integer) and fractions. Positive integers and fractions are collectively referred to as positive rational numbers, and negative integers and fractions are collectively referred to as negative rational numbers. Therefore, the number of rational number sets can be divided into positive rational numbers, negative rational numbers and zero. Since any integer or fraction can be converted into a cyclic decimal, and vice versa, every cyclic decimal can also be converted into an integer or fraction, so a rational number can also be defined as a cyclic decimal.

The set of rational numbers is an extension of the set of integers. In the set of rational numbers, the four operations of addition, subtraction, multiplication and division (divisor is not zero) are unimpeded.

Provisions on the order of magnitude of rational numbers A and B: If a-b is a positive rational number, it is said that when A is greater than B or B is less than A, it is marked as A >；; B or b < a. Any two unequal rational numbers can compare sizes.

An important difference between rational number set and integer set is that rational number set is dense while integer set is dense. After arranging rational numbers in order of size, there must be other rational numbers between any two rational numbers, which is density. Integer sets do not have this feature, so there are no other integers between two adjacent integers.

Rational numbers are compact subsets of real numbers: every real number has a rational number that is arbitrarily close. A related property is that only rational numbers can be transformed into finite continued fractions. Rational numbers have an ordered topology according to their sequences. Rational number is a (dense) subset of real number, so it also has subspace topology. [ 1]

Rational Numbers and Their Classification

According to different standards, rational numbers can be divided into the following two types:

(1) Classification according to the definition of rational numbers: [2]

(2) According to the nature of rational numbers: [2]

Rational number classification

The basic algorithm

Add operation

1, add two numbers with the same sign, take the same sign as the addend, and add the absolute values.

2. Add two numbers with different signs. If the absolute values are equal, the sum of two numbers with opposite numbers is 0; If the absolute values are not equal, take the sign of the addend with the larger absolute value and subtract the smaller absolute value from the larger absolute value.

3. Add two numbers with opposite numbers to get 0.

4. Adding a number to 0 still gets this number.

You can add two opposite numbers first.

6. Numbers with the same sign can be added first.

7. Numbers with the same denominator can be added first.

8. If several numbers can be added to get an integer, they can be added first. [ 1]

subtraction

Subtracting a number is equivalent to adding the reciprocal of this number, that is, the subtraction of rational numbers is converted into addition. [ 1]

multiply operation

1, the same sign is positive, the different sign is negative, and the absolute value is multiplied.

Any number multiplied by zero will get zero.

3. Multiply several numbers that are not equal to zero. The sign of the product is determined by the number of negative factors. When there are odd negative factors, the product is negative, and when there are even negative factors, the product is positive.

When several numbers are multiplied, if one factor is zero, the product is zero.

5. Multiply several numbers that are not equal to zero, first determine the sign of the product, and then multiply it by the absolute value. [ 1]

Division operation

1 divided by a number that is not equal to zero is equal to the reciprocal of this number.

2. Divide two numbers, the same sign is positive and the different sign is negative, and divide by the absolute value. Divide zero by any number that is not equal to zero to get zero.

note:

Zero can't be a divisor and denominator.

Division and multiplication of rational numbers are reciprocal operations.

In the division operation, according to the law that the same sign is positive and the different sign is negative, the sign is determined first and then divided by the absolute value. If there is a fraction in the formula, it is usually calculated by turning it into a false fraction first. If it is not divisible, all division operations are converted into multiplication operations. [ 1]

Real number classification diagram

Power operation

1, the odd power of a negative number is negative, and the even power of a negative number is positive. For example: (-2) 3 (the third power of 2) =-8, (-2) 2 (the second power of 2) =4.

2. Any degree of a positive number is a positive number, and any degree of a positive number is zero. For example: 2 (the second power of 2) =4, 2 (the third power of 2) =8, 0 (the third power of 0) =0.

The zeroth power of zero is meaningless.

4. Because power is a special case of multiplication, the multiplication of rational numbers can be completed by the multiplication of rational numbers.

5. The arbitrary power of 1 is 1, the even power of-1 is1and the odd power is-1. [ 1]

Law of rational number operation

Law of addition:

1, additive commutative law: Two numbers are added, the position of the addend is exchanged, and the sum is unchanged, that is.

2, the law of addition and association: three numbers are added, the first two numbers are added first or the last two numbers are added first, and the sum is unchanged, that is.

Subtraction algorithm:

Law of subtraction: subtracting a number is equal to adding the reciprocal of this number. Namely:

.

Multiplication algorithm:

1, multiplication and conversion law: when two numbers are multiplied, the position of the exchange factor remains unchanged, that is.

2. Multiplication and association law: when three numbers are multiplied, the first two numbers are multiplied first, or the last two numbers are multiplied first, and the product is unchanged, that is.

3. Multiplication and distribution law: multiplying a number by the sum of two numbers is equivalent to multiplying this number by these two numbers respectively, and then adding the products, namely:

Hybrid algorithm

For the mixed operation of addition, subtraction, multiplication and division of rational numbers, if there is no bracket to indicate what operation to do first, it will be performed in the order of "multiplication and division first, then addition and subtraction", and if it is the same level operation, it will be calculated from left to right.

relevant issues

The fallacy of dividing by zero

Improper use of divisor in algebraic operation will lead to invalid proof;

. prerequisite

unequal to

.

From: 0a=0 and 0b=0, 0a=0b is obtained.

Divide both sides by zero to get 0a/0=0b/0.

Simplify and get: a = b.

The above fallacy assumes that a number divided by 0 is allowed, and. [ 1]

Algebraic processing

If a mathematical system obeys the axiom of a field, then dividing by zero in the mathematical system must be meaningless. This is because division is defined as the inverse of multiplication, that is

Value is an equation.

middle

The solution (if any). Ruohypothesis

, equation

Can be written as

Or directly

. Therefore, the equation

No solution (when

When), but any numerical value can also solve this equation (when

When). [ 1]

integer

Integer is the general name of all numbers in the sequence ...,-3,-2,-1,0,1,2,3, ...}, including negative integer, zero (0) and positive integer. Like natural numbers, integers are a countable infinite set. Mathematically, this set is usually expressed as a bold Z or, which comes from the first letter of the German word Zahlen (meaning "number").

In algebraic number theory, these general integers belonging to rational numbers will be called rational integers to distinguish them from concepts such as Gaussian integers.

All integers form a ring about addition and multiplication. The whole ring, zero-factor-free ring and unique decomposition domain in ring theory can be regarded as abstract models of integers.

Z is an additive cyclic group because any integer is the sum of several 1 or-1. 1 and-1 are the only two generators of z, and a cyclic group with infinite elements is isomorphic to (z,+). [ 1]

put right

reference data

[1] Curriculum Textbook Institute, compiled by Development Center of Middle School Mathematics Curriculum Textbook Institute. People's Education Press.2012.

[2] Qu Yixian. List of junior high school mathematics knowledge. Capital Normal University Press/Education Science Press. 2065 438+04 03: Chapter 1 Numbers and Algebra.