Chapter 1 The concept of number and number operation (1) The meaning of integer 1: both natural numbers and 0 are integers. 2 natural numbers: when we count objects, 1, 2, 3 ... the numbers used to represent the number of objects are called natural numbers. There is no object. 0 means. 3 Counting units one (one), ten, one hundred, one thousand, ten thousand, one million, ten million, one hundred million ... are all counting units. The propulsion rate between every two adjacent counting units is 10. This counting method is called decimal counting method. Four digital counting units are arranged in a certain order, and their positions are called numbers. Five numbers. The quotient of division is an integer without remainder, so we say that A can be divisible by B, or that B can be divisible by A. If the number A can be divisible by the number B (b≠0), A is called a multiple of B, and B is called a divisor. Multiplication and divisor are interdependent. Because 35 is divisible by 7, 35 is a multiple of 7. 7 is the divisor of 35. 7. What is the ratio? The division of two numbers is called the ratio of two numbers. For example, the first term and the last term of the ratio of 2÷5 or 36 or 13 are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged. 8. What is the ratio? The formula that two ratios are equal is called ratio. Such as 36 = 965438. The product of two external terms is equal to the product of two internal terms. 10, solution ratio: Finding the unknown term in the ratio is called solution ratio. For example, 3 χ = 9 18 1, and proportional relationship: two related quantities, one of which changes and the other changes. If there is a corresponding proportional relationship between the two quantities, their relationship is called proportional relationship. For example, yx=k(k must be) or kx=y 12, which is inversely proportional: two related quantities, one of which changes and the other changes. If the product of the corresponding two numbers in these two quantities is certain, these two quantities are called inverse proportional quantities. Their relationship is called inverse relationship. For example, x×y=k(k must be) or kx=y percentage: the number indicating that one number is a percentage of another number is called a percentage. Percentage is also called percentage or percentage. 13. To convert a decimal into a percentage, just move the decimal to the right by two places and add hundreds of semicolons. In fact, to convert a decimal into a percentage, just multiply this decimal by 100%. To convert a percentage into a decimal, just remove the percent sign and move the decimal two places to the left. 14, and turn a fraction into a percentage. Usually, the fraction is converted into a decimal (three decimal places are usually reserved when it is not used up), and then it is converted into a percentage. In fact, to turn a fraction into a percentage, we must first turn it into a decimal. Multiply by 100%. For the percentage component number, first, the percentage is rewritten as the component number, and it can be turned into the simplest fraction. 15. Learn how to divide the fractional number into decimals. 16. greatest common divisor: several numbers can be divisible by the same number at the same time, and this number is called the maximum value of these numbers. It is called the common divisor of these numbers. The largest one is called the greatest common divisor. ) 17, prime number: the common divisor has only two numbers 1, which is called prime number. 18, least common multiple: the multiple shared by several numbers is called the common multiple of these numbers. The smallest one is called the least common multiple of these numbers. 19. Comprehensive score: the score of different denominators is changed into the score of the same denominator equal to the original score, which is called comprehensive score. (Generally, the lowest common multiple is used for scores) 20. Approximation: to change a fraction into an equal fraction, but the numerator and denominator are small, which is called reduction. (Approximation uses the greatest common divisor. ) It is called simplest fraction. At the end of the score calculation, the number must be converted into the simplest score. Numbers with digits of 0, 2, 4, 6 and 8 can all be divisible by 2, that is to say, they can be divisible by 2. Numbers with 0 or 5 digits can be divisible by 5, that is to say, they can be divisible by 5. Pay attention to .22, even number and odd number when dividing: the number divisible by 2 is called even number. They are not divisible by 5. If 1 and itself have only two divisors, such numbers are called prime numbers (or prime numbers) .24. Composite number: a number. If there are other divisors besides 1 and itself, such a number is called a composite number. 1 is neither prime nor composite. 28. Interest = principal × interest rate× time (. Interest rate: The ratio of interest to principal is called interest rate. The ratio of interest to principal within one year is called annual interest rate. The ratio of interest to principal in January is called monthly interest rate. 30. Natural number: an integer used to represent the number of objects, called natural number. 0 is also a natural number. 3 1. Cyclic decimal: a decimal, starting from a certain position in the decimal part, a decimal. This decimal is called a recurring decimal. (2) Meaning of decimals 1 Divide integers 1 0, 100, 1000 ... Decimals, percentages, thousandths ... can be expressed in decimals. A decimal place represents a decimal. Three decimal places represent thousands ... A decimal place consists of an integer part, a decimal part and a decimal point part. The point in the number is called the decimal point, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part. In decimals, the series between every two adjacent counting units is 10. The progressive rate between the highest decimal unit "one tenth" of the decimal part and the lowest unit "one" of the integer part is also 10. 2. Decimals whose integer part is zero are called pure decimals. For example, 0.25 and 0.368 are pure decimals. For decimals, the integer part is not. It's called a decimal. For example, 3.25 and 5.26 are all decimals. Finite decimals: The digits in the decimal part are finite decimals, which are called finite decimals. For example, 4 1.7, 25.3 and 0.23 are all finite decimals. Infinite decimal: The digits in the decimal part are infinite decimal. It's called infinite decimal. For example: 4.33...3. 14 15926 ... infinite acyclic decimal: the decimal part of a number, in which the digits are arranged irregularly and the digits are infinite, is called infinite acyclic decimal. Cyclic decimal: The decimal part of a number in which one or more numbers are repeated. This number is called cyclic decimal. For example, the decimal part of 3.555...0.0333 ... cycle decimal, and the repeated numbers in turn are called the cycle segment of this cycle decimal. For example, the cycle part of 3.99 ... is "9" and the cycle part of 0.5454 ... is "54". Pure cyclic decimal: the cyclic part starts from the first position of the decimal part. It is called pure cyclic decimal. For example: 3. 1 1...0.5656 ... mixed cycle decimal: the cycle section does not start from the first decimal part, which is called mixed cycle decimal. 3. 1222 ...0.03333 ...(3) The score is 1. The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are. Divide the unit "1" into several parts to represent a part of the number, which is called decimal unit. 2 Classification of Fractions True Fractions: Fractions with numerator less than denominator are called true fractions. The true score is less than 1. False fraction: A fraction whose numerator is greater than or equal to the denominator is called a false fraction. False score is greater than or equal to 1. With fraction: False fraction can be written as a number composed of integer and true fraction, which is usually called with fraction. 3 Reduced fraction and general fraction A fraction is converted into a fraction equal to it, but the numerator and denominator are compared, which is called a reduced fraction. The denominator of a numerator is a fraction of a prime number, which is called simplest fraction. Different denominator scores are converted into the same denominator score equal to the original score. It's called score. (4) Percentage 1 indicates that one number is the percentage of another number, also called percentage or percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage. Method (1) read and write the number 1, and read the integer: from high to low, step by step. Add a word "100 million" or "10 thousand" after it. The zeros at the end of each level are not read, and only a few zeros of other digits are read. 2. Writing of integers: from high to low, write step by step, where there is no unit, write 0. 3. Read the decimal on that number: When reading the decimal, the integer part is read as an integer. The decimal part reads the number on each digit from left to right. 4. Decimal writing: When writing decimals, the integer part is written as an integer, and the decimal part is written in the lower right corner of each bit. 5. Fraction reading: When reading fractions, read the denominator first, then the numerator, and both the numerator and denominator read integers. 6. fractional writing: write first. Write as an integer. 7. Read Percent: When reading Percent, read Percent first, then read the number before the percent sign, and read the whole number when reading. 8. Writing of percentage: percentage is usually not written as a fraction, but is expressed by adding a percent sign "%"after the original molecule. (2) Rewrite numbers into larger multi-digits, which are often rewritten as numbers in units of "10,000" or "100 million" for convenience of reading and writing. Sometimes, the number after a certain number of this number can be omitted and written as an approximation. 1. Accurate number: In real life, for the convenience of counting, larger numbers can be rewritten as numbers in tens of thousands or hundreds of millions. The rewritten number is the exact number of the original number. Like 65438. 2. Approximation: According to the actual needs, we can also use a divisor to represent a larger number and omit the mantissa after a certain bit. For example, the mantissa after omitting 100 million is 654.38+03024900 15 is 65.438+03 billion. When the number of the highest digit of mantissa is 5 or more, the mantissa is rounded, and the size of this number 1 is added to its previous digit to compare the integer sizes: if the integer sizes are compared, the number with more digits is larger; If the number of digits is the same, it depends on the highest digit; The greater the highest digit, the greater the digit; If the highest digit is the same, look at the next digit, and the digit with the largest digit will be the largest. Compare the sizes of decimals: look at their integer parts first, and the bigger the integer part, the bigger the number; If the integer parts are the same, the tenth largest number is larger; One tenth of the numbers are the same, and the number with the largest number in the percentile is larger ... Compare the sizes of scores: the scores with the same denominator and the scores with large numerator are larger; For numbers with the same numerator, the fraction with the smaller denominator is larger. If the denominator and numerator of a fraction are different, divide them first and then compare the sizes of the two numbers. (3) Exchange numbers 1 with each other. Fractional component number: There are several decimals, so write a few zeros after 1 as the denominator, and remove the decimal point from the original decimal point as the numerator, which can be a fraction. 2. Some of them are inseparable and cannot be converted into finite decimals. Generally, three decimals are reserved. 3. If the denominator contains no other prime factors except 2 and 5, the simplest fraction can be converted into a finite decimal; If the denominator contains prime factors other than 2 and 5, the fraction cannot be converted into a finite decimal. 4. Decimal conversion to percentage: Just move the decimal point to the right and add a hundred semicolons after it. 5. Convert percentage to decimal: Just remove the percent sign and move the decimal point to the left. 6. Fractions are converted into percentages: under normal circumstances, fractions are converted into decimals first (except for the inexhaustible ones, three decimal places are generally reserved). Decimals are then converted into percentages. 7. Conversion of percentage to decimal: First, the percentage is rewritten as a component number, and the divisible offer becomes the simplest fraction. (4) Divide the number by 1. A composite number is usually decomposed into prime factors by short division. First, divide it by a prime number, which can evenly divide the composite number until the quotient is a prime number. Then write the divisor and quotient as multiplication. The way to find the greatest common divisor of several numbers is to divide the common divisors of these numbers continuously until the quotient obtained is only the common divisor 1, and then multiply all the common divisors to get the product, which is the greatest common divisor of these numbers. 3. The way to find the least common multiple of these numbers is: First, use these numbers (. Divide all the way by coprime (or pairwise coprime) and multiply by all the divisors and quotients. This product is the least common multiple of these numbers. 4. Two numbers that become coprime: 1 and any natural number coprime; Two adjacent natural numbers are coprime; When the composite number is not a multiple of the prime number, the composite number and the prime number are coprime; When the common divisor of two composite numbers is only 1, two composite numbers are coprime. (5) Divide method and general divide method: use the common divisor of numerator and denominator (except 1) to remove numerator and denominator; Usually, you have to divide until you get the simplest score. General division method: first find the least common multiple of the denominator of the original fraction, and then turn each fraction into a fraction with this least common multiple as the denominator. Three properties and laws (1) Constant quotient law: Dividend and divisor in division expand or shrink the same factor at the same time. The quotient remains the same. (2) the nature of the decimal system: adding zero or removing zero at the end of the decimal system remains unchanged. (3) The movement of decimal position causes the change of decimal size. (4) The basic nature of the fraction: the numerator and denominator of the fraction are multiplied or divided by the same number (except zero). The size of the score remains the same. (5) Relationship between fraction and division: 1. Divider ÷ Divider = Divider/Divider 2. Because zero is not divisible, the denominator of the fraction cannot be zero. 3. Divider is equivalent to numerator and divisor is equivalent to denominator. The meaning of (1) four operations is 1 integer addition. The added number is called sum. The appendix is a partial figure, and the sum is the total. Complement+complement = and one addend = and-another addend 2 integer subtraction: the operation of finding the other addend by knowing the sum of two addends and one of them is called subtraction. In subtraction, the known sum is called the minuend, the known addend is called subtraction, and the unknown addend is called difference. The minuend is the total. Subtraction and difference are partial numbers respectively. Addition and subtraction are reciprocal operations. Integer multiplication: The simple operation of finding the sum of several identical addends is called multiplication. In multiplication, the same addend and the number of the same addend are called factors. The sum of the same addend is called product. In multiplication, multiplying 0 by any number results in 0. 1 multiplied by any number. One factor × one factor = product = product ÷ another factor 4 integer division: the operation of finding another factor is called division. In division, the known product is called dividend, the known factor is called divisor, and the calculated factor is called quotient. 0 cannot be partitioned. Because 0 is multiplied by any number to get 0, and any number is divided by 0. None of them can get a definite quotient. Divider ÷ Divider = quotient divider = quotient divider = quotient × divider (2) Four decimal operations 1. Decimal addition: Decimal addition has the same meaning as integer addition. It is an operation that combines two numbers into one number. 2. Decimal subtraction: Decimal subtraction has the same meaning as integer. The meaning of multiplying a number by a pure decimal is to find a few tenths, a few percent and a few thousandths of this number. 4. Fractional division: The meaning of fractional division is the same as that of integer division, which is the product of two factors, one of which is known. The operation of finding another factor. 5. The operation of finding the product of several same factors by multiplication is called multiplication. For example, the operation of 3×3=32 (3) is 1. Fractional addition: Fractional addition has the same meaning as integer addition. 2. Fractional subtraction: Fractional subtraction has the same meaning as integer subtraction. The sum of two addends is known. The operation of finding another addend. 3. Fractional multiplication: Fractional multiplication has the same meaning as integer multiplication, that is, a simple operation to find the sum of several identical addends. 4. Two numbers whose product is 1 are called reciprocal. 5. Fractional division: Fractional division has the same meaning as integer division. That is, the product of two factors, one of which is known. The operation of finding another factor. (4) The algorithm is 1. Additive commutative law: When two numbers are added, the position of the addend is exchanged, and their sum is unchanged, that is, A+B = B+A.2. The law of addition and combination: when three numbers are added, the first two numbers are added, and then the third number is added; Or add the last two numbers first, and then add them to the first number, and their sum remains unchanged, that is, (a+b)+c=a+(b+c) 3. Multiplicative commutative law: when two numbers are multiplied, the position of the commutative factor remains unchanged, that is, a× b = b× a.4 Multiplicative associative law: when three numbers are multiplied, the first two numbers are multiplied. Or multiply the last two numbers first, and then multiply the first number, and their products are unchanged, that is, (a×b)×c=a×(b×c) 5. Multiplication and distribution law: when the sum of two numbers is multiplied by a number, you can multiply the two addends by this number respectively, and then add the two products, that is, (a+b )× c = a×. That is a-b-c=a-(b+c). (5) Arithmetic 1. Integer addition calculation rule: the same digit is aligned, starting from the lower digit, and the number whose digits add up to 10 will go to the upper digit. 2. Integer subtraction calculation rules: the same number of digits is aligned, starting from the low bit, and the number of digits that are not reduced enough is from the top. Minus it again. 3. Calculation rules of integer multiplication: multiply the number on each bit of another factor by the number on each bit of the factor, then align the end of the multiplied number with which bit, and then add the multiplied numbers. 4. Calculation rules of integer division: Divide the dividend from the high position first, and the divisor is a few digits, depending on the first few digits of the dividend; If the divisor is not enough, look at one more person and the quotient will be written on it. If the quotient is less than 1 in any position, you should add "0". The remainder of each division should be less than the divisor. 5. Decimal multiplication rule: first calculate the product according to the calculation rule of integer multiplication, and then count from the right of the product according to how many decimal places the factor * * * has. If the number of digits is not enough, use "0" to make it up. 6. Calculation rules of fractional division in which divisor is an integer: First, divide according to the law of integer division, and the decimal point of quotient should be aligned with the decimal point of dividend; If there is a remainder at the end of the dividend, add "0" after the remainder, and then continue the divisor. 7. The division calculation method of divisor is decimal: first move the decimal point of divisor to make it an integer, then move the decimal point of divisor to the right by several digits (add "0" if the number of digits is not enough), and then calculate according to the division in which the divisor is an integer. 8. Calculation method of addition and subtraction of fractions with the same denominator: addition and subtraction of fractions with the same denominator. The denominator remains the same. 9. Calculation method of fractional addition and subtraction with different denominators: divide the fractions first, and then calculate according to the law of fractional addition and subtraction with the same denominator. 0. Calculation method of fractional addition and subtraction: add and subtract the integer part and the decimal part respectively, and then combine the obtained numbers. 1 1. The calculation method of fractional multiplication: multiply the fraction by an integer, and use the product of the numerator of the fraction multiplied by the integer as the numerator and denominator. Fractions are multiplied by fractions, with the product of numerator multiplication as the numerator and the product of denominator multiplication as the denominator. 12. Calculation rule of fractional division: A number divided by B number (except 0) equals the reciprocal of A number multiplied by B number. (6) Operation sequence: The operation sequence of four decimal operations is the same as that of four integer operations. 2. The operation order of four decimal operations is the same as that of four integer operations. 3. no Two-stage operation calculates multiplication and division first, and then addition and subtraction. 4. Mixed operation with brackets: first calculate what is in brackets, and then calculate what is in brackets. Finally, the first-level operation: addition and subtraction are called first-level operation. Secondary operation: multiplication and division is called secondary operation. Area of graphic triangle = base × height ÷2. Formula S=a×h÷2 square area = side length× side length formula S=a×a rectangular area = length× width formula S. 2 formula s = (a+b) sum of interior angles of h ÷ 2: sum of interior angles of triangle = 180 degrees. Cuboid volume = length× width× height formula: V = volume of AAA cuboid (or cube) = bottom area× height formula: V = volume of AAA cube = side length× side length formula: V = V = circumference of AAA circle. The surface (side) area of a cylinder is equal to the perimeter of the bottom multiplied by the height. Formula: S = CH = π DH = 2π RH. The surface area of a cylinder is equal to the perimeter of the bottom multiplied by the height plus the area of the circles at both ends. Formula: S=ch+2s=ch+2πr2. The volume of a cylinder is equal to the bottom area multiplied by the height. Formula: V=Sh. The volume of a cone = 65433. 3Sh law of fractional addition and subtraction: Fractions with the same denominator are added and subtracted, only numerators are added and subtracted, and the denominator remains unchanged. Add and subtract fractions with different denominators, divide first, then add and subtract. Fractional multiplication: Use the product of molecules as molecules. Use the product of the denominator as the denominator. The law of division of fractions: dividing by a number is equal to multiplying the reciprocal of this number. The following definition theorem will be applied to understanding and understanding. 1. Arithmetically, 1, additive commutative law: When two numbers are added, the position of the addend changes and the sum remains unchanged. 2. Law of additive combination: When three numbers are added, the first two numbers are added first, or the last two numbers are added first, and then the third number is added. The position of the exchange factor, the product remains unchanged. 4. Multiplication and association law: when three numbers are multiplied, the first two numbers are multiplied, or the last two numbers are multiplied, and then the third number is multiplied, and their products are unchanged. 5. Multiplication and distribution law: When two numbers are multiplied by the same number, you can multiply the two addends by this number respectively, and then add the two products, and the result remains unchanged. Divider and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged. O divided by any number that is not O, simple multiplication: both the multiplicand and the multiplier have O at the end, you can multiply the one before O first, zero does not participate in the operation, and several zeros are added at the end of the product. 7. What is an equation? An equation in which the value on the left of the equal sign is equal to the value on the right of the equal sign is called an equation. The basic property of the equation is that both sides of the equation are multiplied (or divided) by the same number at the same time, and the equation still holds. 8. What is an equation? A: Equations with unknowns are called equations. 9. What is a linear equation with one variable? Answer: An unknown, unknown degree equation is called a linear equation. Learn the example method and calculation of the one-dimensional linear equation, that is, for example, use χ instead of the formula to calculate. 10, fraction: divide the unit "1" into several parts to represent such a number or fraction, which is called fraction. 10. Only numerator addition and subtraction, denominator unchanged. When fractions of different denominators are added or subtracted, they are divided first and then added or subtracted. 12. Comparison of fraction size: Compared with the fraction of denominator, the numerator is large and the numerator is small. Compare the scores of different denominators, divide them first and then compare them; If the numerator is the same, the denominator is smaller. 13, the fraction is multiplied by an integer, and the denominator remains the same. 14, the fraction is multiplied by a fraction, the denominator is multiplied by a product, and the denominator is. 15, and the fraction is divided by an integer (except 0). It is equal to the fraction times the reciprocal of this integer. 16. True fraction: The fraction with numerator less than denominator is called true fraction. 17. False fraction: the fraction with numerator greater than denominator or the fraction with numerator equal to denominator is called false fraction. False score is greater than or equal to 1. 18. Use fraction: Write false fraction in the form of integer and true fraction. It is called a part of. 19. The basic nature of the fraction: the numerator and denominator of the fraction are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains unchanged. 20. A number divided by a fraction is equal to the number multiplied by the reciprocal of the fraction. 2 1, and the number a divided by the number b (except 0). It is equal to the number A times the reciprocal of the number B. From the calculation formula of quantity relationship, 1, unit price × quantity = total price 2, single output × quantity = total output 3, speed × time = distance 4, work efficiency × time = total workload 5, Addendum+Addendum = and one addend = and+The other addend is addend-Differential Meimei = Differential Meimei = Meimei+Differential Factor × Factor = Product One Factor = Product Divider of Another Factor = Quotient Divider = Quotient × Quotient Example: 90 ÷ 5 ÷. 1 dm2 = 100 cm2 1 cm2 = 100 mm2 1 m3 = 1 dm3 = 1000 cm3 = g/kloc-0. Formulas expressed in letters are called algebraic expressions. For example, 3x=(a+b)*c Application: The method to calculate bud rate, attendance rate, qualified rate, oil yield, survival rate ... is to divide the sprouted trees, attendance rate, qualified number, oil yield weight, and surviving trees. Multiply by their respective sums, and then multiply by 100%. Note: the unit is "65438"