We all learn integers, fractions, percentages, decimals, prime numbers, composite numbers, odd numbers, even numbers and negative numbers. ...
1. natural number
Natural number: When we count objects, 1.2.3 ... is called a natural number. An object is not represented by 0, and 0 is also a natural number.
integer
decimal
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2. Integer
Integer: refers to a number with a decimal part of 0, including positive integers and negative integers.
Integers are divided into odd numbers, even numbers, prime numbers, composite numbers, positive numbers and negative numbers.
Relationship between natural numbers and integers: natural numbers must be integers, and integers are not necessarily natural numbers.
score
Fraction (true fraction, false fraction): divide the unit "1" into several parts on average, and the number representing such one or several parts is called a fraction. True fraction numerator is less than denominator, and false fraction numerator is greater than or equal to denominator.
Scores are divided into true scores, false scores and identical scores.
Relationship between fraction and division: when two integers are divided, their quotient can be expressed by fraction, that is, the numerator represents the dividend, the denominator represents the divisor, and the fractional line equals the divisor. The denominator and divisor of a fraction cannot be 0.
4. Decimals
Decimal: Divide an integer into 65,438+00,65,438+000,65,438+0000. ............................................................................................................................
Decimals are divided into finite decimals and infinite decimals (cyclic decimals).
5. Percentage
Percent: A number indicating that one number is a percentage of another number is called a percentage, also called a percentage or percentage.
Percentage and discount are interchangeable, such as 70% = 30%; 85% = 8.5 fold, which means how much discount.
6. How to read and write numbers (decimal, integer, fraction, percentage)
Integer reading method: from high to low, read one level at a time, without reading the zero at the end of each digit, and other digits read a zero for several consecutive zeros.
Integer writing: write one level at a time from high to low. If there is no unit on any number, write 0 on that number.
Decimal reading: for example, 12. 13 is pronounced as 12. 13.
Percent reading: For example, 67% is read as 67%.
Fractional reading: for example, 4/5 is read as 4/5.
7. Number rewriting (conversion of fractions, decimals and percentages)
Fractional decimal: numerator divided by denominator.
Decimal score: 0.3 Writing: 3/ 10.
Decimal percentage: move the decimal point by two places, add a percentage sign and decimal percentage, and convert the fraction into decimal and then into percentage.
Error-prone exercises and answers of "understanding numbers"
1. Among the numbers 2,6,0,1.2,5,-78,51,32%, -2 1, 3 1, the natural number is (2,6,0. Odd numbers are (5,5 1, -2 1, 3 1), even numbers are (2,6,0, -78), prime numbers are (2,5,31) and composite numbers are (51).
Analysis: 0 is a natural number, it is not a negative number.
2. Two parts per million means (2000000), one tenth means (20), two percent means (0.02) and two thousandths means (0.002).
Analysis: 0 Don't write less
3. A three-digit, one-digit number is even and prime, the tenth digit number is odd and composite, and the hundredth digit number is neither prime nor composite. This three-digit number is (192).
Analysis: Even and prime numbers are both 2.
4.6. 15 hours =369)
Analysis: error in carrying 60.
5. There is no maximum or minimum natural number. …………………………(×)
Analysis: The smallest natural number is 0.
6.960074000 is written in "100 million" (960074 million); Take "1 100 million" as the unit, and keep two decimal places (960 million).
Analysis: When two digits are reserved, 0 cannot be empty.
7,3.3 is (2)
3: 30 (2)3: 00 18 (3: 03)
Analysis 3.3 is 198 minutes, which is 3. 18 hours.
Numbers and algebra
-The operation of numbers
The operations we learn in primary school include addition, subtraction, multiplication and division. Different formulas have different algorithms. Let me explain them one by one.
One or four operational relations and their meanings
1, addition
Meaning of addition: The operation of combining two numbers into one number is called addition.
Step 2 subtract
Meaning: The operation of finding the other addend by knowing the sum of two addends and one of them is called subtraction.
3. Addition relation: addend+addend = sum.
Subtraction relation: minuend-minuend = difference
Negative difference = negative
The relationship found by addition and subtraction: difference+subtraction = minuend.
Sum-Appendix (1)= Appendix (2)
Step 4 increase
Meaning: The simple operation of finding the sum of several identical addends is called multiplication.
Step 5 separate
Meaning: The operation of finding the other factor by knowing the product of two factors and one of them is called division.
Multiplication relation: factor × factor = product
Division relation: dividend γ divisor = quotient
Dividend = divisor
The relationship found by multiplication and division: product factor (1)= factor (2)
Quotient × Divider = Divider
Second, elementary arithmetic order.
1, equivalent operation: in order, from left to right, in turn.
2. Different levels of operation: multiply first, then divide, then add and subtract, and the one in brackets counts first.
Third, the algorithm
1. Integer addition calculation rule: the same digits are aligned, starting from the low order. When the numbers add up to ten, they will advance to the previous number.
2. Integer subtraction calculation rules: align with the same number, starting from the low order. If the number of digits is not reduced enough, subtract it from the last digit, merge it with the standard number, and then subtract it.
3. Calculation rules of integer multiplication: multiply the number on each bit of one factor by the number on each bit of another factor, then multiply the number on which bit of the factor, align the end of the multiplied number with which bit, and then add the multiplied numbers.
4. Calculation rules of integer division: divide from the high order of the dividend, and the divisor is a few digits, depending on the first few digits of the dividend; If the division is not enough, look at another place and the quotient is written on the dividend. If any number is not quotient 1, a "0" placeholder should be added. 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 look at the factor * * *, how many decimals there are, count a few from the right side of the product and point to the decimal point; If the number of digits is not enough, make up with "0".
6. Calculation rules of fractional division with divisor as integer: First, divide according to the rules 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 to continue the division.
7. Divisor is a decimal division calculation rule: first move the decimal point of the divisor to make it an integer, then move the decimal point of the divisor to the right by several digits (if the number of digits is not enough, make up "0"), and then calculate according to the division rule that 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, only addition and subtraction of numerators, and the denominator remains unchanged.
9. Calculation method of addition and subtraction of scores with different denominators: divide the scores first, and then calculate according to the addition and subtraction law of scores with the same denominator.
10. 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. Calculation rules of fractional multiplication: Fractions are multiplied by integers, and the product of fractional numerator and integer multiplication is taken as numerator, with the denominator unchanged; Fractions are multiplied by fractions, the product of numerator multiplication is numerator, and the product of denominator multiplication is 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.
Fourth, the operation law.
1. additive commutative law: When two numbers are added, the positions of addends are exchanged, and the sum is unchanged, that is, A+B = B+A.
2. The law of addition and association: when three numbers are added, the first two numbers are added first, and then the third number is added; Or add the last two numbers first, and then add the first number, and their sum remains the same, 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. Multiplication and association law: multiply three numbers, first multiply the first two numbers and then multiply the third number; Or multiply the last two numbers first, and then multiply them with 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, and then add the two products, that is, (a+b) × c = a× c+b× c.
6. The essence of subtraction: If you subtract several numbers from a number continuously, you can subtract the sum of all subtractions from this number, and the difference is unchanged, that is, a-b-c=a-(b+c).
Friendly reminder: simply calculate more and more, more and more.
Legal summary:
(1) additive commutative law:
(2) Additive associative law:
(3) Multiplicative commutative law:
(4) Multiplicative associative law:
(5) Multiplication and distribution law:
(6) Subtraction operation attribute:
(7) the nature of the division operation:
Numbers and Algebra-Formulas and Equations
1. Algorithm and Representation of Operation Properties
(1) additive commutative law:
(2) Additive associative law:
(3) Multiplicative commutative law:
(4) Multiplicative associative law:
(5) Multiplication and distribution law:
(6) Subtraction operation attribute:
(7) the nature of the division operation:
Second, the writing rules of numbers expressed by letters:
1. When the number is multiplied by letters, letters and letters, the multiplication sign can be recorded as ""or omitted, and the number should be written before the letters. For example:
2. When "1" is multiplied by any letter, "1" is omitted. For example:
3. For the same problem, use the same letter to represent the same quantity, and use different letters to represent different quantities.
For example:,,.
When the answer to the question is expressed by a formula containing letters, the divisor is generally written as the denominator. If there is a plus sign or a minus sign in the formula, first enclose the formula in brackets with letters, and then write the company name after the brackets.
Third, express the quantitative relationship with letters.
(distance = speed and time) (total price = unit price and quantity)
(Total amount of work = working efficiency and working hours) (Total output = cultivated land area per unit output)
Four, said the calculation formula
The calculation formula in mathematics can be expressed concisely by letters. The details are as follows:
Alphabetic formula of alphabetic meaning of names
Rectangular length and width
-perimeter-area
Square length perimeter
-Area
Base height of parallelogram
-Area
Triangle base height
-Area
Trapezoid-upper bottom-lower bottom
Height-area
circle
-Radius-Diameter
-perimeter-area
Cuboid-length-width-height
-Surface area-volume
cube
-Side length-Surface area-Volume
Cylinder height and bottom circumference
-Bottom surface area
-Side area
—surface area
-Volume
Cone height
-Bottom area
-Volume
Verb (abbreviation for verb) equation
1. Concept: An equation with an unknown number is called an equation.
2, the equation must meet the conditions:
(1) must be an equation;
(2) There must be unknowns.
Common knowledge points: the relationship between equations, which are equations, but not necessarily equations, and the relationship between them is graphically represented as:
Sixth, the basis for solving the equation
1, the relationship between the four parts of the operation:
One addend = another addend = minus+minus = minus.
The product dividend of one factor = another factor = divisor quotient divisor = dividend quotient
2, the nature of the equation:
The property of the equation is 1: both sides of the equation add and subtract the same number at the same time, and the equation remains unchanged;
Property 2 of the equation: both sides of the equation are multiplied or divided by the same number at the same time (0 cannot be divided), and the equation remains unchanged.
Common knowledge points: "solving equations" and "solving equations" are two concepts that are easily confused. Note that "solving the equation" is a process of solving the equation, while "solving the equation" is an unknown value in the equation.
Error-prone questions and answers in Formulas and Equations;
1, there is such a set of numbers: 30, 1+30, 2+30, 3+30, ... where the first number is represented by the letter ().
Analysis: carefully observe this group of numbers and find that each number is equal to 30 plus a number smaller than the ordinal number of this number 1.
2. As shown in the figure, one table can seat 6 people, and two tables together can seat 10 people, so keep fighting …
…………
How many people can sit at a table?
Analysis: Careful observation of the figure shows that if four people are added to each table, then each table can be regarded as sitting four people, and each table can sit four people. Finally, two people are added to each end, and the relationship (4+2) can be obtained.
3. Judges:
(1) is an equation. (error)
(2) An equation is an equation. (error)
(3) is an equation. (error)
4. Specify, know and find value.
Analysis: This topic examines the value of algebraic expressions and solves equations. This problem can be solved by a new operation according to the definition, and then the sum of this number is transformed into a general equation through operation, and finally the value is obtained.
Answer: Solution: