Integer concept
The natural numbers when we count objects, 1, 2, 3, 4, 5, ... are called natural numbers. There is no object, which is represented by "0". "0" is also a natural number, which is the smallest natural number. There is no maximum natural number, and the natural number is infinite.
Integer In primary schools, integers usually refer to natural numbers.
Numbers are symbols representing numbers, and numbers are usually called numbers.
Addition The operation of combining two numbers into one number is called addition.
Addendum Two numbers are added together, which is called addend.
And the sum of two addends is called sum.
Subtraction seeks the sum of two numbers, and the operation of adding one and the other is called subtraction.
In subtraction, the known sum is called the minuend.
In subtraction, the known addend of subtraction is called subtraction.
Difference In subtraction, the unknown addend obtained is called difference.
Multiplication is a simple operation to find the sum of several identical addends, which is called multiplication.
In multiplication, the multiplied two numbers are called factors of product.
In multiplication, the result of multiplication is called product.
Division The operation of finding the product of two factors with one factor and the other factor is called division.
The known product of dividend in division is called dividend.
Divider In division, a known factor is called divisor.
Quotient In division, the unknown factor is called quotient.
Counting units of one, ten, one hundred, one thousand, ten thousand, one hundred thousand, one million, ten million and one hundred million are all called counting units.
The ratio between every two adjacent counting units in decimal notation is 10. This counting method is called decimal counting method.
When writing numbers with numbers, the counting units are arranged in a certain order, and their positions are called numbers. Different numbers of a number represent different sizes of the number. The first number is the unit, followed by 10, 100, 1000, 10000, 10000. ......
Divide an integer by another non-zero integer, and there is a remainder after the quotient of the integer. This division is called division with remainder. The remainder is less than the divisor.
We have learned four operations of addition, subtraction, multiplication and division in elementary arithmetic of integers, which are collectively called four operations.
One-level operation Among the four operations, addition and subtraction are called one-level operations.
Secondary operation Among the four operations, the multiplication and division operation is called secondary operation.
Divide two integers equally. If expressed in letters, it can be said that the quotient obtained by dividing an integer A by an integer b(b is not equal to 0) is exactly an integer without a remainder. We can say that A is divisible by B, or B is divisible by A. ..
Factor and multiple If the number A is divisible by B (B is not equal to 0), A is called a multiple of B, and B is called a divisor or factor of A. Multiplication and divisor are interdependent. The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself. The number of multiples of a number is infinite, and the smallest multiple is itself. For example, 15 can be divisible by 3, so we say that 15 is a multiple of 3, and 3 is a divisor of 15.
Even numbers divisible by 2 are called even numbers, because 0 can also be divisible by 2, so 0 is an even number.
An odd number that is not divisible by 2 is called an odd number. For example, 1, 3, 5 and 7. ......
A prime number is a number. If 1 and itself have only two divisors, such numbers are called prime numbers or prime numbers. For example, 2, 3, 5, 7, 1 1 are prime numbers.
Prime numbers are prime numbers.
If a composite number has other factors besides 1 and itself, it is called a composite number. 1 is neither prime nor composite. For example, 4, 6, 8, 9, 10, 12 ... are all complex numbers.
The composite number of each prime factor can be written as the product of several prime numbers. Every prime number is a factor of this composite number, which is called the prime factor of this composite number.
Prime factorization refers to a composite number multiplied by a prime factor, which is called prime factorization. For example: 12=3*2*2
The common divisor of several numbers is called the common divisor of these numbers.
The greatest common divisor of several numbers is called the greatest common divisor of these numbers. For example, 1, 2,4 is the common divisor of 8 and 12; 4 is the greatest common divisor of 8 and 12.
The common divisor of a prime number is only 1, which is called a prime number. For example, 5 and 7 are prime numbers, and 8 and 9 are prime numbers.
The common multiple of several numbers is called the common multiple of these numbers.
The smallest common multiple of several numbers is called the smallest common multiple of these numbers. For example, 12, 24, 36 ... are all common multiples of 4 and 6, and 12 is the least common multiple of 4 and 6.
The unit price, quantity and total price of each commodity, we call it the unit price, how much we bought is called the quantity, and how much we spent is called the total price. Total price = unit price × quantity
Speed, Time and Distance The distance traveled per hour (or every minute or every day) is called speed, and after several hours (or minutes or days) we call it time, and we call it distance. Distance = speed × time
When two numbers in additive commutative law are added, the positions of addends are interchanged, and their sums remain unchanged. This is called additive commutative law. The letter means: a+b = b+a.
The law of addition and association adds three numbers, first adding the first two numbers and then adding the third number; Or add the last two numbers first, and then add them to the first number, and their sum remains the same. This is the so-called law of additive association. The letter means: (a+b)+c=a+(b+c)
Law of Multiplication and Exchange When two numbers are multiplied, the position of the exchange factor and its product remains the same. This is the so-called multiplication commutative law. The letter means: a×b = b×a a a.
Multiplication 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 remain unchanged. This is the so-called law of multiplication and association. The letter means: (a×b)×c=a×(b×c)
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. This is the so-called multiplication distribution rate. The letter means: (a+b) × c = a× c+b× c.
The addition rule of three digits or four digits (1) is the same digit alignment; (2) from the unit; (3) Where the numbers add up to ten, you should go to the previous one and enter one.
Multiplier is the multiplication rule of one digit (1). Starting from one digit, multiply each digit of the multiplicand by the multiplier in turn; (2) Whoever gets dozens of good grades will be promoted to the top one. Multiply 0 by any number to get 0.
The changing law of the sum product of two factors: one factor remains unchanged, the other factor expands (or shrinks) several times, and the product also expands (or shrinks) several times.
In division, the dividend and divisor are expanded (or reduced) by the same multiple (except zero) at the same time, and the quotient remains unchanged.
Multiplication factor × factor = the product of one factor = the relationship between the parts of the product of another factor.
Relationship between the parts of division Divider/Divider = Divider/Divider = Quotient/Divider = Quotient × Divider
Multiplication is calculated by dividing the product by one factor. If another factor is obtained, the multiplication is done correctly.
Division is calculated by multiplying the quotient by the divisor. If you get the dividend, or divide by the quotient, if you get the divisor, the division will be done correctly.
A simple multiplication algorithm is to multiply three numbers. You can multiply the last two numbers first, and then multiply the first number, and the result remains the same. Using this law, it is sometimes easier to multiply a number by two one-digit numbers in succession and become the product of two one-digit numbers. Sometimes it is easier to multiply a number by a two-digit number and replace it with a one-digit number multiplied by two digits continuously. For example: 6×12× 5 = 6× (12× 5) 25×16 = 25× (4× 4) = 25× 4× 4.
A simple division algorithm: one number is divided by two numbers continuously. When you can divide evenly every time, you can multiply two divisors first, and then divide the number by their product, and the result remains the same. Using this law, it is sometimes easier to divide a number into two digits continuously and become the product of these two digits. Sometimes it is easier to divide a number by two digits and then divide it by two digits continuously. For example:1000 ÷ 25 ÷ 4 =1000 ÷ (25× 4) 420 ÷ 35 = 420 ÷ 7 ÷ 5.
The application problem solving step (1) clarifies the meaning of the problem and finds out the known conditions and problems; (2) analyze the relationship between the quantities in the problem, and determine what to calculate first, then what to calculate, and finally what to calculate (3) determine how to calculate each step, list formulas, and work out numbers; (4) Test and write the answer.
According to the meaning of the question, check the application question (1), and check the formula and calculation of each step in turn to see if it is correct. (2) Take the number as the known condition, and calculate it step by step according to the meaning of the question to see if the result meets the original known condition.
Multi-digit writing method (1) starts from the high position and writes down level by level; (2) Write 0 on any number without a number. For example: 70.302 billion writing
Addition = Addendum+Addendum = and-The sum of the relationships between the parts of another addend.
Difference between the parts of subtraction = meimei-meimei = meimei-difference meimei = meimei+difference.
Simple addition and subtraction operation: one number subtracts two numbers continuously, which is equal to the sum of this number MINUS two numbers. For example,130-46-34 =130-80 = 50.
Relationship between parts of division with remainder Divider = quotient × divisor+remainder
The order of sibling operations in an expression, if it only contains sibling operations, should be calculated from left to right.
Operation order of different levels of operations If an expression contains two levels of operations, then the second level of operations should be done first, and then the first level of operations should be done. For example,100-7× 5 =100-35 = 65.
Decimal concept
Decimals are written on the right side of integers, separated by dots, indicating decimals, hundredths, thousands, etc. Call it a decimal. For example, 0.2 means two tenths and 0.02 means two percent.
Counting unit of decimals The counting unit of decimals is one tenth, one hundredth and one thousandth ... written as 0. 1, 0.05438+0, 0.005438+0 respectively. ......
Decimal addition Decimal addition has the same meaning as integer addition, and it is an operation that combines two numbers into one number.
Decimal subtraction Decimal subtraction has the same meaning as integer subtraction. It is the operation of finding the other addend by knowing the sum of two addends and one of them.
The Meaning of Integer Decimal Multiplication The meaning of Integer Decimal Multiplication and Integer Multiplication is the same, and they are both simple operations to find the sum of several identical addends.
Multiplying a number by a decimal means finding a few tenths, a few percent and a few thousandths of this number. ......
Decimal division Decimal division has the same meaning as integer division, and it is an operation to find another factor by knowing the product of two factors and one of them.
Cyclic decimals A decimal in which one or more numbers appear repeatedly from a certain position in the decimal part is called a cyclic decimal.
The fractional part of a cyclic decimal, and the numbers that appear repeatedly in turn, are called the cyclic part of this cyclic decimal.
The cyclic segment of a pure cyclic decimal begins with the first digit of the decimal part, which is called a pure cyclic decimal.
A mixed cycle decimal cycle that does not start with the first decimal part is called a mixed cycle decimal.
The number of digits in the decimal part of a finite decimal is a finite decimal, which is called a finite decimal.
The number of digits in the decimal part of an infinite decimal is an infinite decimal, which is called an infinite decimal. Cyclic decimal is infinite decimal.
The nature of decimals adds or removes 0 at the end of decimals, and the size of decimals remains unchanged. This is the nature of the so-called decimal.
Calculation rules of decimal addition and subtraction To calculate decimal addition and subtraction, first align the decimal points of each number, then calculate according to the rules of integer addition and subtraction, and finally align the horizontal lines in the obtained numbers.
A decimal point on the decimal point. There is a 0 at the end of the decimal part of a number, which is usually removed.
Calculation rules of decimal multiplication To calculate decimal multiplication, first calculate the product according to the rules of integer multiplication, and then look at a factor * * *, how many decimal places there are, just count the digits from the right side of the product and point to the decimal point.
Divider is fractional division of integer, divisor is fractional division of integer, 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 continue the division.
The law of fractional division in which the divisor is a decimal is divided. First, move the decimal point of the divisor to make it an integer. The decimal point of the divisor is shifted to the right by several digits, and the decimal point of the dividend is also shifted to the right by several digits (the digits are not enough, and the end of the dividend is supplemented by "0"); Then it is calculated by fractional division with the divisor being an integer.
How to read decimals When reading decimals, the integer part is read by integer method (the integer part is read as "zero") and the decimal point is read as "dot". The decimal part usually reads the numbers on each digit in sequence.
When writing decimals, the integer part is written as an integer (the integer part is written as a number "0"), the decimal point is written in the lower right corner of the unit, and the decimal part is written on each digit in turn.
Application of decimal nature (1) According to the nature of decimal, when there is a "0" at the end of decimal, the "0" at the end can generally be removed to simplify decimal. (2) Sometimes, as required, you can add "0" after the decimal point, or you can add 0 after the decimal point in the lower right corner of the unit and integer to write the integer as a decimal.
Fraction concept
The score line is in the score, and the horizontal line in the middle is called the score line.
The denominator is in the fraction, and the number below the fraction line is called the denominator, indicating how many copies the unit "1" is divided into.
The numerator is in the score, and the number above the score line is called the numerator, indicating how many copies there are.
Decimal unit divides the unit "1" into equal parts according to the denominator, and the number representing one part is called decimal unit. For example, the unit of six fifths is one sixth.
Fractions with numerator less than denominator are called true fractions. The true score is less than 1.
A false fraction is a fraction whose numerator is greater than the denominator or whose numerator and denominator are equal.
A fraction is called a complex fraction if its numerator contains a fraction, or its denominator contains a fraction, or both numerator and denominator contain fractions.
Numbers consisting of integers and true fractions are usually called fractions. For example, two and a fifth.
Divisor changes a fraction into a fraction equal to it, but the numerator and denominator are relatively small, which is called divisor.
The simplest fraction whose numerator and denominator are prime numbers is called the simplest fraction.
General score is called comprehensive score, which is to change two different denominator scores into the same denominator score equal to the original score. For example, comparing the size of two scores requires a rough score.
Fractional addition Fractional addition has the same meaning as integer addition, and it is an operation that combines two fractions into one fraction.
The significance of fractional subtraction is the same as that of integer subtraction. It is the operation of finding the other addend by knowing the sum of two addends and one of them.
Fractional Multiplication of Integers Fractional multiplication of integers has the same meaning as integer multiplication, and it is a simple operation to find the sum of several identical addends.
The significance of multiplying a number by a fraction is to find out what fraction this number is.
The reciprocal product of two numbers is 1, which is called reciprocal. For example, three-eighths and three-eighths are reciprocal, which means that the reciprocal of three-eighths is three-eighths.
The significance of fractional division is the same as integer division, that is, knowing the product of two factors and one of them, and finding the other factor.
The basic nature of a fraction The numerator and denominator of a fraction are multiplied or divided by the same number (except zero) at the same time, and the size of the fraction remains unchanged. This is called the basic nature of fractions.
The law of addition and subtraction of denominator fraction is the same as that of denominator fraction, the denominator is unchanged, and only the numerator is added and subtracted. The calculation result can be simplified to the simplest fraction, which is a pseudo-fraction and generally needs to be converted into component numbers or integers.
Ratio and proportion
Percentage means that one number is the percentage of another number, which is called percentage. Percentage is also called percentage and percentage.
The extra money paid by the bank when withdrawing interest is called interest.
Money deposited in the bank is called principal.
Interest rate The percentage of interest and principal is called interest rate. The interest rate is set by the bank, calculated annually and monthly.
The calculation formula of interest = principal × interest rate × time.
A few percent is a few tenths, or a few tens. For example, 30% is three tenths, and if it is rewritten as a percentage, it is 30%.
A discount of "several discounts" is just a few tenths, that is, dozens of percent.
The division of two numbers is also called the ratio of two numbers.
The comparison number is indicated by ":"and is pronounced comparison.
The number before the ratio number is called the ratio.
The number after the ratio sign is called the later term of the ratio.
The quotient obtained by dividing the former term by the latter term is called the ratio.
Proportion means that two proportions are equal, which is called proportion.
The terms of the ratio constitute four numbers of the ratio, which are called ratio terms.
Among the four items of proportion, the two items at both ends are called disproportion.
Among the four proportions, the middle two terms are called proportional internal terms. For example, 80:2=200:5, where 2 and 200 are internal items and 80 and 5 are external items.
Solution ratio According to the basic properties of proportion, if any three items in the proportion are known, we can find another unknown item in this proportion. The unknown term of finding proportion is called solution ratio. For example, the solution ratio is 3:8= 15:x solution: 3x= 15×8 x=40 primary school mathematics practice machine, the best primary school mathematics counseling and practice software version 49.0, which can automatically set questions and correct them.
The ratio of the distance on a scale map to the actual distance is called the scale of the map. To simplify the calculation, the scale is usually written as the ratio of 1. Distance on the map: actual distance = scale
A proportional quantity is two related quantities, one of which changes and the other changes. If the ratio of the corresponding two numbers in these two quantities is certain, these two quantities are called proportional quantities, and their relationship is called proportional relationship. For example, distance varies with time, and their ratio (speed) remains the same, so distance and time are proportional quantities.
A quantity in inverse proportion is two related quantities, one of which changes and the other changes with it. If the product of the corresponding two numbers in these two quantities is certain, these two quantities are called inverse proportional quantities, and their relationship is called inverse proportional relationship.
The basic nature of the ratio The first and last items of the ratio are multiplied or divided by the same number (except 0) at the same time, and the ratio remains unchanged. This is the basic nature of the so-called ratio.
The basic properties of proportion In proportion, the product of two external terms is equal to the product of two internal terms. This is the basic nature of the so-called proportion.
Percentages are usually not written in the form of fractions, but expressed by adding a percent sign "%"after the original molecule. For example, 90% is written as 90%.
Percentages and decimals are converted into percentages by moving the decimal point two places to the right and adding hundreds of semicolons after it; To convert percentages to decimals, simply remove the percent sign and move the decimal point two places to the left. For example, 0.25=25% and 27%=0.27.
Percentages and fractions are converted into percentages. Usually, the fractions are converted into decimals first (three decimal places are usually reserved when they are not used up), and then the decimals are converted into percentages; Divide the percentage into components, and rewrite the percentage into components first, so that the quotation that can be lowered can be made into the simplest score.
The method of simplifying the integer ratio Simplification of the integer ratio According to the basic properties of the ratio, the two terms before and after the ratio are divided by the greatest common divisor of the two terms before and after the ratio at the same time, and the simplest ratio is obtained.
Decimal ratio simplification method Decimal ratio simplification According to the basic properties of the ratio, the front and rear terms of the ratio are expanded by the same multiple at the same time, and then the integer ratio is simplified.
The method of simplifying fractional ratio includes simplifying fractional ratio, multiplying the first term and the last term of ratio by the least common multiple of denominator, converting fractional ratio into integer ratio, and then simplifying integer ratio.
Geometric concept
A line segment is obtained by connecting two points with a ruler. These two points are called the endpoints of the line segment. The line segment AB represents a line segment whose endpoints are point A and point B. ..
The basic property of a line segment is the shortest among all the straight lines connecting two points, and the length of the line segment can be measured.
A ray infinitely extends one end of a line segment, and a ray is obtained. A ray has only one endpoint, so it cannot measure its length.
A straight line extends infinitely at both ends of the line segment, and you get a straight line. The straight line has no end point and cannot be measured. You can draw countless straight lines after one o'clock, and only one straight line after two o'clock.
The distance between two points The length of the line segment connecting two points is called the distance between these two points (the length of line segment AB is the distance between point A and point B).
A figure composed of two rays with a common endpoint is called an angle.
The vertex of an angle is called the vertex of an angle.
The two rays that make up an angle are called the edges of the angle. Primary school mathematics practice machine version 49.0, the best primary school mathematics counseling practice software, automatically sets questions and automatically corrects them.
The internal angle of an angle can be regarded as a graph formed by a ray rotating from one position to another around the endpoint. The plane through which the light rotates is the inside of the angle.
The right-angle ray OA rotates around point O. When the end position OC and the starting position OA are in a straight line, the angle formed is called a right angle. The flat angle is 180 degrees.
When the fillet ray OA rotates around the O point and returns to the initial position OA, the angle it forms is called fillet. Its fillet is 360 degrees.
Half of a right angle is called a right angle. The right angle is 90 degrees.
An angle with an acute angle less than a right angle is called an acute angle. The acute angle is less than 90 degrees.
An obtuse angle greater than a right angle and less than a right angle is called an obtuse angle. The obtuse angle is less than 180 degrees and more than 90 degrees.
The bisector ray of an angle divides an angle into two equal angles. This ray is called the bisector of an angle.
Two straight lines are perpendicular to each other when one of the four angles formed by the intersection of two straight lines is a right angle, it is said that the two straight lines are perpendicular to each other. One of the straight lines is called the perpendicular of the other straight line, and their intersection point is called the vertical foot.
A triangle consists of three line segments that are not on the same straight line, and these three line segments are connected end to end in turn. It's called a triangle.
The sides of a triangle form the line segments of the triangle, which are called the sides of the triangle.
In a triangle, the angle formed by two adjacent sides is called the angle of the triangle.
The height of a triangle draws a vertical line from the vertex of the triangle to its opposite side. The line segment between the vertex and the vertical foot is called the height line of the triangle, which is called the height of the triangle for short.
An equilateral triangle is a triangle with three unequal sides.
An isosceles triangle is a triangle with two equal sides.
An equilateral triangle is called an equilateral triangle.
The waist of an isosceles triangle is in an isosceles triangle, and two equal sides are called the waist.
The base of an isosceles triangle In an isosceles triangle, the third side except the two equal sides is called the base.
The vertex of an isosceles triangle is in an isosceles triangle, and the angle between the two waists is called the vertex.
The base angle of an isosceles triangle is in an isosceles triangle, and the angle between the waist and the base is called the base angle.
Acute triangle A triangle with three angles at acute angles is called an acute triangle.
A right triangle is a triangle with a right angle.
A triangle with an obtuse angle is called an obtuse triangle.
The right side and hypotenuse of a right triangle are inside the right triangle. The two sides of the right angle are called the right side, and the opposite side of the right angle is called the hypotenuse.
An isosceles right triangle with two equal right angles is called an isosceles right triangle.
The stability of a triangle, such as nailing it with three wooden sticks and pulling it hard, has not changed the shape of the triangle. Visible triangle is stable.
Area of triangle = base × height ÷2
A quadrilateral composed of four line segments that are not on the same straight line is called a quadrilateral.
Two straight lines whose parallel lines do not intersect in the same plane are called parallel lines.
Parallelogram Two groups of parallelograms with parallel opposite sides are called parallelograms.
Area formula of parallelogram Area of parallelogram = base × height
A rectangle is a parallelogram with right angles.
A rhombus has a set of equilateral parallelograms called a rhombus.
A square has a set of parallelograms with equal adjacent sides and a right angle, which is called a square.
A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid.