7, 2, 1 rational numbers
1. Definition
1. Positive integers, 0, and negative integers are collectively called integers.
2. Positive fractions and negative fractions are collectively called fractions
3. Rational numbers: Integers and fractions are collectively called rational numbers.
2. Classification
(1) According to concept:
(2) According to nature
rational numbers
positive rational numbers
negative rational numbers
(3) Supplement: Wu is not a rational number, Wu is a positive number.
7, 2, 2, number line
1. Number axis
A straight line specifying the origin, positive direction, and unit length
three elements of number line
Positive direction
Right (or upward) indicated by an arrow
2,
Generally speaking, if the number a is a positive number, then the distance between the number a point on the number axis and the origin to the right of the origin is a unit length, and the point representing the number -a is a unit length away from the origin to the left of the origin.
7, 2, 3 opposite numbers
1. Suppose a is a positive number. There are two numbers a on the number axis that are a distance from the origin. They represent -a and a respectively on the left and right of the origin. We say that these two numbers are opposites of each other.
2. Opposite numbers: Two numbers with different signs are called opposite numbers.
3. To find the opposite of a number, add one ( ) in front of the number. The opposite is equal to the number itself, which is 0.
7, 2, 4 absolute value
1. Absolute value: The distance between the point representing the number a and the origin on the number axis is called the absolute value of the number a, which is recorded as 1a1. The absolute value of 0 is 0.
2. Properties of absolute value
1. The absolute value of a positive number is itself
2. The absolute value of a negative number is its opposite.
3. The absolute value of 0 is 0
Comparison of the sizes of rational numbers 7, 2, and 5
1. Regulation: Rational numbers are represented on the number axis. The order from left to right is the order from small to large, that is, the number on the left is smaller than the number on the right.
2. Rules
Positive numbers are greater than 0, 0 is greater than negative numbers, positive numbers are greater than negative numbers
When comparing two negative numbers, the one with a larger absolute value is smaller.
7, 3, 1 rational number addition
(1)
1. Rules
1. Add two numbers with the same sign and add their absolute values.
2. Add two numbers with different signs whose absolute values are not equal, take the sign with the larger absolute value, and subtract the smaller absolute value from the larger absolute value.
3. If you add a number to 0, you still get this number.
2. Method:
Addition calculation method:
② Calculate the absolute value (same sign plus different sign minus)
(2)
Laws of addition of rational numbers
1. Commutative law: a + b = b + a
2. Associative law: (a + b) C = a + (b + C)
Skill
1. Numbers with the same sign can be added
2. Convert decimals into fractions or convert fractions into decimals and then add them. Add the numbers that can be rounded up by adding fractions with the same denominator.
3. Adding two numbers that are opposites of each other
7, 3, 2 rational number subtraction
(1)
1. Rational number subtraction rule: subtracting a number is equal to adding the opposite of the number
2. Compare the size of two numbers using the difference method.
(2)
one,
Pronunciation:
1. The operand twenty plus three plus five minus seven
2. The sum of negative twenty, positive three, positive five and negative
two,
Points AB represent ab respectively, then ab is the length of the distance between two points on the number axis
7, 4, 1 rational number multiplication
(1)
1. Rational number multiplication rule: when two numbers are multiplied, if they have the same sign, they will be positive, if they have different signs, they will be negative, and the absolute values will be multiplied together.
2. Any number multiplied by zero gets 0
step:
②Multiplication of absolute values
(2)
The sign rule of products: the sign of the product of several numbers not equal to zero is absolutely determined by the number of negatives
When there are an even number of negative factors, the product is positive
When there are an odd number of negative factors, the product is negative
Multiply several numbers. If any of the factors is zero, the product is zero.
(3)
Laws of multiplication of rational numbers
1. Commutative law of multiplication: ab=ba
2. Associative law of multiplication: Cab)c=a(bc)
3. Distributive law of multiplication: a (b + c) = ab + ac
a (b ten C ten d) = ab ten ac ten ad
7, 4, 2 rational number division
(1)
Rational division rules
1. Dividing by a number that is not equal to 0 is equivalent to multiplying by the reciprocal of this number.
2. Divide two numbers, if they have the same sign, they will be positive, if they have different signs, they will be negative, and divide the absolute values.
(2)
new operation
example:
a*b=(a十b)÷(1-ab)
(3)
Mixed operations of addition, subtraction, multiplication and division of rational numbers:
1. Multiply and divide first and then subtract.
2. If there are parentheses, first calculate the ones inside the parentheses, then proceed in order of small brackets, square brackets, and curly brackets.
3. Operations at the same level are performed sequentially from left to right.
7, 5, 1 power
(2)
The order of operation of rational numbers: you first calculate the exponentiation, then the multiplication and division, and then the addition and subtraction of two. The same-level operations are performed three times from left to right. If there are parentheses, the ones in the parentheses are calculated first, and then the small brackets, middle brackets, and curly brackets are performed in order.
(1)
1. The multiplication of N identical factors a is written as n
a
2. The operation of multiplying the product of several identical numbers by exponentiation is called exponentiation, and the result of exponentiation is called exponentiation.
3. Sign rules of exponentiation
1. Odd negative numbers of negative numbers
2. The even number of a negative number is a positive number.
3. Any power of a positive number is a positive number.
4. Any positive or negative power of 0 is 0
7, 5, 2 scientific notation
100……00 (n number of zeros after) = n
10
A is greater than or equal to and less than 10, n is a positive integer.
Scientific notation represents an m-digit integer, where the exponent of zero is m minus 1
7, 3, 2 approximate numbers
1. Exact numbers and approximate numbers
2. Accuracy: The degree of closeness between approximate numbers and exact numbers.
odd negative even positive
The number whose reciprocal is equal to itself is ±1
Adding two numbers that are opposites of each other gives 0
a+b=0
ab are opposite numbers to each other.
Non-negative numbers: 0 and positive numbers
Non-positive numbers: 0 and negative numbers
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