Chapter 1 Rational Numbers
1. 1 positive and negative numbers
Books that have been studied before, except 0, are called negative numbers with the minus sign "-"in front of them.
Numbers other than 0 that I learned before are called positive numbers.
The number 0 is neither positive nor negative, it is the dividing line between positive and negative numbers.
In the same question, positive numbers and negative numbers have opposite meanings.
1.2 rational number
1.2. 1 rational number
Positive integers, 0 and negative integers are collectively called integers, and positive and negative fractions are collectively called fractions.
Integers and fractions are collectively called rational numbers.
1.2.2 axis
The straight line that defines the origin, positive direction and unit length is called the number axis.
Function of number axis: All rational numbers can be represented by points on the number axis.
Note: The origin, positive direction and unit length of (1) axis are indispensable.
⑵ The unit length of the same shaft cannot be changed.
Generally speaking, if it is a positive number, the point representing a on the number axis is on the right side of the origin, and the distance from the origin is a unit length; The point representing the number -a is on the left of the origin, and the distance from the origin is one unit length.
1.2.3 reciprocal
Numbers with only two different symbols are called reciprocal.
Two points representing the opposite number on the number axis are symmetrical about the origin.
Add a "-"sign before any number, and the new number represents the antonym of the original number.
1.2.4 absolute value
Generally speaking, the distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A.
The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0.
Rational numbers are represented on the number axis, and the order from left to right is from small to large, that is, the number on the left is smaller than the number on the right.
Compare the sizes of rational numbers: (1) Positive numbers are greater than 0, 0 is greater than negative numbers, and positive numbers are greater than negative numbers.
(2) Two negative numbers, the larger one has the smaller absolute value.
Addition and subtraction of rational number 1.3
1.3. 1 addition of rational numbers
Law of rational number addition:
(1) Add two numbers with the same sign, take the same sign, and add the absolute values.
⑵ Add two numbers with different symbols with unequal absolute values, take the sign of the addend with larger absolute value, and subtract the number with smaller absolute value from the number with larger absolute value. Two opposite numbers add up to 0.
(3) When a number is added to 0, the number is still obtained.
When two numbers are added, the positions of the addend are exchanged and the sum is unchanged.
Additive commutative law: A+B = B+A.
Add three numbers, first add the first two numbers, or add the last two numbers first, and the sum remains the same.
Additive associative law: (a+b)+c = a+(b+c)
1.3.2 subtraction of rational numbers
The subtraction of rational numbers can be converted into addition.
Rational number subtraction rule:
Subtracting a number is equal to adding the reciprocal of this number.
a-b=a+(-b)
Multiplication and division of rational number 1.4
The rational number multiplication of 1.4. 1
Rational number multiplication rule:
Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value.
Any number multiplied by 0 is 0.
Two numbers whose product is 1 are reciprocal.
Multiply several numbers that are not 0. When the number of negative factors is even, the product is positive. When the number of negative factors is odd, the product is negative.
When two numbers are multiplied, the exchange factor and the product are in the same position.
ab=ba
Multiply three numbers, first multiply the first two numbers, or multiply the last two numbers first, and the products are equal.
C=a (BC)
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.
a(b+c)=ab+ac
Writing specification for multiplication of numbers and letters;
(1) Multiplies a number with a letter, omitting the multiplication sign or using "".
(2) Numbers multiplied by letters. When the coefficient is 1 or-1, 1 should be omitted.
(3) The band score is multiplied by letters, and the band score becomes a false score.
If any rational number is represented by the letter X, the product of 2 and x is 2x, and the product of 3 and x is 3x, then the formula 2x+3x is the sum of 2x and 3x, 2x and 3x are the terms of this formula, and 2 and 3 are the coefficients of these two terms respectively.
Generally speaking, when combining formulas with the same letter factor, it is only necessary to combine their coefficients, and the obtained results are used as coefficients, and then multiplied by the letter factor, that is,
ax+bx=(a+b)x
In the above formula, X is the letter factor, and A and B are the coefficients of ax and bx respectively.
Support removal rules:
There is a "+"before the brackets. Remove brackets and the "+"in front of brackets, and nothing in brackets will change its sign.
There is a "-"before the brackets. Remove brackets and the "-"sign in front of brackets, and change all the symbols in brackets.
The factors outside brackets are positive numbers, and the symbols of the items in the formula after removing brackets are the same as those of the corresponding items in the original brackets; The factor outside the bracket is negative, and the sign of each item in the formula after the bracket is opposite to that of the corresponding item in the original bracket.
1.4.2 division of rational numbers
Rational number division rule:
Dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.
a÷b=a？ (b≠0)
Divide two numbers, the same sign is positive, the different sign is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0 to get 0.
Because the division of rational numbers can be converted into multiplication, the operation can be simplified by using the operational nature of multiplication. The mixed operation of multiplication and division often turns division into multiplication first, then determines the sign of the product, and finally calculates the result.
1.5 power of rational number
1.5. 1 power
The operation of finding the product of n identical factors is called power, and the result of power is called power. In, a is called the base and n is called the exponent. When an is regarded as the result of the n power of a, it can also be read as the n power of a. ..
The odd power of a negative number is negative and the even power of a negative number is positive.
Any power of a positive number is a positive number, and any power of a positive integer is 0.
Operation sequence of rational number mixed operation:
(1) first power, then multiply and divide, and finally add and subtract;
(2) the same layer operation, from left to right;
(3) If there are brackets, do the operation in brackets first, and then press brackets, brackets and braces in turn.
1.5.2 scientific counting method
Numbers greater than 10 are expressed in the form of a× 10n (where a is a number with only one integer and n is a positive integer), and scientific notation is used.
Use scientific notation to represent n-bit integers, where the exponent of 10 is n- 1.
1.5.3 divisors and significands
A number that is close to the actual number but still different from the actual number is called a divisor.
Accuracy: an approximate value is rounded to the nearest place, so it is accurate to the nearest place.
From the first non-zero digit to the last digit on the left of a number, all digits are valid digits of this number.
For the number a× 10n expressed by scientific notation, its effective number is specified as the effective number in A. ..