What's the difference between "duo" and "duo"
"Times" refers to the relationship between quantities, which is based on the concept of multiplication. In practical teaching, it is a mathematical concept gradually abstracted from "ge" and "fen".
For example, the white cloth is 8 meters long and the printed cloth is 4 8 meters long; In other words, 8m white cloth is regarded as 1 copy, and the length of printed cloth is 4 copies. The words "ge" and "fen" mentioned here are replaced by mathematical language, that is, the length of cloth is four times that of 8 meters, and the number of meters of cloth is 8×4=32 meters. It can be seen that the emergence of "times" is gradually abstracted from "individuals" and "shares" in life and is based on the concept of multiplication.
"Multiplication" is the relationship between exponent and number, which is based on the big concept of "divisibility of number" and is established at the same time with "divisor" on the premise of clear divisibility.
For example, 28 is a multiple of 7, because 28 is divisible by 7. 28÷7=4, and 28 is four times that of 7. If multiplication is used to express the quantitative relationship between these three numbers, 7×4=28, and four times of 7 is 28. Therefore, the "multiple" of the former is strictly limited within the scope of "divisibility", while the "multiple" of the latter is only reflected in the concept of multiplication, which is the obvious difference between the two.
There is another situation in the application of "multiple" in primary school mathematics textbooks, that is, in proportion teaching, when explaining the relationship between positive and negative proportions, the "multiple" mentioned here is a concept in general division, not a concept within the scope of "divisibility". The multiple in the proportion represents the two most numerous numbers, which are not necessarily integers, but also decimals. Decimals are not allowed when studying multiples in divisible numbers.
The difference between "times" and "multiples"
"Multiplication" refers to the quantitative relationship and is based on the concept of multiplication. For example, 10 rooster and 10 hen. We say that the number of hens is three times that of roosters, or three times that of 10, that is, there are three hens in 10.
"Multiplication" is the relation between exponent and number, which is based on the concept of divisibility. For example, 30 is divisible by 6, and 30 is a multiple of 6. But 30 is five times that of 6, because 6×5=30, "6×5" is five times that of 6.
When expounding the positive-negative proportional relationship, the same "multiple" as "expansion" or "contraction" is mentioned. The meaning of "duo" here is different from that mentioned above. The aforementioned multiple refers to a concept in divisibility, refers to the dividend, and can only be an integer. The latter multiple is a concept in general division and refers to quotient.
In the future, with the increase of our knowledge, the range of multiples will continue to expand. The multiple can be an integer, a fraction or a decimal, and even a percentage can be used to express the multiple relationship between two quantities.
What is the nature of "divisible number"
The divisibility of numbers has many attributes, including the following:
1 If both integers A and B are divisible by C, then the sum of A and B can also be divisible by C. ..
For example: 42÷7=6 56÷7=8.
42+56÷7= 14
42 can be divisible by 7 and 56 can be divisible by 7, so the sum of 42 and 56 and 98 can also be divisible by 7.
On the other hand, if one of the integers A and B is divisible by C and the other is not, then the sum of A and B must not be divisible by C. ..
For example: 36 ÷ 9 = 4 83 ÷ 9 = 9...2
36+83÷9= 13……2
36 is divisible by 9, and 83 is divisible by 9, so the sum of 36 and 83, 1 19, is divisible by 9.
If both integers A and B are divisible by C, then the difference between A and B can also be divisible by C. ..
For example: 88 ÷11= 8,66 ÷11= 6.
88-66÷ 1 1=2
88 can be divisible by 1 1, and 66 can also be divisible by 1 1, so the difference between 88 and 66 can be divisible by1.
On the other hand, if one of the integers A and B is divisible by C and the other is not, then the difference between A and B must not be divisible by C. ..
For example: 91÷13 = 7 30 ÷13 = 2 ...
9 1-30÷ 13=4……9
9 1 can be divisible by 13, and 30 cannot be divisible by 13, so the difference between 9 1 and 30, 6 1, cannot be divisible by 13.
3 If the integers A and B are not divisible by C .. then the sum or difference of A and B may or may not be divisible by C, which is an uncertain conclusion.
For example: 65 ÷ 7 = 9...2 33 ÷ 7 = 4 ...5.
65+33÷7= 14
65-33÷7=4……4
65 is not divisible by 7, and 33 is not divisible by 7. Because the sum of two remainders is 2+5=7, which is exactly equal to the divisor, the sum of 65 and 33 can be divisible by 7. The difference between 65 and 33 is not divisible by 7.
Another example: 85 ÷11= 7 ... 838 ÷1= 3 ... 5.
85+38÷ 1 1= 1 1……2
85-38÷ 1 1=4……3
85 is not divisible by 1 1, and 38 is not divisible by 1 1. In this case, the sum of 85 and 38 or the difference of 47 cannot be divisible by 1 1.
If the integer a is divisible by the natural number c, then multiples of a can also be divisible by C.
For example: 39÷ 13=3
39×4÷ 13= 12
39 is divisible by 13, and four times of 39 156 can also be divisible by 13.
5 If among the three numbers A, B and C, A can be divisible by B and B and C, then A must be divisible by C, which is divisible transitivity.
For example, there are three numbers: 84,217.
84÷24=4 2 1÷7=3
84÷7= 12
84 is divisible by 2 1, and 2 1 is divisible by 7, so 84 must be divisible by 7.
On the other hand, if only one situation between A and B or between B and C is inseparable, then A must not be divisible by C. ..
For example, there are three numbers: 12 1,1and 5.
12 1÷ 1 1= 1 1 1 1÷5=2…… 1
12 1÷5=24…… 1
12 1 can be divisible by 1 1, but 1 1 cannot be divisible by 5, so 12 1 must not be divisible by 5. I guess you like it.