整数是否在减法下关闭?
数字是社会世界中金融、专业以及社会领域中使用的数学数字。数字中的位数和位值以及数字系统的基数决定了数字的值。数字用于各种数学运算,如加法、减法、乘法、除法、百分比等,这些运算用于我们的日常业务和交易活动。
数字
数字是适用于计数、测量和其他算术计算的数学数字或数值。数字的一些例子是整数、整数、自然数、有理数和无理数等。数字系统是将数字表示为不同形式的标准化方法,即数字和文字。它包括不同类型的数,例如素数、奇数、偶数、有理数、整数等。这些数可以根据所使用的数系以形式表示。
数系包括不同类型的数,例如质数、奇数、偶数、有理数、整数等。这些数可以相应地以数字和文字的形式表示。例如,40、65等以数字形式表示的数字,也可以写成40、65。
表示数字的基本系统称为数字系统。它是数字表示的标准化方法,其中数字以算术和代数结构表示。
数字类型
有不同类型的数字按数字系统分类。类型描述如下:
- 自然数:自然数从 1 到无穷大。它们是用“N”表示的正计数数。这是我们通常用于计数的数字。自然数集可以表示为 N = {1, 2, 3, 4, 5, 6, 7, ...}
- 整数:整数从零到无穷大。整数不包括分数或小数。整数集由“W”表示。该集合可以表示为 W = {0, 1, 2, 3, 4, 5, ...}
- 有理数:有理数是可以表示为两个整数之比的数。它包括所有整数,可以用分数或小数表示,用“Q”表示。
- 无理数:无理数是不能用分数或整数比表示的数字。它可以写成小数,小数点后有无穷无尽的不重复数字。它们由“P”表示。
- 整数:整数是一组数字,包括所有正数、零以及从负无穷到正无穷的所有负数。该集合不包括分数和小数。整数集由“Z”表示。示例:Z = {.., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}
- 十进制数:任何由小数点组成的数值都是十进制数。可表示为 2.5、0.567 等。
- 实数:不包括任何虚数并且是所有正整数、负整数、分数和十进制值的组成部分的数字集合是实数。它通常用“R”表示。
- 复数:它们是一组数字,其中包括虚数是复数。它可以表示为a + bi,其中“a”和“b”是实数。它用“C”表示。
整数是否在减法下关闭?
回答:
The subset of the number system that consists of all positive integers including 0 is defined as a whole number. The whole number counts from zero to positive infinity. These numbers are mostly used for counting, measurement of fundamental quantities, and daily calculations.
Whole numbers are the only constituents of natural numbers including zero. The subset is given by {0, 1, 2, 3, 4, 5, …}, the set does not include fractions, decimals, and negative integers. The set of whole numbers is represented by ‘W’. The set can be represented as W = {0, 1, 2, 3, 4, 5, …}
Hence, the standard form of whole number is 0, 1, 2, 3, 4, 90,…
Examples of Whole Numbers
Positive integers are also known as counting numbers including zero are part of whole numbers, such as 0, 1, 2, 3, 4, 5, etc, excluding negative integers, fractions, and decimals.12, 120, 1200, etc all are examples of whole numbers.
Whole numbers are not closed under subtraction operation because when assume any two numbers, and if subtracted one number from the other number. it is not compulsory that the result is a whole number. Recall the definition of the whole number set W, take any two whole numbers a, b ∈ W and then add, subtract, multiply them to check whether the result is also a whole number or not.
示例问题
问题1:闭包属性中的整数是否在减法下关闭?
回答:
In this property of whole numbers, add or multiply any two whole numbers that will also result in a whole number, but when two whole number are subtracted that does not result into whole number. Example,
- 2 + 5 = 7 and 80 + 40 = 120 for addition
- 6 × 5 = 30, and 5 × 6 = 30 for multiplication
Now for subtraction:
1 – 0 = 0 i.e a whole number, and 0 – 1 = -1 i.e not a whole number
So in this property, whole number are not closed under subtraction .
问题2:交换性质中的整数在减法下是否封闭?
回答:
It states that the operation of addition or multiplication on the number does not matter what is the order, it will give the same result even after swapping or reversing their position or, the placement of adding or multiplying numbers can be changed but it will give the same results. This property is valid for addition and multiplication not for subtraction.
- x + y = y + x
- x × y = y × x
Example:
- If add 6 in 2 or add 2 in 6 results will be same,
7 + 2 = 9 = 2 + 7
- Multiply both the real numbers,
6 × 7 = 42 = 7 × 6
- Now for subtraction:
x – y = y – x
7 – 2 = 5 i.e a whole number, or 2 – 5 = -7 is not a whole number.
So once again it proved through commutative property that whole number are not closed under subtraction .
问题 3:在关联属性中,整数在减法下是否闭合?
回答:
This property states that when three or more numbers are added (or multiplied) or the sum (or product) is the same regardless of the grouping of the addends (or multiplicands). The addition or multiplication in which order the operations are performed does not matter as long as the sequence of the numbers is not changed. This is defined as the associative property. That is, rearranging the numbers in such a manner that will not change their value.
- (x + y) + z = x + (y + z)
- (x × y) × z = x × (y × z)
Example:
- (8 + 5) + 6 = 8 + (5 + 6)
19 = 19
- (8 × 5) × 6 = 8 × (5 × 6)
240 = 240
As it can be seen even after changing their order, it gives the same result in both the operations adding as well as multiplication.
Now for subtraction:
Example:
(x – y) – z = x – (y-z)
(6 – 5 ) – 2 = 6 – ( 5 – 2 )
– 1 = 3
So it’s proved that whole number is not closed under subtraction in associative property .
问题4:分配财产中的整数在减法下是否封闭?
回答:
This property helps simplify the multiplication of a number by a sum or difference. It distributes the expression as it simplifies the calculation.
x × (y + z) = x × y + x × z
x × (y – z) = x × y – x × z
Example:
Simplify 20 × (5 + 6)
= 20 × 5 + 20 × 6
= 100 + 120
= 220
Now for subtraction:
x × (y – z) = x × y – x × z
20 × ( 6 – 5 ) = 20 × 6 – 20 × 5
20 × 1 = 120 – 100
20 = 20
It applies same for the subtraction also. Hence, from this property it shows that whole number is closed under subtraction in this property.