即使在创造整数之后，也无法放松！ 10÷5无疑是好的，给出答案2，但8÷5舒服吗?需要数字之间的数字。 8÷5被视为1.6，是1到2之间的数字。但是(-3)÷4在哪里?在0到-1之间。因此，将一个整数除以另一个整数所得到的比率称为有理数。所有有理数的集合由Q表示。
A Rational number is a number of the fractional form a / b, where a and b are integers and b ≠ 0.
Examples: 1 / 4 , 3 / 7 , (-11) / (-6)
- 全自然数，整数，整数和分数是有理数。
- 每个有理数都可以在数字行上表示。
- 0既不是正数也不是负有理数。
有理数的性质
有理数集合Q的闭包性质
- 加法的闭包属性：对于任意两个有理数a和b ，总和a + b也是有理数。
- 乘法的闭包属性：对于任何两个有理数a和b ，总和ab也是有理数
例子：
Take a = 3 / 4 and b = (-1) / 2
Now,
a + b = 3 / 4 + (-1) / 2
= 3 / 4 + (-2) / 4
= (3 – 2) / 4
= 1 / 4 is in Q
Also,
a * b = 3 / 4 * (-1) / 2
= (-3) / 8 is in Q
有理数集合Q的可交换性
- 加法的交换性质：对于任何两个有理数a和b ， a + b = b + a。
- 乘法的交换性质：对于任何两个有理数a和b ， ab = ba。
例子：
Take a = (-7) / 8 and b = 3 / 5
Now,
a + b = (-7) / 8 + 3 / 5
= -7 x 5 + 3 x 8 / 40
= (-35 + 24) / 40
= (-11) / 40
Also
b + a = 3 / 5 + (-7) / 8
= 3 x 8 + (-7) x 5 / 40
= (24 – 35) / 40
= (-11) / 40
Hence addition is Commutative.
Further,
ab = (-7) / 8 x 3 / 5
= (-7 x 3) / (8 x 5)
= (-21) / 40
Also,
ba = 3 / 5 x (-7) / 8
= (3 x 7 ) / (5 x 8)
= (-21) / 40
Hence multiplication is Commutative.
有理数集合Q的结合性质
- 加法的缔合性质：对于任何三个有理数a ， b和c ， a +(b + c)=(a + b)+ c
- 乘法的关联属性：对于任何三个有理数a ， b和c ， a(bc)=(ab)c
例子：
Take rational numbers a,b,c as a = -1 / 2, b = 3 / 5, c = -7 / 10
Now,
a + b = -1 / 2 + 3 / 5
= -5 / 10 + 6 / 10
= -5 + 6 / 10
= 1 / 10
(a + b) + c = 1 / 10 + (-7) / 10
= 1 – 7 / 10
= -6 / 10
= -3 / 5………………………………………………….( 1 )
Also,
b + c = 3 / 5 + (-7) / 10
= 6 / 10 + (-7) / 10
= 6 – 7 / 10
= -1 / 10
a + (b + c) = -1 / 2 + (-1) / 10
= -5 / 10 + (-1) / 10
= -5 – 1 / 10
= -6 / 10
= -3 / 5 ………………………………………………..( 2 )
(1) and (2) shows that (a + b) + c = a + (b + c) is true for rational numbers.
Similarly,
a ∗ b = -1 / 2 ∗ 3 / 5
= -3 / 10
(a ∗ b)∗ c = -3 / 10 ∗ -7 / 10
= -3 ∗ (-7) / 100
= 21 / 100 …………………………………………( 3 )
Also,
b∗ c = 3 / 5 ∗ (-7) / 10
= -21 / 50
a ∗ ( b ∗ c ) = -1 / 2 ∗ (-21) / 50
= 21 / 100 …………………………………….( 4 )
( 3 ) and ( 4 ) shows that (a∗ b)∗ c = a ∗ ( b ∗ c ) is true for rational numbers. Thus,the associative property is true for addition and multiplication of rational numbers.
有理数集合Q的标识属性
- 加法的标识属性：对于任何有理数a，都有一个唯一的有理数0 ，使得0 + a = a = a + 0 。
- 乘法的恒等性质：对于任何有理数a，都存在唯一的有理数1 ，使得a * 1 = a = a * 1。
例子：
Take a = 3 / (-7) that is a = -3 / 7
Now,
-3 / 7 + 0 = -3 / 7 = 0 + (-3) / 7
Hence,0 is the additive identity for -3 / 7
Also,
-3 / 7 ∗ 1 = -3 / 7 = 1 ∗ 3 / 7
Hence,1 is the multiplicative identity for -3 / 7.
有理数集合Q的逆性质
- 加性逆属性：对于任何有理数a，存在一个唯一有理数-a使得a +(-a)=(-a)+ a = 0.在这里， 0是加法恒等式。
- 乘法逆性质：对于任何有理数b，存在唯一的有理数1 / b，使得b ∗ 1 / b = 1 / b ∗ b = 1。在这里， 1是乘法恒等式。
例子：
Take a = -11 / 23
Now, -a = -(-11) / 23
= 11 / 23
So,
a + (-a) = -11 / 23 + 11 / 23
= -11 + 11 / 23
= 0 / 23
=0
Also,
(-a) + a = 11 / 23 + (-11) / 23
= 11-11 / 23
= 0 / 23
= 0
Hence a + (-a) = (-a) + a = 0 is true.
Also,
Take b = -17/29
Now,
1 / b = 29 / (-17) = -29 / 17
b ∗ 1 / b = -17 / 29 ∗ -29 / 17 = 1
Also,
1 / b ∗ b = 29 / 17 ∗ -17 / 29 = 1
Hence, b ∗ 1 / b = 1 / b ∗ b = 1 is true.
有理数集合Q的分布性质
对于有理数的收集，乘法是加法的分布。对于任何三个有理数a，b和c，分布律为a ∗(b + c)=(a ∗ b)+(a ∗ c)
例子：
Take rational number a, b, c as a = -7 / 9, b = 11 / 18 and c = -14 / 27
Now,
b + c = 11 / 18 + (-14) / 27
= 33 / 54 + (-28) / 54
= 33 – 28 /54
= 5 / 54
a ∗ ( b + c ) = -7 / 9 ∗ 5 / 54
= (-7) ∗ 5 / 9 ∗ 54
= -35 / 486………………………………………………………………….(1)
Also,
a ∗ b = -7 / 9 ∗ 11 / 18
= (-7) ∗ 11 / 9 ∗ 18
= -77 / 9 ∗ 9 ∗ 2
a ∗ c = (-7) / 9 ∗ (-14) / 27
= 7 ∗ 14 / 9 ∗ 9∗ 3
= 98 / 9 ∗ 9 ∗ 3
(a ∗ b) + (a ∗ c) = (-77 / 9 ∗ 9 ∗ 2 ) + ( 98 / 9 ∗ 9 ∗ 3)
= (-77) ∗ 3 + 98 ∗ 2 / 9 ∗ 9 ∗ 2 ∗ 3
= -231 + 196 / 486
= (-35) / 486…………………………………………………………….(2)
( 1 ) and( 2 ) shows that a ∗ ( b + c ) = ( a ∗ b ) + ( a ∗ c ). Hence,multiplication is distributive over addition for the collection Q of rational numbers.