问题1：验证以下有理数对中每对有理数加法的可交换性
(i)-11/5和4/7
解决方案：
Commutativity property is verified if two rational numbers a and b, a + b = b + a
In this case a = -11 / 5 and b = 4 / 7
a + b = (-11) / 5 + 4 / 7
LCM of 5 and 7 is 35
= (-11 × 7 + 4 × 5) / 35
= (-77 + 20) / 35
= (-57) / 35
b + a = 4 / 7 + (-11) / 5
LCM of 5 and 7 is 35
= (4 × 5 + (-11 × 7)) / 35
= (20 – 77) / 35
= (-57) / 35
Hence, a + b = b + a
Therefore, commutativity is verified in this case.
(ii)4/9和7 / -12
解决方案：
Here a = 4 / 9, b = (-7) / 12
a + b = 4 / 9 + (-7) / 12
LCM of 9 and 12
9 = 3*3
12 = 3 * 2 * 2
LCM = 3 * 3 * 2 * 2 = 36
= (4 * 4 + (-7 * 3)) / 36
= (16 – 21) / 36
= (-5) / 36
Now, b + a = (-7) / 12 + 4 / 9
= ((-7 * 3) + 4 * 4) / 36
= (-21 + 16) / 36
= (-5) / 36
Hence, a + b = b + a
Therefore, commutativity is verified in this case.
(iii)-3 / 5和-2 / -15
解决方案：
Here a = -3 / 5, b = -2 / -15 = 2 / 15
a + b = (-3) / 5 + 2 / 15
LCM is 15
= (-3 * 3 + 2 * 1) / 15
= (-9 + 2) / 15
= (-7) / 15
Now,
b + a = 2 / 15 + (-3) / 5
= (2 × 1 + (-3 * 3)) / 15
= (2 – 9) / 15
= (-7) / 15
Hence, a + b = b + a
Therefore, commutativity is verified.
(iv)2 / -7和12 / -35
解决方案：
a = 2 / -7 = -2 / 7
b = 12 / -35 = -12 / 35
a + b = -2 / 7 + (-12) / 35
LCM of 7 and 35 is 35
= (-2 × 5 + (-12 × 1)) / 35
= (-10 – 12) / 35
= (-22) / 35
b + a = (-12) / 35 + (-2) / 7
LCM of 7 and 35 is 35
= (-12 × 1 + (-2 × 5)) / 35
= (-12 – 10) / 35
= (-22) / 35
a + b = b + a
Therefore, commutativity is verified.
(v)4和-3 / 5
解决方案：
a = 4 / 1
b = -3 / 5
a + b = 4 / 1 + (-3) / 5
LCM of 1 and 5 is 5
= (4 * 5 + (-3 * 1)) / 5
= (20 – 3) / 5
= 17 / 5
b + a = (-3) / 5 + 4 / 1
= (-3 + 4 * 5) / 5
= (-3 + 20) / 5
= (17) / 5
a + b = b + a
Therefore, commutativity is verified.
(vi)-4和4 / -7
解决方案：
a = -4 / 1
b = -4 / 7
a + b = (-4) / 1 + (-4) / 7
LCM of 1 and 7 is 7
= (-4 * 7 + (-4 * 1)) / 7
= (-28 – 4) / 7
= (-32) / 7
b + a = (-4) / 7 + (-4) / 1
= (-4 * 1 + (-4 * 7)) / 7
= (-4 – 28) / 7
= (-32) / 7
a + b = b + a
Therefore, commutativity is verified.
问题2：在以下情况下验证有理数加法的关联性，即(x + y)+ z = x +(y + z)
(i)x = 1/2，y = 2/3，z = -1 / 5
解决方案：
To verify associativity solving the LHS
(1 / 2 + 2 / 3) + (-1 / 5)
Bracket needs to be solved first
LCM of 2 and 3 is 6
= (1 * 3 + 2 * 2) / 6 + (-1 / 5)
= (3 + 4) / 6 + (-1 / 5)
= 7 / 6 + (-1 / 5)
Now solving these,
LCM of 5 and 6 is 30
= (7 * 5 + (-1 * 6)) / 30
= (35 – 6) / 30
= 29 / 30
Now solving the RHS
1 / 2 + (2 / 3 + (-1 / 5))
Solving the bracket first, LCM of 3 and 5 is 15
= 1 / 2 + (2 * 5 + (-1 * 3)) / 15
= 1 / 2 + (10 – 3) / 15
= 1 / 2 + 7 / 15
LCM of 2 and 15 is 30
= (1 * 15 + 7 * 2) / 30
= (15 + 14) / 30
= 29 / 30
Hence LHS = RHS, associativity property is verified
(ii)x =-/ 2/5，y = 4/3，z = -7 / 10
解决方案：
According to the property x + (y + z) = (x + y) + z
Solving LHS
-2 / 5 + (4 / 3 + (-7 / 10))
Solving bracket first
LCM of 3 and 10 is 30
= -2 / 5 + (4 * 10 + (-7 * 3)) / 30
= -2 / 5 + (40 – 21) / 30
= -2 / 5 + 19 / 30
Now, LCM of 5 and 30 is 30
= (-2 * 6 + 19 * 1) / 30
= (-12 + 19) / 30
= 7 / 30
Now, considering RHS
(-2 / 5 + 4 / 3) + (-7 / 10)
LCM of 5 and 3 is 15
= (-2 * 3 + 4 * 5) / 15 + (-7 / 10)
= (-6 + 20) / 15 + (-7 / 10)
=(14) / 15 + (-7) / 10
LCM of 15 and 10 is 30
= (14 * 2 + (-7 * 3)) / 30
= (28 – 21) / 30
= 7 / 30
Hence, LHS = RHS
Therefore, associativity property is verified
(iii)x = -7 / 11，y = 2 / -5，z = -3 / 22
解决方案：
According to the property,
LHS is (-7 / 11 + (-2 / 5)) + (-3 / 22)
Solving bracket first
LCM of 11 and 5 is 55
= (-7 * 5 + (-2 * 11)) / 55 + (-3 / 22)
= (-35 – 22) / 55 + (-3 / 22)
= (-57) / 55 + (-3 / 22)
LCM of 55 and 2 is 110
= (-57 * 2 + (-3 * 5)) / 110
= (-114 – 15) / 110
= (-129) / 110
Now, solving RHS
= -7 / 11 + (-2 / 5 + (-3) / 22)
LCM of 22 and 5 is 110
= -7 / 11 + (-2 * 22 + (-3 * 5)) / 110
= -7 / 11 + (-44 – 15) / 110
= -7 / 11 + (-59) / 110
LCM of 11 and 110 is 110
= (-7 * 10 + (-59 * 1)) / 110
= (-70 – 59) / 110
= (-129) / 110
Hence, LHS = RHS, associativity is verified.
(iv)x = -2，y = 3/5，z = -4 / 3
解决方案：
According to the property,
LHS is (-2 / 1 + 3 / 5) + (-4 / 3)
LCM of 1 and 5 is 5
= (-2 * 5 + 3 * 1) / 5 + (-4 / 3)
= (-10 + 3) / 5 + (-4 / 3)
= (-7) / 5 + (-4 / 3)
LCM of 5 and 3 is 15
= (-7 * 3 + (-4 * 5)) / 15
= (-21 – 20) / 15
= (-41) / 15
Now solving RHS
-2 / 1 + (3 / 5 + (-4 / 3))
LCM of 5 and 3 is 15
= -2 / 1 + (3 * 3 + (-4 * 5)) / 15
= -2 / 1 + (9 – 20) / 15
= -2 / 1 + (-11) / 15
LCM of 1 and 15 is 15
= (-2 * 15 + (-11 * 1)) / 15
= (-30 – 11) / 15
= (-41) / 15
Hence, LHS = RHS
Therefore, associativity is verified.
问题3：写下以下各项的加法逆
(i)-2 / 17
Additive inverse is a number which when added to the given number gives 0. Therefore, it is negative of the number given.
Additive inverse of -2 / 17 = -(-2 / 17)
= 2 / 17
(ii)3 / -11
Additive inverse of -3 / 11 is 3 / 11
(iii)-17 / 5
Additive inverse of -17 / 5 is 17 / 5
(iv)-11 / -25
It can be written as 11 / 25
Additive inverse is -11 / 25
问题4：写下以下每个的负(加)逆
(i)-2 / 5
Negative (Additive) inverse is 2 / 5
(ii)7 / -9
It can be written as -7 / 9
Negative(Additive) inverse is 7 / 9
(iii)-16/13
Negative(Additive) inverse is 16 / 13
(iv)-5 / 1
Negative(Additive)inverse is 5
(v)0
0 is neutral number
Negative(Additive) inverse is 0
(六)1
Negative(Additive) inverse is -1
(vii)-1
Additive inverse is 1
问题5：使用有理数相加的可交换性和联想性，将以下各项表示为有理数
(i)2/5 + 7/3 + -4 / 5 + -1 / 3
灵魂：
According to commutativity order of numbers can be changed, so writing numbers with same denominators together.
(2 / 5 + -4 / 5) + (7 / 3 + -1 / 3)
= (2 – 4 / 5) + (7 – 1 / 3)
= -2 / 5 + 6 / 3
LCM of 5 and 3 is 15
= (-2 × 3 + 6 × 5) / 15
= (-6 + 30) / 15
= 24 / 15
3 is a common number so can be simplified further
= 8 / 5
(ii)3/7 + -4 / 9 + -11/7 + 7/9
灵魂：
According to the commutativity order of numbers can be changed, so writing numbers with the same denominators together.
= (3 / 7 + -11 / 7) + (-4 / 9 + 7 / 9)
= (3 – 11) / 7 + (-4 + 7) / 9
= -8 / 7 + 3 / 9
LCM of 7 and 9 is 63
= (-8 × 9 + 3 × 7) / 63
= (-72 + 21) / 63
= (-51) / 63
3 is the common number so can be simplified further
= -17 / 21
(iii)2/5 + 8/3 + -11/15 + 4/5 + -2 / 3
灵魂：
According to the commutativity order of numbers can be changed, so writing numbers with the same denominators together.
= (2 / 5 + 4 / 5) + (8 / 3 + -2 / 3) + (-11 / 15)
= (6) / 5 + 6 / 3 + (-11 / 15)
Applying associativity
LCM of 5 and 3 is 15
= (6 × 3 + 6 × 5) / 15 = () + (-11) / 15
= (18 + 30) / 15 + (-11) / 15
= 48 / 15 + (-11) / 15
= 37 / 15
(iv)4/7 + 0 + -8 / 9 + -13/7 + 17/21
灵魂：
According to the commutativity order of numbers can be changed, so writing numbers with the same denominators together.
(4 / 7 + -13 / 7) + -8 / 9 + 17 / 21
= -9 / 7 + (-8 / 9) + 17 / 21
LCM of 7, 9 and 21
7 = 7×1
9 = 3×3
21 = 3×7
LCM is 3 × 3 × 7 = 63
= (-9 × 9 + (-8 × 7) + 17 × 3) / 63
= (-81 – 56 + 51) / 63
= (-86) / 63
问题6：适当重新排列并在以下每个中求和
(i)11-12 + -17/3 + 11/2 + -25/2
灵魂：
According to the commutativity order of numbers can be changed, so writing numbers with the same denominators together.
= 11 / 12 + -17 / 3 + (11 – 25) / 2
= 11 / 12 + (-17) / 3 + (-14) / 2
LCM of 3 and 2 is 6
= 11 / 12 + (-17 × 2 + (-14 × 3)) / 6
= 11 / 12 + (-34 – 42) / 6
= 11 / 12 + (-76) / 6
LCM of 6 and 12 is 12
= (11 + (-76 × 2)) / 12
= (11 – 152) / 12
= (-141) / 12
(ii)-6/7 + -5 / 6 + -4 / 9 + -15/7
灵魂：
According to the commutativity order of numbers can be changed, so writing numbers with the same denominators together.
= (-6 / 7 + -15 / 7) + -5 / 6 + -4 / 9
= (-6 – 15) / 7 + -5 / 6 + -4 / 9
= -21 / 7 + -5 / 6 + -4 / 9
= -3 / 1 + -5 / 6 + -4 / 9
LCM of 6 and 9 is
6 = 3 × 2
9 = 3 × 3
LCM is 18
= (-3 × 18 + (-5 × 3) + (-4 × 2)) / 18
= (-54 – 15 – 8) / 18
= -77 / 18
(iii)3/5 + 7/3 + 9/5 + -13/15 + -7/3
灵魂：
According to the commutativity order of numbers can be changed, so writing numbers with the same denominators together.
= (3 / 5 + 9 / 5) + (7 / 3 + -7 / 3) + -13 / 15
= (3 + 9) / 5 + (7 – 7) / 3 + -13 / 15
= 12 / 5 + 0 / 3 + -13 / 15
= 12 / 5 + 0 + -13 / 15
= 12 / 5 + -13 / 15
LCM of 5 and 15 is 15
= (12 × 3 + (-13 × 1)) / 15
= (36 – 13) / 15
= 23 / 15
(iv)4/13 + -5 / 8 + -8/13 + 9/13
灵魂：
According to commutativity order of numbers can be changed, so writing numbers with same denominators together.
= (4 / 13 + -8 / 13 + 9 / 13) + -5 / 8
= (4 – 8 + 9) / 13 + -5 / 8
= (5) / 13 + -5 / 8
LCM of 13 and 8 is 104
= (5 × 8 + (-5 × 13)) / 104
= (40 – 65) / 104
= -25 / 104
(v)2/3 + -4 / 5 + 1/3 + 2/5
灵魂：
According to the commutativity order of numbers can be changed, so writing numbers with the same denominators together.
= (2 / 3 + 1 / 3) + (-4 / 5 + 2 / 5)
= (2 + 1) / 3 + (-4 + 2) / 5
= (3) / 3 + (-2) / 5
LCM of 3 and 5 is 15
= (3 × 5 + (-2 × 3)) / 15
= (15 – 6) / 15
= 9 / 15
= 3 / 5
(vi)1/8 + 5/12 + 2/7 + 7/12 + 9/7 + -5 / 16
灵魂：
According to the commutativity order of numbers can be changed, so writing numbers with the same denominators together.
= (5 / 12 + 7 / 12) + (2 / 7 + 9 / 7) + 1 / 8 + -5 / 16
= (5 + 7) / 12 + (2 + 9) / 7 + 1 / 8 + -5 / 16
= 12 / 12 + 11 / 7 + 1 / 8 + -5 / 16
= 1 / 1 + 11 / 7 + 1 / 8 + -5 / 16
LCM of 1, 7, 8 and 16 is 112
= (112 + 11 × 16 + 14 + (-5 × 7)) / 112
= (112 + 176 + 14 – 35) / 112
= (267) / 112