问题1：定义一个无理数。
解决方案：
A real number that cannot be expressed in the form of fractions i.e. p/q, where p and q are integers and q ≠ 0. It is a non-terminating or non-repeating decimal. i.e. for example:
1.1000120010211…..
问题2：说明，非理性数字与有理数有何不同?
解决方案：
An irrational number is a real number that cannot be expressed in the form of fractions i.e. p/q, where p and q are integers and q ≠ 0 i.e it cannot be expressed as a ratio of integers. It is a non-terminating or non-repeating decimal.
For example, √2 is an irrational number
A rational number is a real number that can be expressed as a fraction and as a decimal i.e. it can be expressed as a ratio of integers. It is a terminating or repeating decimal.
For examples: 0.101 and 5/4 are rational numbers
问题3：检查以下数字是有理还是无理：
(i)√7
(ii)√4
(iii)2 +√3
(iv)√3+√2
(v)√3+√5
(vi)(√2– 2) 2
(vii)(2 –√2)(2 +√2)
(viii)(√3+√2) 2
(ix)√5– 2
[x)√23
(西)√225
(xii)0.3796
(xiii)7.478478……
(xiv)1.101001000100001……
解决方案：
(i) √7
Given: √7
Since, it is not a perfect square root,
Therefore, it is an irrational number.
(ii) √4
Given: √4
Since, it is a perfect square root of 2.
Therefore, 2 can be expressed in the form of 2/1, thus it is a rational number.
(iii) 2 + √3
Given: 2 + √3
Here, 2 is a rational number, and √3 i is not a perfect square thus it is an irrational number.
Since, the sum of a rational and irrational number is always an irrational number.
Therefore, 2 + √3 is an irrational number.
(iv) √3 + √2
Given: √3 + √2
Here, √3 is not a perfect square thus it is an irrational number.
Similarly, √2 is not a perfect square, thus it is an irrational number.
Since, the sum of two irrational numbers is always an irrational number.
Therefore, √3 + √2 is an irrational number.
(v) √3 + √5
Given: √3 + √5
Here, √3 is not a perfect square thus it is an irrational number
Similarly, √5 is not a perfect square thus it is an irrational number.
Since, the sum of two irrational numbers is always an irrational number.
Therefore, √3 + √5 is an irrational number.
(vi) (√2 – 2)2
Given: (√2 – 2)2
(√2 – 2)2 = 2 + 4 – 4 √2
= 6 – 4 √2
Here, 6 is a rational number but 4√2 is an irrational number.
Since, the sum of a rational and irrational number is always an irrational number.
Therefore, (√2 – 2)2 is an irrational number.
(vii) (2 – √2)(2 + √2)
Given: (2 – √2)(2 + √2)
(2 – √2)(2 + √2) = ((2)2 − (√2)2) [As, (a + b)(a – b) = a2 – b2]
= 4 – 2
= 2 or 2/1
Since, 2 is a rational number,
Therefore, (2 – √2)(2 + √2) is a rational number.
(viii) (√3 + √2)2
Given: (√3 + √2)2
(√3 + √2)2 = (√3)2 + (√2)2 + 2√3 x √2 [ As, (a + b)2 = a2 – 2ab + b2 ]
= 3 + 2 + 2√6
= 5 + 2√6
Since, the sum of a rational and irrational number is always an irrational number.
Therefore, (√3 + √2)2 is an irrational number.
(ix) √5 – 2
Given: √5 – 2
Here, √5 is an irrational number but 2 is a rational number.
Since, the difference between an irrational number and a rational number is an irrational number.
Therefore, √5 – 2 is an irrational number.
(x) √23
Given: √23
√23 = 4.795831352331…
Since, the decimal expansion of √23 is non-terminating and non-recurring
Therefore, √23 is an irrational number.
(xi) √225
Given: √225
√225 = 15 or 15/1
Since, √225 can be represented in the form of p/q and q ≠ 0.
Therefore, √225 is a rational number
(xii) 0.3796
Given: 0.3796
Since, the decimal expansion is terminating.
Therefore, 0.3796 is a rational number.
(xiii) 7.478478……
Given: 7.478478……
Since, the decimal expansion is a non-terminating recurring decimal.
Therefore, 7.478478…… is a rational number.
(xiv) 1.101001000100001……
Given: 1.101001000100001……
Since, the decimal expansion is non-terminating and non-recurring.
Therefore, 1.101001000100001…… is an irrational number
问题4：将以下内容标识为有理数或无理数。给出有理数的十进制表示形式：
(i)√4
(ii)3√18
(iii)√1.44
(iv)√9/ 27
(v)–√64
(六)√100
解决方案：
(i) √4
Given: √4
Since, √4 = 2 = 2/1, it can be written in the form of a/b.
Therefore, √4 is a rational number.
The decimal representation of √4 is 2.0
(ii) 3√18
Given: 3√18
3√18 = 9√2
Since, the product of a rational and an irrational number is always an irrational number.
Therefore, 3√18 is an irrational number.
(iii) √1.44
Given: √1.44
Since, √1.44 = 1.2, it is a terminating decimal.
Therefore, √1.44 is a rational number.
The decimal representation of √1.44 is 1.2
(iv) √9/27
Given: √9/27
Since, √9/27 = 1/√3, as the quotient of a rational and an irrational number is an irrational number.
Therefore, √9/27 is an irrational number.
(v) – √64
Given: – √64
Since, – √64 = – 8 or – 8/1, as it can be written in the form of a/b.
Therefore, – √64 is a rational number.
The decimal representation of – √64 is –8.0
(vi) √100
Given: √100
Since, √100 = 10 = 10/1, as it can be written in the form of a/b.
Therefore, √100 is a rational number.
The decimal representation of √100 is 10.0
问题5：在以下方程式中，找出哪些变量x，y，z等表示有理数或无理数：
(i)x 2 = 5
(ii)y2 = 9
(iii)z 2 = 0.04
(iv)u 2 = 17/4
(v)v 2 = 3
(vi)w 2 = 27
(vii)t 2 = 0.4
解决方案：
(i) x2 = 5
Given: x2 = 5
When we take square root on both sides, we get,
x = √5
Since, √5 is not a perfect square root,
Therefore, x is an irrational number.
(ii) y2 = 9
Given: y2 = 9
When we take square root on both sides, we get,
y = 3
Since, 3 = 3/1, as it can be expressed in the form of a/b
Therefore, y is a rational number.
(iii) z2 = 0.04
Given: z2 = 0.04
When we take square root on both sides, we get,
z = 0.2
Since, 0.2 = 2/10, as it can be expressed in the form of a/b and is a terminating decimal.
Therefore, z is a rational number.
(iv) u2 = 17/4
Given: u2 = 17/4
When we take square root on both sides, we get,
u = √17/2
Since, the quotient of an irrational and a rational number is irrational,
Therefore, u is an irrational number.
(v) v2 = 3
Given: v2 = 3
When we take square root on both sides, we get,
v = √3
Since, √3 is not a perfect square root,
Therefore, v is an irrational number.
(vi) w2 = 27
Given: w2 = 27
When we take square root on both sides, we get,
w = 3√3
Since, the product of a rational and irrational is always an irrational number.
Therefore, w is an irrational number.
(vii) t2 = 0.4
Given: t2 = 0.4
When we take square root on both sides, we get,
t = √(4/10)
t = 2/√10
Since, the quotient of a rational and an irrational number is always an irrational number.
Therefore, t is an irrational number.
问题6：举两个不合理的数字，每个例子：
(i)有理数差异
(ii)无理数的差异
(iii)总数合理
(iv)总和是不合理的数字
(v)产品数量合理
(vi)产品编号不合理
(vii)有理数的商
(viii)无理数的商
解决方案：
(i) Difference in a rational number
√5 is an irrational number
Since, √5 – √5 = 0
Here, 0 is a rational number.
(ii) Difference in an irrational number
Let the two irrational number be 5√3 and √3
Since, (5√3) – (√3) = 4√3
Here, 4√3 is an irrational number.
(iii) Sum in a rational number
Let the two irrational numbers be √5 and -√5
Since, (√5) + (-√5) = 0
Here, 0 is a rational number.
(iv) Sum is an irrational number
Let the two irrational numbers be 4√5 and √5
Since, 4√5 + √5 = 5√5
Here, 5√5 is an irrational number.
(v) Product in a rational number
Let the two irrational numbers be 2√2 and √2
Since, 2√2 × √2 = 2 × 2 = 4
Here, 4 is a rational number.
(vi) Product in an irrational number
Let the two irrational numbers be √2 and √3
Since, √2 × √3 = √6
Here, √6 is an irrational number.
(vii) Quotient in a rational number
Let the two irrational numbers be 2√2 and √2
Since, 2√2 / √2 = 2
Here, 2 is a rational number.
(viii) Quotient in an irrational number
Let the two irrational numbers be 2√3 and 2√2
Since, 2√3 / 2√2 = √3/√2
Here, √3/√2 is an irrational number.
问题7：给出介于0.232332333233332和0.212112111211112之间的两个有理数。
解决方案：
Let a = 0.212112111211112
Let b = 0.232332333233332
Here aa has digit 1 and b has digit 3.
If the second decimal place is considered as 2 then it lies between a and b.
Therefore, Let x = 0.22
and y = 0.22112211…
Thus, a < x < y < b
Hence, x and y are the rational numbers required.
问题8：给出两个介于0.515115111511115和0.5353353335之间的有理数
解决方案：
Let a = 0.515115111511115
Let b = 0.5353353335
Here aa has digit 1 and b has digit 3.
If the second decimal place is considered as 2 then it lies between a and b.
Therefore, Let x = 0.52
and y = 0.520520…
Thus, a < x < y < b
Hence, x and y are the rational numbers required.
问题9：找到一个介于0.2101和0.2222之间的无理数…
解决方案：
Let a = 0.2101
and b = 0.2222…
Here aa has digit 1 and b has digit 2.
If the third decimal place is considered as 1 then it lies between a and b.
Therefore, Let x = 0.2110110011…
Thus, a < x < b
Hence, x is the irrational number required.
问题10：找出介于0.3030030003…和0.3010010001…之间的有理数和无理数。
解决方案：
Let a = 0.3010010001…
and b = 0.3030030003…
Here aa has digit 1 and b has digit 3.
If the third decimal place is considered as 2 then it lies between a and b.
Therefore, Let x = 0.302
and y = 0.302002000200002…
Thus, a < x < y < b
Hence, x and y are the rational and irrational numbers required respectively.
问题11：找到两个介于0.5和0.55之间的无理数。
解决方案：
Let a = 0.5
and b = 0.55
Here a
If the second decimal place is considered between1 to 4 then it lies between a and b.
Therefore, Let x = 0.510510051000…
and y = 0.53053530…
Thus, a < x < y < b
Hence, x and y are the irrational numbers required.
问题12：找出两个介于0.1到0.12之间的无理数。
解决方案：
Let a = 0.1
and b = 0.12
Here a
If the second decimal place is considered 1 then it lies between a and b.
Therefore, Let x = 0.11011011000…
and y = 0.11100010100…
Thus, a < x < y < b
Hence, x and y are the irrational numbers required.
问题13：证明√3+√5是一个无理数。
解决方案：
Let √3 + √5 be a rational number equal to x.
Therefore, x = √3 + √5
x2 = (√3 + √5)2
x2 = (√3)2 + (√5)2 + 2 √3 √5
= 3 + 5 + 2√15
= 8 + 2√15
x2 – 8 = 2√15
(x2 – 8)/2 = √15
Here, (x2 – 8)/2 is a rational but √15 is an irrational number.
Therefore, √3 + √5 is an irrational number.