0到1之间有多少个有理数？

数制是表示数轴数字的系统方式。这些数字使用一组符号和规则来描述。这些数字位于属于 0-9 的区间内。这些数字也称为数字，可以使用各种数学运算进行操作。所有的计算，如计数、输入和操作，都可以使用数轴的计算来执行。

有理数

有理数是以分数形式表示的数，p/q，其中 p 和 q 是整数，并且 q 不等于 0。数字系统中的整个有理数集由字母表示问：换句话说，如果一个数可以表示为分数，其中分子和分母都是整数值，则该数称为有理数。这些有理数还可以进一步简化以获得纯整数或十进制值。

有理数的例子

有理数的一些例子可以是：

• 1/2
• -3/4
• 0.3 或 3/10
• -0.7 或 -7/10
• 0.141414… 或 14/99

0到1之间有多少个有理数？

解决方案：

There are infinite sets of rational numbers contained between 0 and 1. To prove this, let us consist of an interval, 

[0 , 1]

Now dividing it into two parts, we obtain, [0,1/2] and [1/2, 1]

Taking A as [0,1/2], we can further divide it into [0, 1/4] and [1/4, 1/2]

Taking A’ as [0,1/4], we can further divide it into [0, 1/8] and [1/8, 1/4]

Similarly, we can infinitely divide these intervals exponentially into further smaller intervals. 

There is no upper bound to the number of intervals that can be created. Each of the obtained intervals at least adds one number to the count of rational numbers between 0 and 1.

Hence, there are infinite rational numbers between 0 and 1.

Extending this argument further, we get, 

Note: Between any two real numbers a and b, there are an infinite number of real numbers.

类似问题

问题 1. 找出 2/3 和 3/4 之间的有理数？

解决方案：

To find a rational number between 2/3 and 3/4

Find the average of these two rational numbers

Average = (2/3 + 3/4)/2

LCM of 3 and 4 = 12

Average = ((4×2)+(3×3)/12)/2

Average = (8+9/12)/2

Average = 17/12/2

Average = 17/12 × 1/2

Average = 17/24

Therefore,

The rational number between 2/3 and 3/4 is 17/24.

问题 2. 从这些分数中找出有理数；

1/7、99/0、7/6、0、2/7、5/8、5。

解决方案：

Here we have to find out the rational numbers

1/7, 99/0, 7/6, 0, 2/7, 5/8, 5.

As we know that

Rational numbers are in the form of a/b, where b ≠ 0

Therefore,

1/7 is a rational number 

7/6 is a rational number 

2/7 is a rational number 

5/8 is a rational number 

问题 3. 找出 1/2 和 2/3 之间的五个有理数？

解决方案：

First make the denominators same

Taking LCM of 2 and 3

LCM of 2 and 3 = 6

Now the rational numbers will be

1×3/2×3 and 2×2/3×2

3/6 and 4/6

Now multiply both numerator and denominator by 10 of both the rational numbers

3×10/6×10 = 30/60

4×10/6×10 = 40/60

Now,

The rational numbers between 1/2 and 2/3 will be

31/60, 32/60, 33/60, 34/60, 35/60

问题 4. 写出 -4/5 到 6/7 之间的任意十个有理数？

解决方案：

First find the LCM of 5 and 7

LCM of 5 and 7 = 35

Now,

-4×7/5×7 = -28/35

6×5/7 ×5 = 30/35

Thus,

The rational numbers between -4/5 and 6/7 are -27/35, -26/35, -25/35, -24/35, -23/35, -22/35, -21/35, -20/35, -19/35, -18/35,….., 29/35.