证明 0.3333… = 0.3，可以表示为 p/q
自从人类有了感官，他们就开始记录事物。这导致了数字系统的产生。最初，数字系统中只有自然数。随着人类大脑的发展，越来越多的研究被完成，新事物被发明和引入。随着时间零的出现，发明了负整数、有理数、无理数、实数和复数。
不同类型数字的符号如下：
N ⇢ represents natural numbers.
W ⇢ represents whole numbers.
Z ⇢ represents integers.
Q ⇢ represents rational numbers.
P ⇢ represents irrational numbers.
R ⇢ represents real numbers.
z ⇢ represents a number in complex plane.
下图显示了数字系统的层次结构，
有理数
“理性”一词源自“比率”一词。如果一个数可以用 p/q 的形式表示，则称它是有理数，其中 p 和 q 是整数，q ≠ 0。所有有理数都是实数。所有正整数、正分数、零、负整数和负分数都是有理数集的一部分。有理数集用Q表示。
有理数的性质
- Basic arithmetic can be performed, operations like addition, subtraction, multiplication, and division on rational numbers. The end result of these operations will result in rational numbers.
- All rational numbers display commutative, associative and distributive properties for the operations of addition and multiplication.
- The rational numbers have 0 and 1 as identity elements for the operation of addition and multiplication respectively.
- For every rational number x there exist a rational number -x so that x+(-x)=0.
- For every rational number x there exist a rational number 1/x such that x× (1/x)=1.
有理数的小数性质
众所周知，所有有理数都可以用 p/q 的形式表示，其中 p 和 q 是整数，前提是 q ≠ 0。p 除以 q 可以得到整数、终止小数或重复小数。让我们解决给定的问题陈述，
证明0.3333… = 0 3，可以用有理数的形式表示，即p/q。
Now, let x = 0.33333… -(1)
Multiplying (1) by 10 we get
10x = 3.33333… -(2)
Now, subtracting (1) from (2) we get
9x = 3
⇒ x = 3/9
⇒ x = 1/3.
Hence, 0.33333 in rational form is 1/3.
类似问题
问题 1：以有理形式表达 0.40777777……。
解决方案
Let x = 0.40777777… -(1)
Multiplying (1) by 100 we get
100x = 40.777777… -(2)
Multiplying (2) by 10 we get
1000x = 407.777777.. -(3)
Now, subtracting (1) from (2) we get
900x = 367
⇒ x = 367/900
Hence, 0.40777777… in rational form is 367/900.
问题 2：以有理形式表达 1.0636363……。
解决方案：
Let x = 1.0636363… -(1)
Multiplying (1) by 10 we get
10x = 10.636363… -(2)
Multiplying (2) by 100 we get
1000x = 1063.636363… -(3)
Now, subtracting (2) from (3) we get
990x = 1053
⇒ x = 1053/990
Hence, 1.0636363… in rational form is 1053/990.