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We are now going to consider while two numbers are given, which one is the greater one. The idea is indeed from a bigger group and a smaller group of apples. Suppose that, a basket contains 5 apples, while another one contains 3 ones, then we will conclude that the former basket has more apples than the latter does. We do as follow:
(a) We choose an apple from each basket, then these baskets has 4 and 2 apples left, respectively.
(b) We again choose an apple from each basket, then 3 and 1 apple are still left.
(c) Repeatedly choose an apple, then the first has 2 apples left while none in the second remains.
It is similar for performing on pure numbers, say, 5 and 2. We might find from above experiment that in order to pick out the greater one, we have to count from 1. Note taht while the numbers are different, the counting will firstly reach the smaller one, and after a while, the greater one.
Another way to determine this is that, we can count from one of the two number and to reach the other number. Note that while the two numbers are different, only and exactly one of the counting would success. In the case of 2 and 5, if we count from 2, then we read, 3, 4, and 5, which is our goal. However, if we start from 5, then we read 6, 7, 8, 9, 10, ... , but it becomes farer and farer from 2. It doesn't have the tendency to reach 2.
The notation is, 5>2, read "five is greater than two", "five is larger than two", or "five exceeds two", and while 5>2, we also say that 2<5, which presents the relative relations, and read "two is smaller than five". While the given numbers are the same, for example, 13 and 13, 7 and 7, 91 and 91, or 8281 and 8281, etc, we write 13=13, 7=7, 91=91, 8281=8281, which are read "13 is 13", or "13 equals to 13", and so on, and say that they are the same, or they are "equal".
The next thing to do is: If the pair of numbers are larger, or very large, for example, 23660 and 23661, or 31415926535897 and 31416777216666, how to determine the "exceeder".
According to experience of counting, that is, imagine that we still determine by counting, during a very long time, and observe properties in the process, then the following gives us a quicker way:
(a) Write down two numbers with their digits in correspondence from small digits to large digits.
(b) Compare these digit from the largest digits to the smallest digits. If the digits of the two numbers are different (e.g. 367 has 3 digit, while 11122 has 5 digits), then that which contains the more digits exceeds.
(c) At the first time that the larger digit appears, the number containing this digit will be the larger number.
(d) If all corresponding digits are the same, then they are the same number, that is, they are equal.
For some example. (i) Comparing 1357 and 246, since 1357 has one more digit than 246 does, we see that 1357>246. (ii) Comparing 2366000 and 2366606, since they have the same number of digit, hence we must compare them digit by digit. Since 2=2, 3=3, 6=6, 6=6, 0<6, we see that 2366000<2366606.