论文部分内容阅读
我们用整数的二进制表示及穷举法证明了整数平方的一个特性。这个特性使我们可以看出一些古老命题的直观性。一、特性的推导设m为任意正整数,并表示成二进制数: m=m_nm_(n-1)……m_(?)……m_3m_2m_1m_0 (1) 式中任一项m_1为1或0。无论正整数m有多大,它的最低四位二进制数m_3m_2m_1m_0却只有16种可能的情况,即0000、0001、0010、0011、0100、0101、0110、0111、1000、1001、1010、1011、1100、11011110和1111。下面我们在证明这个特性时,虽然只讨论了最低四位的16种情况,但实际适合于所有的正整数。这就是穷举法的思想。现求m~2,以最低四位m_3m_2m_1m_0=1101为例,列竖式:
We use a binary representation of the integer and an exhaustive method to prove a property of the square of the integer. This feature allows us to see the intuitiveness of some ancient propositions. First, the characteristics of the derivation Let m be any positive integer, and expressed as a binary number: m = m_nm_(n-1) ... m_(?) ... m_3m_2m_1m_0 (1) where m_1 is either 1 or 0. No matter how large the positive integer m is, its lowest four-bit binary number m_3m_2m_1m_0 has only 16 possible cases, namely 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 11011110 and 1111. In the following, when we prove this feature, although only the lowest four fourteen cases are discussed, it is actually suitable for all positive integers. This is the idea of an exhaustive method. Now find m~2, taking the lowest four m_3m_2m_1m_0=1101 as an example, column vertical: