1957 AMC 12 第 19 题

Question

十进制计数系统以十为底，例如

123=1\cdot10^2+2\cdot10+3

。二进制以二为底，其中最前面的五个正整数为

1

、

10

、

11

、

100

、

101

。那么二进制数

10011

写成十进制为：

The base of the decimal number system is ten, meaning, for example, that

123=1\cdot10^2+2\cdot10+3.

In the binary system, which has base two, the first five positive integers are

1,

10,

11,

100,

101.

The numeral

10011

in the binary system would then be written in the decimal system as:

19

40

10011

11

7

将

2^4,2^3,\ldots,2^0

依次对应到五个二进制数位

Assign powers

2^4,2^3,\ldots,2^0

to the five binary places

只有第一、第四和第五个数位上的数字非零

Only the first, fourth, and fifth digits are nonzero

按二进制位值展开，得到

\begin{aligned} (10011)_2 &=1\cdot2^4+1\cdot2^1+1\cdot2^0\\ &=16+2+1=19 \end{aligned}\text{。}

因此，正确答案是 A。

Expanding by binary place value,

\begin{aligned} (10011)_2 &=1\cdot2^4+1\cdot2^1+1\cdot2^0\\ &=16+2+1=19. \end{aligned}

Therefore, the correct answer is A.

其他年份的第 19 题