Question

(Printing the Decimal Equivalent of a Binary Number) Input an integer containing only 0 s and \(1 s\) (i.e., a "binary" integer) and print its decimal equivalent. Use the modulus and division operators to pick off the "binary" number's digits one at a time from right to left. Much as in the decimal number system, where the rightmost digit has a positional value of \(1,\) the next digit left has a positional value of \(10,\) then \(100,\) then \(1000,\) and so on, in the binary number system the rightmost digit has a positional value of \(1,\) the next digit left has a positional value of \(2,\) then \(4,\) then \(8,\) and so on. Thus the decimal number 234 can be interpreted as \(2^{*} 100+3^{*} 10+4^{*} 1 .\) The decimal equivalent of binary 1101 is \(1^{*} 1+0^{*} 2+1^{*} 4+1^{*} 8\) or \(1+0+4+8\), or \(13 .[\) Note: To learn more about binary numbers, refer to Appendix D.]

Short answer

To convert binary to decimal, multiply each binary digit by its corresponding power of 2 and sum the results. For the binary number 1101, the decimal equivalent is 1*2^3 + 1*2^2 + 0*2^1 + 1*2^0 which is 13.

Step-by-step solution

Understanding the Binary System

Recognize that in binary, each digit (bit) represents an increasing power of 2 from right to left, starting with 2^0 (which is 1). Each successive bit to the left represents 2^1 (2), 2^2 (4), 2^3 (8), and so on.

Isolate Each Binary Digit

Use the modulus operator (%) to isolate the rightmost binary digit. Divide the binary number by 10 and take the remainder. This will give you the rightmost digit (either 0 or 1).

Convert Binary to Decimal

Multiply the isolated binary digit by the corresponding power of 2. For example, the rightmost digit is multiplied by 2^0, the next by 2^1, and so forth. Record this value as part of the total decimal value.

Move to the Next Binary Digit

Divide the binary number by 10 (using integer division to discard the remainder) to move one position to the left. Repeat the isolation and conversion process for this next digit.

Sum the Decimal Values

Continue repeating the isolation, conversion, and movement for each binary digit until there are no more digits. Sum all the decimal values obtained from the conversion of each binary digit.

Output the Decimal Equivalent

Once all binary digits have been processed and their decimal equivalents summed, output the total sum, which is the decimal equivalent of the binary number.

Key concepts

C++ Programming

C++ is a robust programming language that is widely used for system/software development and game programming. It supports both procedural and object-oriented programming, allowing developers to build complex applications efficiently. At the heart of C++ programming are its core concepts, such as control structures (if-statements, loops), data types, functions, and operators. In tackling problems like binary to decimal conversion, C++ provides a reliable framework for writing efficient algorithms.

Understanding how to use C++ effectively involves utilizing its language features to manipulate binary numbers. For instance, when converting binary to decimal, programmers can employ loops to iterate over each digit of the binary number and perform operations using C++ built-in operators, like modulus and division. Proper handling of these operations, combined with bitwise manipulation provided by C++ (not needed in this particular problem but useful in many binary operations), ensures an accurate conversion. By mastering such techniques in C++, you are well-equipped to solve a wide range of programming challenges.

Binary Number System

The binary number system is foundational to digital electronics and computing. It consists of only two digits, 0 and 1, known as bits. Unlike the decimal system, which is base 10 and uses ten digits (0-9), the binary system is base 2, where each bit represents an exponent of two. Starting from the right, the least significant bit represents 2 to the power of 0, and each bit to the left represents twice the value of the bit before it.

For example, the binary number 1010 translates to the decimal number 10 because it is made up of 2³ (8) plus 2¹ (2). Binary representation is central to computer systems as they operate on electrical signals which are naturally binary, with 'on' and 'off' states. Understanding how to work with binary numbers is crucial for tasks involving low-level data processing or where storage and performance efficiency are important.

Modulus Operator

The modulus operator, denoted by %, is an arithmetic operator used to find the remainder of a division of two integers in C++ and many other programming languages. It is a key tool for problems that require examination of individual digits within a number, such as when separating binary digits for conversion to decimal.

In binary to decimal conversion, the modulus operator is used to isolate the rightmost digit by computing the remainder when the binary number is divided by 10. For instance, if we have the binary number 1101, using 1101 % 10 would give us 1, which is the rightmost digit. The ability to single out each digit using the modulus operator allows for a systematic approach to break down and convert binary numbers.

Integer Division

Integer division is a fundamental concept in C++ when dealing with integers. When two integers are divided, the result is also an integer, with any remainder discarded. This is different from floating-point division, which can produce a fractional result. In the context of converting binary to decimal, integer division is used to shift the focus from one binary digit to the next by dividing the number by 10.

If you have a binary number like 1101 and you perform integer division by 10, the result is 110. This effectively 'shifts' all digits to the right, and the rightmost digit is 'dropped'. Repeating this process and using the modulus operator sequentially allows you to navigate through each binary digit to carry out the binary to decimal conversion—one bit at a time.