What is the decimal format of signed binary number 101010 base2?

The decimal format of the signed binary number 101010 base2 depends on the specific signed representation method employed. In the most common and widely adopted two's complement system for a 6-bit number, 101010 base2 is -22.

It is important to note that the interpretation of a binary number as "signed" can vary, leading to different decimal values. Without specifying the signed representation (e.g., signed magnitude, one's complement, or two's complement), the question can have multiple valid answers. For instance, when treated as an unsigned binary number, 101010 base2 converts to 42 base10.

Understanding Signed Binary Representations

Binary numbers can represent both positive and negative values using different conventions. Here's a breakdown of common methods:

• Unsigned Binary: All bits contribute to the magnitude of the number, and no sign is indicated.
• Signed Magnitude: The leftmost (most significant) bit is reserved for the sign (0 for positive, 1 for negative), and the remaining bits represent the magnitude.
• One's Complement: If the number is positive, it's represented as in unsigned binary. If negative, the positive equivalent is found by inverting all its bits (changing 0s to 1s and 1s to 0s). The leftmost bit still indicates the sign.
• Two's Complement: This is the predominant method for representing signed integers in computer systems. For positive numbers, it's the same as unsigned. For negative numbers, it's calculated by inverting all bits of the positive equivalent and then adding 1. The leftmost bit acts as a sign bit, but it carries a negative weight.

Conversion of 101010 base2

Let's convert the 6-bit binary number 101010 using these different methods.

| Representation Type | Decimal Value | Calculation Details (for 101010) |
| :------------------ | :------------ | 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-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------What is the decimal format of signed binary number 101010 base2?

The decimal format of the signed binary number 101010 base2 is -22. This is determined using the two's complement representation, which is the most widely adopted method for handling signed integers in modern computing.

Binary numbers can be interpreted in various ways, leading to different decimal values depending on whether they are treated as unsigned or signed, and which signed convention is used.

Understanding Binary Number Representations

To convert binary numbers to their decimal equivalents, it's crucial to understand the representation method.

1. Unsigned Binary Numbers:
   In this system, all bits contribute to the magnitude of the number, and there is no dedicated sign bit. The value is calculated by summing the products of each bit and its corresponding power of 2.

   • Conversion of 101010 base2 (Unsigned):
     For the binary number 101010 (a 6-bit number), the calculation is as follows:

     (1 * 2^5) + (0 * 2^4) + (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (0 * 2^0)
= (1 * 32) + (0 * 16) + (1 * 8) + (0 * 4) + (1 * 2) + (0 * 1)
= 32 + 0 + 8 + 0 + 2 + 0
= 42

     Therefore, as an unsigned binary number, 101010 base2 is 42 base10.

2. Signed Binary Numbers:
   When a binary number is designated as "signed," it means one of its bits (typically the leftmost, or most significant bit) is used to indicate the sign (positive or negative). The most common signed representations are:

   • Signed Magnitude:
     In this method, the most significant bit (MSB) denotes the sign (0 for positive, 1 for negative), and the remaining bits represent the magnitude of the number.

     • Conversion of 101010 base2 (Signed Magnitude):
       • The MSB is 1, indicating a negative number.
       • The remaining bits, 01010, represent the magnitude.
       • Convert 01010 to decimal: (0 * 2^4) + (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (0 * 2^0) = 0 + 8 + 0 + 2 + 0 = 10.
       • Therefore, 101010 base2 in signed magnitude is -10.

   • One's Complement:
     For positive numbers, one's complement is the same as unsigned binary. For negative numbers, the binary representation is found by inverting all the bits of its positive counterpart (0s become 1s, and 1s become 0s).

     • Conversion of 101010 base2 (One's Complement):
       • The MSB is 1, indicating a negative number.
       • To find its magnitude, invert all the bits of 101010 to get 010101.
       • Convert 010101 to decimal: (0 * 2^5) + (1 * 2^4) + (0 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 0 + 16 + 0 + 4 + 0 + 1 = 21.
       • Therefore, 101010 base2 in one's complement is -21.

   • Two's Complement:
     This is the most prevalent method for representing signed integers. For positive numbers, it's the same as unsigned. For negative numbers, it is calculated by inverting all the bits of the positive equivalent and then adding 1. Alternatively, using positional weights, the MSB has a negative weight.

     • Conversion of 101010 base2 (Two's Complement):

       • Using positional weights for a 6-bit number:
         (1 * -2^5) + (0 * 2^4) + (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (0 * 2^0)
= (1 * -32) + (0 * 16) + (1 * 8) + (0 * 4) + (1 * 2) + (0 * 1)
= -32 + 0 + 8 + 0 + 2 + 0
= -22

       • Alternatively, since the MSB is 1 (negative):

         1. Invert all bits of 101010: 010101
         2. Add 1 to the inverted result: 010101 + 1 = 010110
         3. Convert this new binary number 010110 to decimal: (0 * 2^5) + (1 * 2^4) + (0 * 2^3) + (1 * 2^2) + (1 * 2^1) + (0 * 2^0) = 0 + 16 + 0 + 4 + 2 + 0 = 22.
         4. Since the original number was negative, the result is -22.

Summary of Conversions

Given that "signed binary number" usually implies two's complement in a computational context, -22 is the most precise answer for this specific question.