Convert -1.125 To IEEE 754: Step-by-Step Guide

by Dimemap Team 47 views

Hey guys! Ever wondered how computers store decimal numbers? It's all about a cool standard called IEEE 754. Today, we're going to dive deep and break down how to convert the decimal number -1.125 into its IEEE 754 representation. It might sound a bit technical, but trust me, we'll make it super easy to understand. So, buckle up and let's get started!

Understanding IEEE 754

Before we jump into the conversion, let's quickly chat about what IEEE 754 actually is. Think of it as a universal language for computers to represent floating-point numbers (that's numbers with decimal points). This standard ensures that different computers can understand and process these numbers in the same way, avoiding any weird discrepancies. The IEEE 754 standard defines how these numbers are stored using three main components: the sign, the exponent, and the mantissa (also called the significand).

  • Sign: This is the simplest part. It's just one bit (either 0 or 1) that tells us whether the number is positive or negative. 0 means positive, and 1 means negative. So, for -1.125, the sign bit will definitely be 1.
  • Exponent: This part represents the power of 2 that the mantissa is multiplied by. It's a bit trickier because it's stored in a biased form. This means a bias value is added to the actual exponent to ensure it's stored as a positive number. We'll get into the nitty-gritty of this later.
  • Mantissa: This is the fractional part of the number, also known as the significand. It represents the significant digits of the number. In IEEE 754, the mantissa is normalized, meaning it's written in the form 1.xxxx, where xxxx are the fractional bits. The leading 1 is usually implicit (not actually stored) to save space, but we'll bring it back during the conversion.

Now that we have a basic understanding of IEEE 754, let's get our hands dirty and convert -1.125. Ready? Let's do this!

Step 1: Convert the Decimal to Binary

The first thing we need to do is convert our decimal number, -1.125, into its binary equivalent. We'll handle the whole number part (-1) and the fractional part (.125) separately.

  • Converting the Whole Number: Converting 1 to binary is super easy: it's just 1. Since we're dealing with -1, we'll keep that negative sign in mind for later.
  • Converting the Fractional Part: This is where it gets a little more interesting. To convert .125 to binary, we'll use the multiplication method. We repeatedly multiply the fractional part by 2 and note the integer part (0 or 1) of the result. Let's see how it works:
      1. 125 * 2 = 0.25 (Integer part: 0)
      1. 25 * 2 = 0.5 (Integer part: 0)
      1. 5 * 2 = 1.0 (Integer part: 1)

We stop when the fractional part becomes 0. Now, we read the integer parts from top to bottom: 001. So, .125 in decimal is 0.001 in binary.

Combining the whole number and fractional parts, we get -1.001 in binary. Awesome! We're one step closer.

Step 2: Normalize the Binary Number

Next up, we need to normalize the binary number. Normalizing means expressing the number in the form 1.xxxx * 2^y, where 1.xxxx is the mantissa and y is the exponent. This form makes it easy to represent the number in IEEE 754.

Our current binary number is -1.001. To normalize it, we need to move the decimal point so that there's only one non-zero digit to the left of it. In this case, we move the decimal point zero places.

So, -1.001 can be written as -1.001 * 2^0. Notice that we moved the decimal point 0 places. This 0 will be important for calculating the biased exponent later.

Step 3: Determine the Sign, Exponent, and Mantissa

Now, let's break down our normalized binary number into its three components for IEEE 754 single-precision (32-bit) format:

  • Sign: Since our number is negative (-1.001), the sign bit is 1.
  • Exponent: This is where the bias comes in. For single-precision IEEE 754, the bias is 127. We add this bias to our exponent (which is 0 in this case) to get the biased exponent: 0 + 127 = 127. Now, we need to convert 127 to binary. 127 in binary is 01111111. So, our exponent field is 01111111.
  • Mantissa: The mantissa is the fractional part of our normalized binary number (1.001). We take the digits after the decimal point (001) and pad them with zeros to fill the 23 bits allocated for the mantissa in single-precision format. So, our mantissa becomes 00100000000000000000000.

Step 4: Combine the Components

We've got all the pieces! Now, let's put them together in the correct order. In IEEE 754 single-precision format, the order is:

  1. Sign bit (1 bit)
  2. Exponent (8 bits)
  3. Mantissa (23 bits)

So, we combine our sign bit (1), exponent (01111111), and mantissa (00100000000000000000000) to get:

1 01111111 00100000000000000000000

Step 5: Express in Hexadecimal (Optional)

Often, the binary representation is grouped into sets of 4 bits and expressed in hexadecimal for easier readability. Let's do that:

1011 1111 1001 0000 0000 0000 0000 0000

Converting each group of 4 bits to hexadecimal, we get:

BF 90 00 00

The Answer!

So, the IEEE 754 single-precision representation of -1.125 is:

10111111100100000000000000000000

Or, in hexadecimal:

BF900000

Looking at the options you provided, the correct answer is:

d. 10111111110010000000000000000000

Wrapping Up

And there you have it! We've successfully converted -1.125 into IEEE 754 format. It might seem like a lot of steps, but once you get the hang of it, it becomes second nature. Understanding how numbers are represented in computers is super important for anyone diving into computer science or programming. You guys did great following along!

If you have any more questions or want to try another conversion, just let me know. Keep exploring and keep learning! You've got this!