In this article, we deal with the first two of the following key skills that build up to understanding the floating point system:

- Define and identify the most and least significant bits in a word. Count how many combinations are possible in a word given its bit length.
- Encode a positive integer in binary. Decode an unsigned integer to decimal.
- Encode an integer in two's complement notation. Decode a signed integer to decimal.
- Use sign extension to expand the number of bits in a word. Explain the mathematics.
- Convert a decimal fraction to binary. Explain why this does not require any particular encoding scheme, such as a floating point standard.

# The MSB and the LSB. Unsigned Representation Practice

Definition: The *most significant bit* (MSB) is, in unsigned representation, the bit position associated with the greatest absolute value. Also, in two's complement representation, the bit position that changes the sign of a number.

Definition: The *least significant bit* (LSB) is the bit position that has the least value associated, that is, the value of 1.

Let's say we have a 32-bit machine that deals with 32-bit words. Each word is a string of four octets (bytes), so there are 32 bits in it. Then, generally, bit position 0 is the LSB, and bit position 31 is the MSB.

Bit position *n* has the value of 2^*n*, so, for instance, the number `1010`

(unsigned) will be equal to 2^3 + 2^1 = 8 + 1 = 9. In this instance, the rightmost bit is the LSB and the leftmost bit the MSB.

##
Number of Combinations In a Word That Has *n* Bits

In unsigned notation like this, there are a total of 2^*n* combinations of the digits where *n* is the number of bits in a word. If there are 32 bits in a word, then 2^32 numbers can be represented. If we made a sequence of these numbers, it will start from 0, so the last number is exactly 1 less than the number of combinations, that is, 2^*n* - 1. So, in unsigned notation, numbers [0, 2^*n* - 1] are represented.

## Practice

**QUESTION 1** Represent the decimal number 254 in unsigned binary notation. What are the most and least significant bits?

**ANSWER**: The decimal number 254 is closest to a *smaller (or equal)* power of two, 128, or 2^7 . Divide 254 with that power. (Since the quotient is always 1, we replace the division by subtraction.) The remainder is 126. Do the same: The closest smaller (or equal) power of two is 64, or 2^6 . The difference is 62. With 2^5 = 32, the difference is 30. With 2^4 = 16, the difference is 14. With 2^3 = 8, the difference is 6. With 2^2 = 4, the difference is 2. With the equal power of two, 2^1 = 2, the difference is 0. The exponents of two are where the 1's are. They are bit positions 1, 2, 3, 4, 5, 6 and 7.

Let's use an octet (an eight-bit string). The leftmost is the MSB, so it is bit position 7. The rightmost is the LSB, and its bit position is 0. Now, arrange it: `1111 1110`

. This is our representation.

The MSB is 1, and the LSB is 0.

Let's use a 32-bit word. Again, the leftmost is the MSB, and its bit position is 31. The rightmost is the LSB, and its position is 0. Now, arrange it: `0000 0000 0000 0000 0000 0000 1111 1110`

. The MSB is changed: It is now 0. The LSB is still 0.

**QUESTION 2** Represent 142 in unsigned representation.

**Answer**: Remember to choose the closest smaller (or equal) power of two, until the remainder (or difference) is 0. Then, record the bit positions---that's our method.

142 - 128 = 14

14 - 8 = 6

6 - 4 = 2

2 - 2 = 0

The powers are: 1, 2, 3, 7. Use an octet. The leftmost is the MSB, bit position 7, and the rightmost is the LSB, bit position 0. Arrange: `1000 1110`

.

**QUESTION 3** Decode `1001 1010`

in unsigned rep. to decimal. The leftmost is the MSB.

**Answer**: It is a sum of powers of two. The 7th, 4th, 3rd and 1st powers are represented. So, add them up: 2^7 + 2^4 + 2^3 + 2^1 = 154.

# Challenge

## Challenge Problem 1

Convert the positive integer 1,257,281 to binary:

- Is a byte sufficient to store the representation? A 2-byte word? A 32-bit word? A 64-bit word?

## Challenge Problem 2

Consider the representation `1111 1111`

in unsigned notation in a byte:

- What is its decimal value?
- If the most significant bit was assigned the value of -2^7 , instead of 2^7 ,then what is its new decimal value?

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