Storing Positive Integers

Section 5.2 Storing Positive Integers

Computers speak in binary, which is how integers are stored in memory.

Let’s start by reviewing binary numbers:

Figure 5.1.

Investigate 5.3.

What is the decimal value of 01000001?

Computers "speak" in binary (base-2) only, which includes only 1s and 0s.

Bit: one binary digit (0 or 1)
Byte: number of bits used to represent a character, almost always 8 bits
Word: natural unit of memory for a given computer design; unit of data that can pass across the bus to or from main memory at one time

For example, in the 8-digit binary number 01100110, the left-most bit is the Most Significant Bit, and the right-most bit is the Least Significant Bit.

In decimal (base-10), the number 135 can be represented as:

\begin{equation*} 135 = 1\cdot 10^2 \; + \; 3\cdot 10^1 \; + \; 5\cdot 10^0 \end{equation*}

where \(10^2\) is the "hundreds" place, \(10^1\) is the "tens" place, and \(10^0\) is the "ones" place.

Instead of base-10, what if we have base-b?

\begin{equation*} (number)_b = a_nb^n + a_{n-1}b^{n-1} + ... + a_0b^0 \end{equation*}

\(b\text{:}\) base (binary: 2, decimal: 10, ...)
\(a_k\text{:}\) digits (binary: 0-1, decimal: 0-9, in general 0-(b-1))
\(n\text{:}\) position of "most significant digit" (MSD)

Special case: base \(b = 2\) (binary) -- note: in binary there are only two different digits: 0 and 1

Note: Digits have different weights (place values)

Check Your Understanding Check Your Understanding

1.

Find the decimal value of each of the following binary numbers:

00110001
11111110
10011001

2.

How many bits are there in one byte?

Next, we’ll learn how to convert between different binary and decimal representations:

Investigate 5.4.

What is the remainder when dividing 50 by 3?

How do we convert from one base to another?

In particular, how do we find the binary representation (base-2) of a decimal number (base-10)?

Example: What is 135 decimal in binary or base-2?

Hint: The easiest way to do this is to divide the number by 2 repeatedly and keep track of the remainder.

135    /2       = 67 R1     * 2^0
 67    /2       = 33 R1     * 2^1
 33    /2       = 16 R1     * 2^2
 16    /2       =  8 R0     * 2^3
  8    /2       =  4 R0     * 2^4
  4    /2       =  2 R0     * 2^5
  2    /2       =  1 R0     * 2^6
  1    /2       =  0 R1     * 2^7

 135              -> 1000 0111

Does it work? 128 + 4 + 2 + 1 = 135

and... does this work for any base?

Example: what does the algorithm say 135 is in... base-10?

135    /10      = 13 R5     * 10^0    LSD
 13    /10      =  1 R3     * 10^1
  1    /10      =  0 R1     * 10^2    MSD

 135              -> 135

Not very surprising...

There are other ways to find the binary representation of a number.

Recall the binary place values:

So...

135 = 1 * 128 + 7
  7 = 0 * 64 + 0 * 32 + 0 * 16 + 0 * 8 + 1 * 4 + 3
  3 = 1 * 2 + 1
  1 = 1 * 1
135 = 1 * 128 + 0 * 64 + 0 * 32 + 0 * 16 + 0 * 8 + 1 * 4 + 1 * 2 + 1 * 1

Thus 135 is 10000111 in binary.

Check Your Understanding Check Your Understanding

1.

What is the remainder when dividing 36 by 15?

2.

Find the 8-bit binary representation of the decimal number 222:

Prev Top Next