Starting scenario: You are setting up a network and have to get licenses for various software. You just received the figures of the budget from the accounting dept. = $9,979 total, and heard that you need 219 licenses. The price of each license is at least $40. You have to instantly calculate if you have enough and how much money you have available to spend per license. Your computer is currently down and you have no calculator available to you, what do you need to do?

What have we done? Rounding to Place Values:

1. Look at the place values: 9 9 7 9
2. 1000s 100s 10s 1^s: place
3. Look at the place values: 2 1 9
4. 100s 10s 1^s: place

A Review of the Decimal System – Positional Weight

Anthropologists think that counting on 10 fingers resulted in the decimal system. The base is 10, ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
You count using the decimal number system, you get the next number by adding. How do we count past 9: 9
1 +
10

The decimal system is a positional system: The place value of a digit is determined by its position. I.e.: The digits 1 by itself are worth 1, but the digit 1 followed by a 0 is worth 10:

We have used the decimal system for so long that when we see a number like "123", we don’t think about the value 123; rather you generate mental image how many items this value represents. However, in reality, the number 123 represents:

So: Each digit appearing to the left of the decimal point represents a value between zero and nine times an increasing power of 10:

Another example: 6019 =

When we use the decimal system, we can also say: We are using base 10 (written as #₁₀). Besides base 10, we can also use other bases, i.e. base 2 (written as #₂), base 3 (#₃), base 4 … base16. We will discuss base 2 (= binary system) and base 16 (hexadecimal) (written as #₁₆)in more detail. Choosing a base is a matter of choice. Before going into the base the computer uses lets just look at some base options:

How does the computer manipulate numbers and letters? It makes use of electrical circuits, so we need some type of conversion. People use symbols to write as 2, G, 8, etc.

A computer is an electronic device, so it cannot write down the data it works with. A computer needs electrical signals to represent data: How it represents data depends on whether it is an analog device or a digital device.

Analog device: Operates continuously varying data, i.e. a dimmer switch, cassette tape, records
Digital device: Uses separate digits: on and off are the only possibilities.

With digital electronic circuits, information is first converted into a group of pulses. This code consists of a series of voltages. The early digital systems constructed in the 1950s made use of a decimal code that used ten levels of voltages, with each of these voltages corresponding to one of the ten digits in the decimal number system (0=0 Volt, 1 = 1 V, 2=2 V, 3 = 3 V, up to 9 = 9 V). The circuits that had to manage these decimal codes, however, were very complex and had to sense the difference between all ten voltage levels. This complexity led to inaccuracy since some circuits would periodically confuse one voltage level for a different voltage level. The solution to this problem of circuit complexity and inaccuracy was solved by adopting a two-state system instead of a ten-state system.

All information or data is encoded in HIGH and LOW voltages, in which the HIGH voltages are called "1s" (ones) and the LOW voltages are called "0s" (zeros). The computer represents values using two voltage levels (0v and +5V) and employs the binary numbering system. The binary system = base 2.

Each number or letter is represented by a series of electrical signals. Each 1 or 0 that represents data is referred to as a bit. In most computers, a string (series) of 8 bits is called a byte and represents a character: number, letter, or symbol.

The binary numbering system works just like the decimal numbering system, with two exceptions: in the binary number system, there are only two digits: 0 and 1 (rather than 0-9) and binary uses powers of two instead of powers of ten. The numeral 2 cannot be used in the binary system, so when you run out of options after writing 1, instead of writing 2 you would carry over the 1 and write "1" in front of the 0: =10.

To understand how the binary system works, think about the decimal system first.

Counting with the binary number system is similar to counting with the decimal system, but you can only use two digits, 1 and 0. You begin counting with 1. To get to the next number, you must add one. However, in the binary system 1 + 1 cannot equal 2 because there is no 2 digit. So, just as in the decimal system, when you run out of digits, you carry a 1 to the next column: 1

1 +
10

The second difference with the decimal system is that binary uses the powers of two instead of the powers of ten. For each "1" in the binary string, add in 2ⁿ where "n" is the zero-based position of the binary digit. We will use the example 10001100₂

We need to first set up a table of binary place values, starting with the number 1, which we put in the right most column. Since 8 bits make up one byte, we use 8-place values: As mentioned above, the place values are the powers of 2.

Using pencil and paper, create the following table:

Let’s use our table by converting the binary number 10001100 in a decimal number:

To convert decimal to binary is slightly more difficult. You must find those powers of two which, when added together, produce the decimal result. The easiest method is to work from the larger power of two down to 2⁰. Consider the decimal value 73:

Write the number 73
Find the largest binary place value that you can subtract from the decimal number, in this case 64

Perform the subtraction: : 73
64 -
9 remainder.

Remainder is 8: What is the largest place value you can subtract from the remainder? = 8:

Place a 1 in column 8: (Another remainder left: 9-8 = 1)

The remainder of 1 can only be placed in the first column: Now there are no remainders left.

In the rest of the columns: add 0 in each column.
In the bottom row multiply 64 x 1 = 64; 8 x 1 = 8; 1 x 1 = 1
Add: 64 + 8 + 1 = 73 to check your answer.

Summarized steps:

Find the largest place value that will fit into a decimal number
Subtract that place value from the decimal number
Put a 1 in the column for that place value
Repeat the process until there is no remainder.

Try out different numbers: i.e. decimal value 1359 to practice.

Ø 2¹⁰ =1024, 2¹¹=2048. So 1024 is the largest power of two less than 1359. Subtract 1024 from 1359 and begin the binary value on the left with a "1" digit. Binary = "1", decimal result is 1359 – 1024 = 335

Ø The next lower power of two (2⁹ = 512) is greater than the result from above, so add a "0" to the end of the binary string. Binary = "10", decimal result is still 335

Ø The next lower power of two is 256 (2⁸). Subtract this from 335 and add a "1" digit to the end of the binary number. Binary ="101". Decimal result is 79.

Ø 128 (2⁷) is greater than 79, so tack a "0" to the end of the binary string. Binary = "1010", decimal result remains 79.

Ø The next lower power of two (2⁶ = 64) is less than 79, so subtract 64 and append a "1" to the end of the binary string. Binary = "10101", Decimal result is 15.

Ø 15 is less than the next power of two (2⁵ = 32) so simply add a "0" to the end of the binary string. Binary = "1010100", Decimal result is still 15

Ø 16 (2⁴) is greater than the remainder so far, so append a "0" to the end of the binary string. Binary = "1010100" , decimal result is 15

Ø 2³ (eight) is less than 15, so stick another "1" digit on the end of the binary string. Binary = "10101001", decimal result is 7.

Ø 2² is less than seven, so subtract four from seven and append another one to the binary string. Binary = "101010011" , decimal result is 3.

Ø 2¹ is less than 3, so append a one to the end of the binary string and subtract 2 from the decimal value. Binary = "1010100111", Decimal result is now 1.

Ø Finally, the decimal result is one, which is 2⁰, so add a final "1" to the end of the binary string. The final binary result is "10101001111"

In the United States, most people separate every three digits with a comma to make larger numbers easier to read.. For example, 1,023,435,208 is much easier to read and comprehend than 1023435208. A similar convention is adopted for binary numbers. We can separate each group of four binary bits with a space. For example, the binary value 1010111110110010 will be written as 1010 1111 1011 0010

Computers generally work with specific numbers of bits: Common collections on the 80x86 microprocessor are:

1. Single bits = smallest unit of data on a binary computer, capable of representing only two values, zero and one

2. Nibbles = groups of four bits. With a nibble, we can represent up to 16 distinct values.

3. Bytes = group of eight bits and it is the smallest addressable data item. Bits in a byte are normally numbered from zero to seven. One byte represents a character, i.e. "A"

4. Words = groups of 16 bits. The bits in a word start from zero on up to fifteen. A word exactly contains two bytes.

A big problem with the binary system is verbosity. To represent the value 1359₁₀ (see example above) requires 11 binary digits. The decimal version requires only 4 digits, and thus represents numbers much more compactly than does the binary numbering system. The hexadecimal (hex) system is used as a sort of shorthand. Furthermore, although we can convert between decimal and binary, this is not an easy task. The hexadecimal (base 16) numbering system solves this problem. Hexadecimal numbers offer two features: they are very compact, and it is simple to convert them to binary and vice versa.

Each hexadecimal digit can represent one of sixteen values between 0 and 15_10.: 0-9 and beyond 9, hex makes use of the letters A₁₆(=10₁₀), B₁₆(=11₁₀), C₁₆(=12₁₀), D₁₆(=13₁₀), E₁₆=(14₁₀) and F₁₆(=15₁₀). Since the base of a hex number is 16, each hex digit to the left of the hex point represents some value times a successive power of 16:

For example, the number 1234₁₆ =

Ø The largest power of sixteen (256) is subtracted once from 425 and therefore a 1 is placed in the 256 column, leaving a remainder of 169.

Ø the next largest power of 16 is 16 which can be subtracted ten times from 169. Therefore the hexadecimal equivalent of ten, which is A, is placed in the sixteen’s column, leaving a remainder of 9.

Ø Since nine 1s can be subtracted from the remainder of 9, the units column is advanced nine times, giving us our final hexadecimal result of 1A9

Convert decimal 4525 to its hexadecimal equivalent:

=4525 – 4096 = 429, 429 – 256 = 173,

173 – 16 – 16 – 16 – 16 –16 –16 – 16 – 16 –1 6 – 16 = 13,

13 – 1 –1 –1 –1 –1 –1 –1 – 1 – 1 – 1 – 1 – 1 – 1 = 0

As mentioned earlier, hexadecimal is used as a short-hand for representing large groups of binary digits. To illustrate this we can represent a 16-bit binary word in four hexadecimal digits. A four bit binary word can have any value from 0₁₀ (0000₂)to 15₁₀ (1111₂). To convert from hexadecimal to binary, we simply do the opposite. Since each hexadecimal digit represents four binary digits, a 4 digit hexadecimal number will convert to a 16 bit binary word.

Use the table below for your conversion. The table provides all information to convert any hexadecimal number into a binary number and vice versa.

Copyright© 2001. Sigrid Zuñiga