Number Systems

TYPES OF NUMBER SYSTEMS

In the world of electronics math is very important. In order to know how to calculate voltage, current, power, resistance, and much more you must be able to use different types of math.

In these tutorials we will cover the basics that will help to introduce you to numbering systems that will help you in your future electronics projects.

Until now, you have probably used only one system, the decimal system. You may also be familiar with the Roman numeral system, even though you seldom use it.

THE DECIMAL NUMBER SYSTEM

In this section you will be studying modern systems. You should realize that these systems have certain things in common. These common terms will be defined using the decimal system as our base. Each term will be related to each system as that system is introduced.

Each of the number systems you will study is built around the following components: the UNIT, NUMBER, and BASE (RADIX).

Unit and Number

The terms unit and number when used with the decimal system are almost self-explanatory. By definition the unit is a single object; that is, an apple, a dollar, a day. A number is a symbol representing a unit or a quantity. The figures 0, 1, 2, and 3 through 9 are the symbols used in the decimal system. Thesesymbols are called Arabic numerals or figures. Other symbols may be used for different systems.

For example, the symbols used with the Roman numeral system are letters - V is the symbol for 5, X for 10, M for 1,000, and so forth. We will use Arabic numerals and letters in the number system discussions in this section.

Base (Radix)

The base, or radix, of a number system tells you the number of symbols used in that system. The base of any system is always expressed in decimal numbers. The base, or radix, of the decimal system is 10. This means there are 10 symbols - 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 - used in the system.

A system using three symbols - 0, 1, and 2 - would be base 3; four symbols would be base 4; and so forth. Remember to count the zero or the symbol used for zero when determining the number of symbols used in a system.

The base of a number system is indicated by a subscript (decimal number) following the value of the number. The following are examples of numerical values in different bases with the subscript to indicate the base:

7592 base 10

214 base 5

123 base 4

656 base 7

You should notice the highest value symbol used in a number system is always one less than the base of the system. In base 10 the largest value symbol possible is 9; in base 5 it is 4; in base 3 it is 2.

Positional Notation and Zero

You must observe two principles when counting or writing quantities or numerical values. They are the POSITIONAL NOTATION and the ZERO principles.

Positional notation is a system where the value of a number is defined not only by the symbol but by the symbol’s position. Let’s examine the decimal (base 10) value of 427.5. You may know from experience that this value is four hundred twenty-seven and one-half. Now examine the position of each number:

If 427.5 is the quantity you wish to express, then each number must be in the position shown. If you exchange the positions of the 2 and the 7, then you change the value.

You can see that the power of the base is multiplied by the number in that position to determine the value for that position.The following graph illustrates the progression of powers of 10:

All numbers to the left of the decimal point are whole numbers, and all numbers to the right of the decimal point are fractional numbers. A whole number is a symbol that represents one, or more, complete objects, such as one apple or $5. A fractional number is a symbol that represents a portion of an object,such as half of an apple (.5 apples) or a quarter of a dollar ($0.25). A mixed number represents one, or more, complete objects, and some portion of an object, such as one and a half apples (1.5 apples).

When you use any base other than the decimal system, the division between whole numbers and fractional numbers is referred to as the RADIX POINT. The decimal point is actually the radix point of the decimal system, but the term radix point is normally not used with the base 10 number system.

Just as important as positional notation is the use of the zero. The placement of the zero in a number can have quite an effect on the value being represented. Sometimes a position in a number does not have a value between 1 and 9. Consider how this would affect your next paycheck.

If you were expecting a check for $605.47, you wouldn’t want it to be $65.47. Leaving out the zero in this case means a difference of $540.00. In the number 605.47, the zero indicates that there are no tens. If you place this value on a bar graph, you will see that there are no multiples of 101.

Most Significant Digit and Least Significant Digit (MSD and LSD)

Other important factors of number systems that you should recognize are the MOST SIGNIFICANT DIGIT (MSD) and the LEAST SIGNIFICANT DIGIT (LSD).

The MSD in a number is the digit that has the greatest effect on that number.

The LSD in a number is the digit that has the least effect on that number.

Look at the following examples:

You can easily see that a change in the MSD will increase or decrease the value of the number the greatest amount. Changes in the LSD will have the smallest effect on the value.

The nonzero digit of a number that is the farthest LEFT is the MSD, and the nonzero digit farthest RIGHT is the LSD, as in the following example:

In a whole number the LSD will always be the digit immediately to the left of the radix point.

Carry and Borrow Principles

Binary Number Systems

MSD and LSD

Binary Addition

Binary Addition Continued

Binary Subtraction

Complimentary Subtraction

Complimentary Subtraction Continued

Octal

Octal Number Addition

Octal Number Subtraction

Hexadecimal

Hex Number Addition

Hex Number Subtraction

Conversion of Bases

Base Conversions Continued

Decimal to Octal

Decimal to Hex

Binary Conversion

Binary to Hex, Hex to Binary, and Octal to Hex

Octal Conversion

Hex to Decimal

Binary Coded Decimal

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