Number Systems

Base 2: The Binary Number System

Base 8: The Octal Number System

Base 16: The Hexadecimal Number System

Learning Objectives

•At the end of the lesson the student should be able to:

– Identify the different number base system

– Convert base ten numbers to base two, eight or sixteen

– Convert base two, eight or sixteen numbers to base ten

– Perform basic operations on various base numbers

Number Base

•What is a number base?

Anumber base is a specific collection of symbols on which a number system can be built.

•The number base familiar to us is base 10, upon which the decimal number system is built.There are ten symbols - 0 to 9 - used in thesystem.

Place Value

•What is the concept of place value?

Place value means that the value of a digit in a number depends not only on its own natural value butalso on its location in the number. It is used interchangeably with the term positional notation.

•Place value tells us that the two 4s in the number 3474 have different values, that is, 400 and 4, respectively.

A Review of the Decimal Number System

•The word “decimal” comes from the Latin word

decem, meaning ten.

•Thus, the number base of the decimal number system is base 10.

•Since it is in base 10, ten symbols are used in the decimal number system. {0,1,2,3,4,5,6,7,8,9}

•This means that only the digits in the above set can be used for each position in every place value in a given decimal number.

A Review of the Decimal Number System

270

•Note that the highlighted place value can be filled by the digits in the set {0,1,2,3,4,5,6,7,8,9}.

•Thus, it can be increased by 1 until it reaches -

279

•At this point, the symbols that can be used to fill the highlighted position has been exhausted. Increasing it further causes a shift in place value, and resets the initial place value to zero.Thus -

280

A Review of the Decimal Number System

•Case Study: 3474

•Using place values, the number 3474 is understood to mean,

3000 + 400 + 70 + 4 = 3474

This can also be expressed as –

(3x1000) + (4x100) + (7x10) + 4 = 3474

Note that each digit is multiplied by powers of

10, so that the above is equal to –

(3x103) + (4x102) + (7x101) + (4x100) = 3474

Note that the rightmost exponent starts from zero and increases by 1 as the place value increases.

Hence, the decimal number system is said to be in base 10.

Base 2: The Binary Number System

•The word “binary” comes from the Latin word

bis, meaning double.

•Thus, the number base of the binary number system is base 2.

•Since it is in base 2, two symbols are used in the binary number system.

•This means that only the digits in the above set can be used foreach position ineveryplace value in a given binary number.

Base 2: The Binary Number System

1100

•Note that the highlighted place value can be

filled by the digits in the set {0,1}.

•Thus, it can be increased by 1 until it reaches -

1101

•At this point, the symbols that can be used to fill

the highlighted position has been exhausted. Increasing it further causes a shift in place value,

and resets the initial place value to zero.Thus -

1110

Base 2: The Binary Number System

•To avoid confusion, one should write a binary number with base 2 as its subscript whenever necessary.

•Thus, the binary number 10110 should be written as - 101102

•It should be read as “one-zero-one-one-zero base two” and NOT “ten-thousand one-hundred ten” since each phrase denotes an entirely different number. Base 2: The Binary Number System

•Case Study: 101102

•We know that the decimal number 3474 can be expressed as powers of 10 – (3x103) + (4x102) + (7x101) + (4x100) = 347410

•In the same manner, the binary number 101102

can...