IS 310 Number Systems 3

Base

The number of different digits, including zero, in a number system.

Base 10

Decimal number system with 10 digits: 0 through 9.

Binary

Base 2 number system using only the digits 0 and 1.

Bit

Short for binary digit; a single 0 or 1.

Octal

Base 8 number system using digits 0 through 7.

Hexadecimal

Base 16 number system using digits 0 through 9 and A through F.

Numeric data

Numbers that can be used in arithmetic operations.

Why do digital devices use binary?

Digital devices represent numeric data using the binary number system, or base 2.

How is decimal 2 represented in binary?

10.

What does lowercase b represent?

A bit.

What does uppercase B represent?

A byte.

Byte

A group of 8 bits.

Bits per second (bps)

Typically used to describe network access speed.

Bytes

Typically used to describe storage capacity.

Kilobit

1,024 or 2^10 bits.

Kilobyte

1,024 or 2^10 bytes.

Megabit

1,048,576 or 2^20 bits.

Megabyte

1,048,576 or 2^20 bytes.

Gigabit

2^30 bits.

Gigabyte

2^30 bytes.

Terabyte

2^40 bytes.

Petabyte

2^50 bytes.

Exabyte

2^60 bytes.

Hexadecimal A

Decimal 10.

Hexadecimal B

Decimal 11.

Hexadecimal C

Decimal 12.

Hexadecimal D

Decimal 13.

Hexadecimal E

Decimal 14.

Hexadecimal F

Decimal 15.

Why is hexadecimal used in computers?

It is easy to convert to binary, works well with bytes, and uses fewer digits than binary.

Positional notation

The value of a digit depends on its position and the base of the number system.

Decimal positional notation

Each position represents a power of 10.

Binary positional notation

Each position represents a power of 2.

Hexadecimal positional notation

Each position represents a power of 16.

What power does the rightmost digit use?

The base raised to the power of 0.

How do you convert binary to decimal?

Multiply each bit by its power of 2 and add the results.

How do you convert hexadecimal to decimal?

Multiply each digit by its power of 16 and add the results.

Can the same number be represented in multiple bases?

Yes, every number can be represented in multiple bases.

What methods can convert decimal to binary?

Subtraction method or division method.

Subtraction method

Subtract powers of 2 from the decimal number to determine the binary digits.

When is the subtraction method easiest?

With smaller numbers.

Division method for decimal to binary

Repeatedly divide by 2 and record the remainders.

How are division-method remainders read?

Read the remainders from bottom to top.

Decimal to binary using DIV/MOD

Use repeated division by 2; DIV gives the quotient and MOD gives the remainder.

Does binary to decimal use DIV and MOD?

No. It uses positional notation and powers of 2.

Binary to decimal positional notation

Multiply each binary digit by its corresponding power of 2 and add.

Relationship between base size and symbols

A larger base requires more different symbols.

Relationship between base size and digits

A larger base requires fewer digits to represent the same value.

How many possible values can 1 bit represent?

2 values: 0 and 1.

How many possible values can 4 bits represent?

16 values.

How many possible values can 8 bits represent?

256 values.

How many possible values can 10 bits represent?

1,024 values.

How many possible values can 16 bits represent?

65,536 values.

How many possible values can 20 bits represent?

1,048,576 values.

How many possible values can 32 bits represent?

4,294,967,296 values.

Counting in binary after 0

1.

Counting in binary after 1

10.

Counting in binary after 10

11.

Counting in binary after 11

100.

Binary 100 in decimal

4.

Binary 101 in decimal

5.

Binary 110 in decimal

6.

Binary 111 in decimal

7.

Binary 1000 in decimal

8.

Binary 1001 in decimal

9.

Binary 1010 in decimal

10.

Binary addition rule: 0 + 0

0.

Binary addition rule: 0 + 1

1.

Binary addition rule: 1 + 0

1.

Binary addition rule: 1 + 1

10.

How do you add binary numbers?

Line up the digits and carry when necessary.

Why is binary arithmetic useful for computers?

Its simplicity allows digital circuits to perform arithmetic operations.

How does binary multiplication work?

Multiply like regular multiplication, shift by place value, then add the binary rows.

What happens when a binary number is multiplied by 0?

The result is 0.

What happens to rows in binary multiplication?

Bits are shifted to line up with the multiplier's place value.

How does binary division work?

It uses regular long division with binary numbers.