Chapter 2. Binary Values and Number Systems

Chapter 2

Binary Values and Number

Systems

Numbers

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Natural numbers, a.k.a. positive integers

Zero and any number obtained by repeatedly adding one to it.

Examples: 100, 0, 45645, 32

Negative numbers

A value less than 0, with a – sign Examples: -24, -1, -45645, -32

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Integers

A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32

Rational numbers

An integer or the quotient of two integers Examples: -249, -1, 0, 3/7, -2/5

Real numbers

In general cannot be represented as the quotient of any two integers. They have an infinite # of fractional digits.

Example: Pi = 3.14159265…

2.2 Positional notation

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How many ones (units) are there in 642?

600 + 40 + 2 ? Or is it

384 + 32 + 2 ? Or maybe…

1536 + 64 + 2 ?

Positional Notation

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642 is 600 + 40 + 2 in BASE 10

The base of a number determines how many digits are used and the value of each digit’s position.

To be specific:

• In base R, there are R digits, from 0 to R-1

• The positions have for values the powers of R, from right to left: R⁰, R¹, R², …

Positional Notation

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In our example:

642 in base 10 positional notation is:

6 x 10² = 6 x 100 = 600 + 4 x 10¹ = 4 x 10 = 40

+ 2 x 10º = 2 x 1 = 2 = 642 in base 10

This number is in base 10

The power indicates the position of the digit inside the

number

Positional Notation

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d_n * R^n-1 + d_n-1 * R^n-2 + ... + d₂ * R + d₁

Formula:

642 is 6₃ * 10² + 4₂ * 10 ⁺ 2₁

R is the base of the number

n is the number of digits in the number

d is the digit in the i^th position

in the number

Positional Notation reloaded

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d_n * R^n-1 + d_n-1 * R^n-2 + ... + d₂ * R + d₁

… but, in CS, the digits are numbered from zero, to match the power of the base:

d_n-1 * R^n-1 + d_n-2 * R^n-2 + ... + d₁ * R¹ + d₀ * R⁰

The text shows the digits numbered like this:

Positional Notation

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What if 642 has the base of 13?

642 in base 13 is equal to 1068 in base 10 642₁₃ = 1068₁₀

+ 6 x 13² = 6 x 169 = 1014 + 4 x 13¹ = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2

= 1068 in base 10

Positional Notation

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In a given base R, the digits range from 0 up to R – 1

R itself cannot be a digit in base R

Trick problem:

Convert the number 473 from base 6 to base 10

Binary

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Decimal is base 10 and has 10 digits:

0,1,2,3,4,5,6,7,8,9

Binary is base 2 and has 2 digits:

0,1

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Converting Binary to Decimal

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What is the decimal equivalent of the binary number 1101110?

1101110₂ = ???₁₀

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Converting Binary to Decimal

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What is the decimal equivalent of the binary number 1101110?

1 x 2⁶ = 1 x 64 = 64 + 1 x 2⁵ = 1 x 32 = 32 + 0 x 2⁴ = 0 x 16 = 0 + 1 x 2³ = 1 x 8 = 8 + 1 x 2² = 1 x 4 = 4 + 1 x 2¹ = 1 x 2 = 2 + 0 x 2º = 0 x 1 = 0

= 110 in base 10

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QUIZ:

10011010

= ???

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Bases Higher than 10

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How are digits in bases higher than 10 represented?

Base 16 (hexadecimal, a.k.a. hex) has 16 digits:

0,1,2,3,4,5,6,7,8,9,

A,B,C,D,E

F

Converting Hexadecimal to Decimal

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What is the decimal equivalent of the hexadecimal number DEF?

D x 16² = 13 x 256 = 3328 + E x 16¹ = 14 x 16 = 224 + F x 16º = 15 x 1 = 15

= 3567 in base 10

QUIZ:

2AF

= ???

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Converting Octal to Decimal

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What is the decimal equivalent of the octal number 642?

642₈ = ???₁₀

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Converting Octal to Decimal

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What is the decimal equivalent of the octal number 642?

6 x 8² = 6 x 64 = 384 + 4 x 8¹ = 4 x 8 = 32 + 2 x 8º = 2 x 1 = 2

= 418 in base 10

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Are there any non-positional number systems?

Hint: Why did the Roman civilization have no contributions to mathematics?

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Today we’ve covered pp.33-39 of the text (stopped before Arithmetic in Other Bases)

Solve in notebook for next class:

1, 2, 3, 4, 5, 20, 21, 22

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QUIZ: Convert to decimal

1101 0011

= ???

AB7

= ???

513

= ???

692

= ???

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Addition in Binary

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Remember that there are only 2 digits in binary, 0 and 1

1 + 1 is 0 with a carry

Carry Values 0 1 1 1 1 1

1 0 1 0 1 1 1 +1 0 0 1 0 1 1 1 0 1 0 0 0 1 0

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Addition QUIZ

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Carry values go here

1 0 1 0 1 1 0 +1 0 0 0 0 1 1

14 Check in base ten!

Subtraction in Binary

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Remember borrowing? Apply that concept here:

1 2 0 2 0 2

1 0 1 0 1 1 1 1 0 1 0 1 1 1 - 1 1 1 0 1 1 - 1 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0

15 Borrow values

Check in base ten!

Subtraction QUIZ

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1 0 1 1 0 0 0 - 1 1 0 1 1 1

15 Borrow values

Check in base ten!

Converting Decimal to Other Bases

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While (the quotient is not zero)

Divide the decimal number by R

Make the remainder the next digit to the left in the answer

Replace the original decimal number with the quotient

Algorithm for converting number in base 10 to any other base R:

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A.k.a. repeated division (by the base):

Converting Decimal to Binary

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Example: Convert 179₁₀ to binary 179  2 = 89 rem. 1

 2 = 44 rem. 1

 2 = 22 rem. 0

 2 = 11 rem. 0

 2 = 5 rem. 1

 2 = 2 rem. 1

 2 = 1 rem. 0

179₁₀ = 10110011₂  2 = 0 rem. 1

Notes: The first bit obtained is the rightmost (a.k.a. LSB)

The algorithm stops when the quotient (not the remainder!) becomes zero

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LSB MSB

Repeated division QUIZ

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Convert 42₁₀ to binary 42  2 = rem.

42₁₀ = ₂

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The repeated division algorithm can be used to convert from any base into any other base, but we

use it only for 10 → 2

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SKIP

Converting Decimal to Octal

Converting Decimal to Hex

Converting Binary to Octal

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• Mark groups of three (from right)

• Convert each group

10101011 10 101 011 2 5 3

10101011 is 253 in base 8

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Converting Binary to Hexadecimal

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• Mark groups of four (from right)

• Convert each group

10101011 1010 1011 A B

10101011 is AB in base 16

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Counting

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Note the patterns!

Converting Octal to Hexadecimal

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End-of-chapter ex. 25:

Explain how base 8 and base 16 are related

10 101 011 1010 1011 2 5 3 A B

253 in base 8 = AB in base 16

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Binary Numbers and Computers

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Computers have storage units called binary digits or bits

Low Voltage = 0

High Voltage = 1

All bits are either 0 or 1

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Binary and Computers

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Word= group of bits that the computer processes at a time

The number of bits in a word determines the word length of the computer. It is usually a multiple of 8.

1 Byte = 8 bits

• 8, 16, 32, 64-bit computers

• 128? 256?

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Individual work

• Read Grace Hopper’s bio, the trivia frames, chapter review questions, and ethical issues

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Who am I?

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I wrote the world’s first compiler in 1952!

From the history of computing: bi-quinary

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The front panel of the legendary IBM 650 IBM 650 (source: Wikipedia)

Roman abacus (source: MathDaily.com)

Chapter Review questions

• Describe positional notation

• Convert numbers in other bases to base 10

• Convert base-10 numbers to numbers in other bases

• Add and subtract in binary

• Convert between bases 2, 8, and 16 using groups of digits

• Count in binary

• Explain the importance to computing of bases that are powers of 2

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Homework

Due next Friday, Feb. 3:

• End-of-chapter ex. 23, 26, 28, 29, 30, 31, 38

• End-of-chapter thought question 4 (paragraph-length answer required)

The latest homework assigned is always available on the course webpage

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