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Binary Numbers

Computer Programming I Mr. Kindt

Spring 2014

The Powers of Two

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

Critical Fact All positive integers can be formed by powers of two or a sum of powers of

two!

Integer Sum

1 1

2 2

3 1 + 2

4 4

5 4 + 1

6 4 + 2

Integer Sum

7 4 + 2 + 1

8 8

9 8 + 1

10 8 + 2

11 8 + 2 + 1

12 8 + 4

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Process: Rewriting as a Sum of Powers

1. List the powers of two (from 128 down to 1). 2. Subtract the largest possible power of two. If 128 is larger than the number, cross it out on your list. Same with 64, 32… 3. Continue to subtract the largest number from the remainder, crossing out numbers you do not use.

128 64 32 16 8 4 2 1

Number: 117

Test it Out!

On your scrap paper, write down any positive integer between 40 and 255 on your paper. Pass it on to someone else and see if they can re-write it as a sum of the powers of two.

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

Number Sum of Powers of Two

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What is a Binary Number?

•  A binary number is made up of at least one binary digit (a.k.a. bit).

•  All bits are either zero (0) or

one (1). •  The bit furthest to the right is

known as “bit 0” (B0), the next one to the left is “bit 1” (B1), and so on.

•  Eight bits together form a byte.

€

01011001B4 B3 B2 B1 B5 B6 B7

The decimal value of this binary number is 89.

It is a byte because it has eight bits.

B8

The Decimal System

€

2194

2000+100+ 90+ 4

2 ⋅103( ) + 1⋅102( ) + 9 ⋅101( ) + 4 ⋅100( )

Decimal (Base 10) System

Place 0: How many 100 (ones) are there? Place 1: How many 101 (tens) are there?

Place 2: How many 102 (hundreds) are there? Place 3: How many 103 (thousands) are there?

Each place can hold an integer from 0 to 9.

P3 P2 P1

Recall: Exponent Laws --Any number raised to the power of zero is equal to 1. --Any number raised to a power of one is equal to itself.

P4

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The Binary System

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101

1⋅22( ) + 0 ⋅21( ) + 1⋅20( )4 + 0+1= 5

Binary (Base 2) System

Bit 0: How many 20 (ones) are there? Bit 1: How many 21 (twos) are there? Bit 2: How many 22 (fours) are there?

Each bit can be from 0 to 1.

B3 B2 B1

Binary a.k.a. “Base 2” Can be re-written as a sum of powers

with a base of 2.

Decimal a.k.a. “Base 10” Can be re-written as a sum of powers

with a base of 10.

Process: Converting Binary to Decimal

Go in reverse! I suggest listing the powers of two for each bit and writing out your sum.

Binary: 10001100 ➭ Decimal: _______

Binary 1 0 0 0 1 1 0 0 128 64 32 16 8 4 2 1

Sum

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Practice: Converting Binary to Decimal Determine the decimal value of these binary numbers.

Binary: 11011 ➭ Decimal: _______

Binary: 10000001 ➭ Decimal: _______

Binary: 10110101 ➭ Decimal: _______

Process: Converting Decimal to Binary

The binary number is a coded way of listing your sum of powers of two. Instead of listing the numbers in the sum, you will simply place a “1” in the bit position if you use that power of two and a “0” if you do not!

Decimal: 117 ➭ Binary: 1110101

128 64 32 16 8 4 2 1 Power of 2 27 26 25 24 23 22 21 20

Remainder ---- 53 21 5 5 1 1 0 Binary 0 1 1 1 0 1 0 1

Note: We ignore zeroes to the left of the first one.

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Practice: Converting Decimal to Binary

Decimal: 241 ➭ Binary: ____________ 128 64 32 16 8 4 2 1

Power of 2 27 26 25 24 23 22 21 20

Remainder

Binary

Decimal: 59 ➭ Binary: ____________

128 64 32 16 8 4 2 1 Power of 2 27 26 25 24 23 22 21 20

Remainder

Binary

Who Cares???

Computer scientists do! Pressing a key activates a unique set of electrical currents, opening some gates (1) and leaving others closed (0). The CPU interprets this, converts it to binary, and uses this to signal a character to the monitor! http://resources.kaboose.com/brain/comp_les3.html

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ASCII (Ask-kee)

American Standard Code for Information Interchange

Original 7-bit code chart http://en.wikibooks.org/wiki/Digital_Electronics/ASCII

OMG!

1001111 1001101 1000111 0100001

Encode-a-Text

Write a three-five letter message in binary code using the ASCII numbering system. Then, see if your partner can decode it.

!