Math Review Lecture Notes, Lecture notes of Vector Analysis

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Lecture 4:

Math Review

Administrative

HW0 out, due 2/

HW1 out soon

This Lecture: Goal

• Not a “Linear algebra in one lecture” – that’s

impossible.

• You should have seen this before
• Aimed at reviving your knowledge and plugging any

gaps

• Aimed at giving you intuitions
• I’ll intentionally go fast ; intent is for slides to be a

reference for you to review later

Floating Point Arithmetic

Free Drinks in Vegas

Each game/variable has mean $0.10, std $

100 games is

uncertain and

fun!

100K games is

guaranteed profit:

99.999999% lowest

value is $0.064.

$0.01 for drinks

$0.054 for profits

Let’s make it big

• Suppose we take the average of 50M normally

distributed numbers (mean=31, std=1)

• Theory : Average is a normally distributed random

variable with mean 31 and std

(

)*+

≈ 10

,)

• Practice :

Result:

17.4695??

What is a number?

7

6

5

4

3

2

1

0

Adding two numbers

“Integers” on a computer are integers modulo 2

k

Carry

Flag

Result

8

7

6

5

4

3

2

1

0

Some Gotchas

42

32 + 9 x 40 / 10 =

32 + 104 / 10 =

Overflow

32 + (9 x 40) / 10 =

uint

32 + (9 x 40) / 10 = 68

math

Why 104?

9 x 40 = 360

360 % 256 = 104

Should be:

9x4=

32 + 10 =

42

What is a number?

7

6

5

4

3

2

1

0

How can we do fractions?

5

4

3

2

1

0

• 1

• 2

Floating Point Numbers

Sign (S) Exponent (E)

Fraction (F)

𝑺

𝑬#𝒃𝒊𝒂𝒔

𝟑

7 - 7

0

Bias: allows exponent to be negative (bias = - 127 for float32)

Note: fraction = significant = mantissa;

exponents of all ones or all zeros are special numbers

Floating Point Numbers

Sign Exponent

Fraction

• 2

0

x 1.00 = - 1

0/

• 2

0

x 1.125 = - 1.

1/

• 2

0

x 1.25 = - 1.

0 1 0

2/

• 2

0

x 1.75 = - 1.

• 2

0

x 1.875 = - 1.

6/

7/

7 - 7=

(-bias)

• 1

−𝟏

𝑺

𝟐

𝑬#𝒃𝒊𝒂𝒔

𝟏 +

𝑭

𝟐

𝟑

Floating Point Numbers

Sign Exponent

Fraction

• 2

0

x 1.00 = - 1

• 2

0

x 1.125 = - 1.

• 2

2

x 1.00 = - 4

• 2

2

x 1.125 = - 4.

Gap between numbers is relative , not absolute

Adding Floating Point Numbers

Sign Exponent Fraction

• 2

2

x 1.00 = - 4

• 2
  • 1

x 1.00 = - 0.

• 2

2

x 1.125 = - 4.

Actual implementation is complex