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- To find the magnitude of a negative twos compliment number, flip all of the bits and add one.
-

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![](./assets/logic-gate)

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Any poduct that contains a logic circuit is classified as digital hardware.
- Moore's Law states that the number of a transistors on a chip doubles every two years
- The International Technology Roadmap for Semiconductors (ITRS) forecasts technology, including the number of transistors on a chip
- Multiple integrated circuits can be connected using a printed circuit board, or PCB.
- *Standard chips* conform to an agreed upon standard for functionality and physical configuration. They are usually less than 100 transistors in size, and provide basic building blocks for logic.
- These chips are combined to form a larger logic circuit
- They were popular until the 1980s
- As ICs improved, it became inefficient space-wise to have separate chips for each logical building block
- The functionality of these chips is fixed, and they do not change.
# Programmable Logic Devices
Programmable logic devices (PLDs) include a number of programmable switches that can configure the internal circuitry of a chip
- The most common type of PLD is a Field Programmable Gate Array (FPGA)
- FPGAs are widely available, but come with the drawback that they're limited in speed and performance
# Application Specific Integrated Circuits
Application Specific Integrated Circuits (ASICs) have higher maximum performance and transistor density compared to FPGAs, but the cost of production is very high.
- A logic circuit is made of connected logic gates
# Binary Numbers
In base 10, a value is expressed by an n-tuple with n digits
$$ D = d_{n-1}d_{n-2} \cdots d_1 d_0 $$
This represents the value
$$ V(D) = d_{n-1} * 10^{n-1} + d_{n - 2} * 10^{n-2} + \cdots + d_1 * 10^1 + d_0 * 10^0 $$
In a binary or base 2 number system, each digit can be a zero or one, called a *bit*.
$$ D = d_{n-1}d_{n-2} \cdots d_1 d_0 $$
To determine the integer value, a very similar formula can be used.
$$ V(B) = b_{n-1} * 2^{n-1} + b_{n-2} * 2^{n-2} \cdots b_{1} * 2^1 + b_0 * 2^0 $$
- The base of a number is often notated in the format of $(n)_b$, EG a base 10 number might be $(14)_{10}$, and a binary number might be $(10)_2$.
- The *least significant bit* (LSB) is usually the right-most bit. The highest value bit, or the *most significant bit* (MSB).
- A nibble is 4 bits, and a byte is 8 bits
## Conversions
### Base 10 to Binary
Repeatedly divide by 2, and track the remainder.
As an example, the below table shows how one might convert from $(857)_{10}$ to base 2.
| Equation | Remainder |
| --------------- | --------- |
| $857 / 2 = 428$ | $1$ |
| $428 / 2 = 214$ | $0$ |
| $214 / 2 = 107$ | $0$ |
| $107 / 2 = 53$ | $1$ |
| $53 / 2 = 26$ | $1$ |
| $26 / 2 = 13$ | $0$ |
| $13 / 2 = 6$ | $1$ |
| $6 / 2 = 3$ | $0$ |
| $3 / 2 = 1$ | $1$ |
| $1 / 2 = 0$ | $1$ |
The final answer is $1101011001$.
# Definitions
- **Xtor** is an abbreviation for *transistor*
- **Moore's Law** states that the number of transistors on a chip doubles every two years.
- A tuple is a finite and ordered list of things

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# Introduction
Every mathematical function can be thought of as a set of ordered pairs, or an input value and an output value.
- Examples include $f(x) = x^2 + 2x + 1$, and $\{(1, 3), (2, 5), (4, 7)\}$.
**A limit describes how a function behaves *near* a point, rather than *at* that point.***
- As an example, given a *well behaved function* $f(x)$ and the fact that:
- $f(1.9) = 8.41$
- $f(1.999) = 8.99401$
- $f(2.1) = 9.61$
- $f(2.01) = 9.061$
- $f(2.0001) = 9.0006$
We can note that the smaller the distance of the input value $x$ to $2$, the smaller the distance of the output to $9$. This is most commonly described in the terms "As $x$ approaches $2$, $f(x)$ approaches $9$", or "As $x \to 2$, $f(x) \to 9$."
Limits are valuable because they can be used to describe a point on a graph, even if that point is not present.
# Standard Notation
The standard notation for a limit is:
$$ \lim_{x \to a} f(x) = L $$
- As $x$ approaches $a$, the output of $f(x)$ draws closer to $L$. In the above notation, $x$ and $a$ are not necessarily equal.
- When plotted, the hole is located at $(a, L)$.
# Definitions
| Term | Definition |
| --------------------- | ----------------------------------------------------------------------------- |
| Well behaved function | A function that is continuous, has a single value, and is defined everywhere. |

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## Exprimental
- A systematic way of testing a hypothesis
## Epidemiological
- Observations of the occurrence, distribution, and associations
# Practice Final
- To reduce blood pressure, one should follow the DASH diet
- Bleeding gums, easy bruising, and poor wound healing that characterize scurvey are all related to vitamin C's role in synthesis of collagen.
- Vitamin K is a fat soluble vitamin that can be made in the GI tract
- Parathyroid hormone enhances the body's ability to retain calcium.
- Males have a higher basal metabolic rate than females
- On the intuitive eating spectrum, awareness, balance, moderation, and variety would exist in the middle. Apathy is on the left and anxiety is on the right
- The term appetite describes a biological need for food
- The 10 principals of intuitive eating are:
1. Reject Diet Culture
2. Honor Your Hunger
3. Make Peace with Food
4. Discover the Satisfaction Factor
5. Feel Your Fullness
6. Challenge the Food Police
7. Cope with Your Emotions with Kindness
8. Respect Your Body
9. Movement -- Feel the Difference
10. Honor Your Health with Gentle Nutrition
- Observations of the occurrence, distribution, and associations