Brush up before you prove it.

Arithmetic¹ Level 1

Rules

• Order of operations: brackets first, then multiplication and division (left to right), then addition and subtraction (left to right). So \(6+2\times3=12\), not 24.
• Division with a remainder: \(23\div4\) is 5 remainder 3, because \(4\times5=20\) and \(23-20=3\). The remainder is always smaller than the number you divide by.
• Making change: change = amount paid − total cost. Add up everything bought first.
• Rounding to the nearest ten: look at the ones digit. 5 or more rounds up (67 becomes 70), less rounds down (63 becomes 60).

Algebra¹ Levels 2–3

Fractions, decimals and percentages

• Fractions: to add or subtract, rewrite over a common denominator. To divide, multiply by the reciprocal.
• Percent means "per hundred": \(p\%\) of \(x\) is \(\tfrac{p}{100}\cdot x\).

Linear equations and inequalities

• Solve by doing the same inverse operation to both sides until \(x\) is alone.
• Multiplying or dividing an inequality by a negative number flips its direction.
• A line is \(y=mx+b\): slope \(m\), y-intercept \(b\).

Quadratics

• Factoring \(x^2+bx+c\): find \(p,q\) with \(p+q=b\) and \(pq=c\), then \(x^2+bx+c=(x+p)(x+q)\).
• Difference of squares: \(a^2-b^2=(a-b)(a+b)\).
• Zero-product rule: if \(AB=0\) then \(A=0\) or \(B=0\). So \((x-5)(x-2)=0\) gives \(x=5\) or \(x=2\).
• The graph of \(y=ax^2+bx+c\) is a parabola: it opens up if \(a>0\), down if \(a<0\), with its vertex at \(x=-\tfrac{b}{2a}\).

Logarithms

Precalculus¹ Level 4

Functions and trigonometry

• Composition: \((f\circ g)(x)=f(g(x))\). Apply \(g\) first.
• Radians: \(\pi\text{ rad}=180^\circ\).
• Unit circle: \(\sin\tfrac{\pi}{6}=\tfrac12\), \(\cos\tfrac{\pi}{3}=\tfrac12\), \(\sin\tfrac{\pi}{4}=\cos\tfrac{\pi}{4}=\tfrac{\sqrt2}{2}\), \(\sin 0=0\), \(\cos 0=1\).

Complex numbers

• \(i^2=-1\). Powers of \(i\) repeat every four: \(i,\,-1,\,-i,\,1\).
• The conjugate of \(a+bi\) is \(a-bi\), and \((a+bi)(a-bi)=a^2+b^2\).

Calculus 1² Level 5

Limits and continuity

Limit:

\(\lim_{x\to a}f(x)=L\) means \(f(x)\) gets arbitrarily close to \(L\) as \(x\) approaches \(a\). It exists only if both one-sided limits exist and are equal.

Continuous at \(a\):

\(f(a)\) is defined and \(\lim_{x\to a}f(x)=f(a)\).

Derivatives

Theorems

Calculus 2² Level 6

Integrals

Sequences and series

Linear algebra³ Level 6

Linearly independent:

no vector in the set is a combination of the others.

Rank:

the number of linearly independent rows (equivalently, columns).

Eigenvector:

a nonzero \(v\) with \(Av=\lambda v\). The scalar \(\lambda\) is its eigenvalue.

Advanced topics^2,4,5 Level 7

Hard questions at this level are mostly conceptual: what a theorem says and what it implies.

Multivariable calculus and differential equations

Abstract algebra

Group:

a set with an operation that is closed and associative, with an identity element and an inverse for every element.

Field:

a set where you can add, subtract, multiply, and divide by anything nonzero, such as \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\).

Analysis and number theory

Computer basics⁴ Level 1

CPU:

the processor that executes program instructions.

RAM:

fast working memory for running programs. It's volatile: it loses its contents when power is off.

Storage (SSD, hard drive):

keeps files when the power is off (non-volatile), but is slower than RAM.

Operating system:

manages hardware, memory, files and running programs (Android, Windows, macOS, Linux).

Bit and byte:

a bit is a 0 or 1. A byte is 8 bits.

IP address and DNS:

every device on the internet has an IP address. DNS turns names like example.com into IP addresses.

Programming basics⁴ Level 2

• Binary place values are powers of 2: \(1011_2=8+0+2+1=11\).
• Loops: in Python, range(n) gives \(0,1,\dots,n-1\), so the body runs \(n\) times.
• Arrays are usually indexed from 0: the last of \(n\) items is at index \(n-1\).
• Linear search checks items one by one: up to \(n\) comparisons.

x = 3
for i in range(4):
    x += i
print(x)

Data structures^1,2 Levels 2–6

• Load factor \(\alpha=n/m\) (keys per bucket). Collisions are handled by chaining (lists in buckets) or open addressing (probing for a free slot).
• In a heap stored in an array from index 0, node \(i\) has children \(2i+1\) and \(2i+2\), and parent \(\lfloor (i-1)/2\rfloor\).

Search algorithms^1,2 Levels 2–6

• Linear search: \(O(n)\), works on any list.
• Binary search: \(O(\log n)\), but only on sorted data. It halves the range each step, so it needs at most \(\lfloor\log_2 n\rfloor+1\) comparisons.
• Two pointers: on sorted data, move inward from both ends, \(O(n)\).
• BFS and DFS visit every vertex and edge of a graph: \(O(V+E)\).

Sorting^1,2 Levels 3–6

• Stable: equal keys keep their original order.
• Quicksort hits \(O(n^2)\) with consistently bad pivots, such as always picking the last element of already sorted input.
• Insertion sort is fast on nearly sorted data: its work is proportional to the number of out-of-order pairs (inversions).
• Counting sort \(O(n+k)\) and radix sort \(O(d(n+k))\) don't compare elements.

Trees^1,2 Levels 4–6

Height:

edges on the longest path from the root to a leaf (a single node has height 0).

Binary search tree (BST):

every key in the left subtree is smaller than the node, and every key in the right subtree is larger. Search and insert take \(O(h)\).

AVL tree:

a BST where subtree heights differ by at most 1, restored by rotations, so \(h=O(\log n)\).

• A tree with \(n\) nodes has \(n-1\) edges. A binary tree of height \(h\) has at most \(2^{h+1}-1\) nodes.
• Preorder: root, left, right. Inorder: left, root, right (sorted order for a BST). Postorder: left, right, root. Level order: breadth-first.
• Deleting a BST node with two children: replace it with its inorder successor (the smallest key in its right subtree).

Graphs¹ Levels 5–6

• Handshake lemma: the degrees add up to \(2|E|\). A complete graph on \(n\) vertices has \(\tfrac{n(n-1)}{2}\) edges.
• Adjacency list: \(O(V+E)\) space. Adjacency matrix: \(O(V^2)\) space, \(O(1)\) edge lookup.
• BFS finds shortest paths by edge count. Dijkstra finds shortest paths with non-negative weights, \(O((V+E)\log V)\) with a binary heap.
• Bellman-Ford allows negative weights and detects negative cycles, \(O(VE)\).
• A minimum spanning tree (Kruskal, Prim) of a connected graph has \(V-1\) edges.
• Topological sort orders a directed acyclic graph so every edge points forward. A graph is bipartite exactly when it has no odd cycle.

Theory of computation³ Level 7

P:

decision problems solvable in polynomial time.

NP:

decision problems whose "yes" answers can be verified in polynomial time, given a certificate. \(P\subseteq NP\).

NP-hard:

every NP problem reduces to it in polynomial time. It need not be in NP itself.

NP-complete:

both in NP and NP-hard. Examples: SAT, 3-SAT, Hamiltonian cycle, subset sum.

Decidable:

some Turing machine always halts with the right yes/no answer.

Coding¹

• Recursion needs a base case that stops it. Each call should move toward that case.
• Integer division and modulo in Python: 7 // 2 == 3, 7 % 2 == 1, and -7 // 2 == -4 (it rounds down).
• Short-circuiting: a and b skips b if a is false. a or b skips b if a is true.
• Off-by-one: range(1, 5) is 1, 2, 3, 4. The end isn't included.
• Object-oriented ideas: encapsulation (hide state behind methods), inheritance (a subclass reuses a parent), polymorphism (one call, behaviour chosen by the object's type).
• Git: a commit is a snapshot, a branch is a movable pointer to commits, and a merge joins two branches' histories.

def f(n):
    return 1 if n <= 1 else n * f(n - 1)

Object-oriented programming⁵

Class and object:

a class is a blueprint, and an object is one instance built from it.

Encapsulation:

bundle data with the methods that use it, and hide the data behind those methods (private fields, public methods).

Abstraction:

expose what an object does and hide how it does it.

Inheritance:

a subclass reuses and extends a parent class ("is-a": a Dog is an Animal).

Polymorphism:

one call, different behaviour depending on the object's actual type.

• Overriding: a subclass replaces a parent's method with the same signature. Overloading: several methods share a name but take different parameters.
• Interface: declares methods without implementing them. Abstract class: may mix implemented and unimplemented methods, and can't be instantiated.
• Composition ("has-a": a Car has an Engine) is often preferred to inheritance because it couples classes less.
• Access modifiers: public (anyone), protected (the class and its subclasses), private (the class only). Exact rules vary by language.
• SOLID: single responsibility, open-closed, Liskov substitution, interface segregation, dependency inversion.

class Shape:
    def area(self): return 0
class Square(Shape):
    def __init__(self, s): self.s = s
    def area(self): return self.s * self.s

shapes = [Shape(), Square(3)]
print(sum(x.area() for x in shapes))

Databases and SQL²

• WHERE filters rows before grouping. HAVING filters groups after GROUP BY.
• INNER JOIN keeps matching rows only. LEFT JOIN keeps every left row, with NULLs where there's no match.
• Primary key: unique and not null. Foreign key: must match a key in another table.
• NULL isn't equal to anything, even NULL: test it with IS NULL. COUNT(col) skips NULLs, and COUNT(*) counts every row.
• Indexes speed up reads but slow down writes. Normalization removes duplicated data.
• ACID: atomicity, consistency, isolation, durability. The isolation levels, from weakest: read uncommitted, read committed, repeatable read, serializable.²

SELECT customer, SUM(amount) FROM orders
GROUP BY customer HAVING SUM(amount) > 20;

Frontend³

• Box model: content, then padding, border and margin. With box-sizing: border-box, width includes padding and border.
• Specificity (strongest first): inline styles, IDs, then classes, attributes and pseudo-classes, then elements.³
• Flexbox: justify-content aligns along the main axis, align-items along the cross axis. Grid lays out rows and columns together.
• rem is relative to the root font size, em to the element's font size (the parent's, when setting font-size itself).
• Most DOM events bubble from the target up through its ancestors.
• In JavaScript, === compares without type conversion: 0 == "0" is true but 0 === "0" is false.
• Accessibility: use semantic elements, give images alt text, and label form fields.

Backend⁴

• Idempotent methods give the same result when repeated: GET, PUT and DELETE are, POST isn't. GET is also safe: it changes nothing.⁴
• Authentication proves who you are. Authorization decides what you may do.
• Store passwords as slow, salted hashes (bcrypt, Argon2), never as plain text or fast hashes.⁴
• Horizontal scaling adds machines behind a load balancer. Vertical scaling makes one machine bigger.
• Common web vulnerabilities: injection, cross-site scripting (XSS), broken access control.

System design basics^3,6 Medium–Hard

Load balancer:

spreads incoming requests across several servers, and stops sending to ones that fail health checks.

Reverse proxy:

sits in front of servers and receives clients' requests for them (for TLS, caching, routing). A forward proxy sits in front of clients and sends requests on their behalf.

Cache:

a fast copy of data close to where it's needed. A TTL (time to live) sets how long an entry stays. Invalidation removes entries that are out of date, and a stale cache serves old data.

CDN (content delivery network):

servers around the world that cache content near users, cutting latency and load on the origin server.

DNS:

turns domain names into IP addresses.

TLS and HTTPS:

TLS encrypts and authenticates a connection, and HTTPS is HTTP over TLS. SSL is TLS's older, deprecated predecessor, which is why people still say "SSL/TLS".

FTP and SFTP:

FTP transfers files with no encryption (port 21). SFTP transfers files over an encrypted SSH connection (port 22).

API gateway:

a single entry point that routes requests to back-end services and handles shared concerns like authentication and rate limiting.

Microeconomics¹

Core ideas

Scarcity:

resources are limited but wants are not, so every choice gives something up.

Opportunity cost:

the value of the next best alternative given up. An evening spent studying instead of working a $100 shift costs $100.

Marginal analysis:

compare the extra benefit of one more unit with its extra cost. Keep going while marginal benefit is at least marginal cost.

Sunk cost:

money already spent that can't be recovered. It shouldn't affect the next decision.

Comparative advantage:

producing something at a lower opportunity cost than others. Trade based on it helps both sides, even when one side has the absolute advantage (can produce more of everything).

Supply and demand

• Law of demand: when the price rises, the quantity demanded falls. Law of supply: when the price rises, the quantity supplied rises.
• Equilibrium is where supply equals demand. Above it, a surplus pushes the price down. Below it, a shortage pushes the price up.
• Movement vs shift: a change in the good's own price moves along the curve (a change in quantity demanded). Anything else shifts the whole curve (a change in demand).
• Demand shifters: income, tastes, prices of related goods, expectations, number of buyers. Supply shifters: input costs, technology, taxes and subsidies, expectations, number of sellers.
• A normal good sells more when incomes rise. An inferior good (instant noodles, bus rides) sells less.
• Substitutes (coffee and tea): if one gets pricier, demand for the other rises. Complements (printers and ink): if one gets pricier, demand for the other falls.
• A price ceiling below equilibrium (rent control) causes a shortage. A price floor above it (minimum wage) causes a surplus. A ceiling above equilibrium, or a floor below it, does nothing.

Elasticity

• Elastic (\(|E_d|>1\)): buyers react strongly. Goods with many substitutes, luxuries, and big purchases. Inelastic (\(|E_d|<1\)): necessities with few substitutes, like gasoline or insulin.
• Total revenue test: if demand is elastic, cutting the price raises revenue. If it's inelastic, raising the price raises revenue. Revenue peaks where \(|E_d|=1\).
• Income elasticity is positive for normal goods (above 1 for luxuries) and negative for inferior goods.
• Cross elasticity is positive for substitutes and negative for complements.
• Tax incidence: whichever side of the market is less elastic pays more of a tax, no matter who it's collected from.

Surplus and efficiency

Consumer surplus:

what buyers were willing to pay minus what they paid. The area under the demand curve and above the price.

Producer surplus:

the price sellers get minus the lowest price they'd accept. The area above the supply curve and below the price.

Total surplus:

consumer plus producer surplus. A competitive market at equilibrium makes it as large as possible.

Deadweight loss:

total surplus lost when a tax, price control, monopoly or externality pushes quantity away from the efficient level.

• A tax drives a wedge between what buyers pay and what sellers keep. Fewer trades happen, creating deadweight loss.

Costs of production

Fixed cost:

doesn't change with output, like rent.

Variable cost:

rises with output, like materials.

Short run:

at least one input is fixed. Long run: every input can change.

Diminishing marginal returns:

adding more of one input to fixed inputs eventually adds less and less output.

Economic profit:

revenue minus explicit and implicit (opportunity) costs. Accounting profit subtracts explicit costs only. Zero economic profit (normal profit) still covers the owner's opportunity cost.

Economies of scale:

long-run average cost falls as output grows. With diseconomies of scale, it rises.

• Marginal cost crosses average total cost and average variable cost at their lowest points.

Market structures

• Firms maximize profit where marginal revenue = marginal cost. In perfect competition, \(P=MR=MC\). In the long run, entry and exit push price to the minimum of \(ATC\), so economic profit is zero.
• Shutdown rule: in the short run, stop producing if \(P\) is below minimum \(AVC\). In the long run, leave the industry if \(P\) is below minimum \(ATC\).
• A monopoly's \(MR\) is below its price, because selling one more unit means lowering the price on every unit. It produces less and charges more than a competitive market (\(P>MC\)), causing deadweight loss.
• A natural monopoly (a water utility) can serve the whole market more cheaply than several firms could. Price discrimination charges different buyers different prices for the same good, like student discounts.

Dominant strategy:

the best move no matter what the other player does.

Nash equilibrium:

no player can do better by changing strategy alone.

Prisoner's dilemma:

each player's dominant strategy (cheat) leaves both worse off than if they'd cooperated. It's why cartel members are tempted to break agreed output limits.

Market failures

Externality:

a cost or benefit that falls on people outside the trade. With a negative one (pollution), the market produces too much. With a positive one (vaccines, education), too little.

Pigouvian tax:

a tax equal to the external cost, so producers pay the full social cost. A subsidy does the same for a positive externality.

Coase theorem:

if property rights are clear and bargaining is cheap, the people involved can fix an externality themselves.

Free rider:

someone who benefits without paying. This is why markets underprovide public goods.

Tragedy of the commons:

a shared resource gets overused because nobody owns it, like overfishing.

Asymmetric information:

one side knows more. Adverse selection happens before the deal (the used-car "lemons" problem). Moral hazard happens after it (insured people take more risks).

Macroeconomics¹

Measuring output

GDP:

the market value of all final goods and services produced inside a country in a period.

Left out of GDP:

intermediate goods (they'd be counted twice), used goods, buying stocks or bonds, transfer payments like pensions, unpaid housework, and the underground economy.

GNP:

output by a country's residents and firms, wherever in the world it's produced.

Nominal vs real GDP:

nominal uses current prices. Real GDP uses base-year prices, so it only grows when output grows.

GDP per capita:

GDP divided by population. A rough measure of living standards.

• In GDP, investment (\(I\)) means new equipment, buildings, housing and inventories, not buying shares. Government spending (\(G\)) leaves out transfer payments.

Inflation

Inflation:

a sustained rise in the overall price level. Deflation is a falling price level. Disinflation is inflation slowing down but still positive.

CPI:

the Consumer Price Index tracks the cost of a fixed basket of goods a typical household buys. The base year is 100.

Core inflation:

inflation without food and energy prices, which jump around.

Demand-pull vs cost-push:

demand-pull is too much spending chasing too few goods. Cost-push comes from rising input costs, like oil.

• Surprise inflation hurts lenders, savers and people on fixed incomes. It helps borrowers with fixed-rate loans, who repay in cheaper money.
• The CPI tends to overstate inflation: people switch to cheaper goods, and quality improvements look like price rises.

Unemployment

Labor force:

people who are employed plus people who are unemployed and actively looking. Retirees, full-time students and discouraged workers (who stopped looking) aren't in it.

Frictional:

between jobs or just starting out. Structural: skills or location don't match the jobs available. Cyclical: caused by a recession.

Natural rate:

frictional plus structural unemployment. "Full employment" means unemployment is at this rate, not at zero.

Aggregate demand and supply

• Aggregate demand slopes down: a lower price level makes people's money worth more (wealth effect), lowers interest rates, and makes exports cheaper.
• Short-run aggregate supply slopes up because some wages and prices are slow to change. Long-run aggregate supply is vertical at potential GDP, which depends on resources and technology, not prices.
• A recessionary gap is output below potential. An inflationary gap is output above it.
• Stagflation is rising prices and falling output together, usually from a supply shock like the 1970s oil crises.

Business cycles and growth

• The four phases of a business cycle are expansion, peak, contraction (recession) and trough.
• A common rule of thumb calls two quarters in a row of falling real GDP a recession. In the US, the NBER officially dates them using wider data such as jobs and income.
• Leading indicators (stock prices, building permits) move before the economy does. Lagging indicators (the unemployment rate) move after.
• Long-run growth comes from more physical capital, more human capital (education and skills), better technology, and institutions like property rights. Productivity, meaning output per hour worked, drives living standards.

Fiscal and monetary policy

• Fiscal policy uses government spending and taxes. Monetary policy uses interest rates and the money supply.
• Expansionary policy (more spending, tax cuts, lower rates) fights recessions. Contractionary policy (the opposite) fights inflation.
• Automatic stabilizers, like unemployment insurance and progressive taxes, push against the cycle without any new law.
• Crowding out: heavy government borrowing can raise interest rates and reduce private investment.
• A budget deficit is one year's spending above revenue. The national debt is the total of past deficits minus surpluses.
• The US Federal Reserve has a dual mandate: maximum employment and stable prices. It aims for 2% inflation over the long run.²
• The short-run Phillips curve shows a trade-off between inflation and unemployment. In the long run it's vertical at the natural rate, so there's no lasting trade-off.

Trade and exchange rates

Trade balance:

exports minus imports. A trade deficit means a country imports more than it exports.

Balance of payments:

a country's record of transactions with the rest of the world. The current account (trade, investment income, transfers) and the financial account (buying and selling assets) roughly cancel out. A current account deficit is paid for by foreigners buying domestic assets.

Appreciation:

a currency buys more foreign currency than before. Exports get pricier abroad and imports get cheaper. Depreciation is the opposite.

Floating vs fixed:

a floating exchange rate is set by the market. A fixed (pegged) rate is held by the central bank.

Purchasing power parity:

in the long run, exchange rates tend to move until the same basket costs the same everywhere.

Tariff:

a tax on imports. A quota limits the quantity imported.

Banking^1,2

Money

Functions of money:

a medium of exchange, a unit of account and a store of value.

M1:

cash in circulation plus checking deposits (and, since 2020 in the US, savings deposits). M2 adds small time deposits and retail money market funds.

Liquidity:

how quickly something can be turned into cash without losing value. Cash is the most liquid asset. A house is illiquid.

How banks work

• Fractional reserve banking: banks lend out most deposits. Each loan ends up deposited at another bank, which lends part of it again, so lending creates money.
• The simple money multiplier is \(\tfrac{1}{\text{reserve ratio}}\). In practice money grows less, because banks keep extra reserves and people hold cash.
• A bank balance sheet: assets = liabilities + equity. Loans, reserves and bonds are the bank's assets. Deposits and borrowed money are its liabilities.
• Banks earn the gap between the interest on loans and the interest paid on deposits, called the net interest margin.
• Maturity mismatch: banks borrow short (deposits can leave any day) and lend long (30-year mortgages).
• Commercial banks take deposits and make loans. Investment banks help companies issue stocks and bonds, advise on mergers and trade securities. Credit unions are owned by their members.

Bank capital:

the equity cushion that absorbs losses. Under Basel III, banks must hold common equity of at least 4.5% of risk-weighted assets, or 7% with the required conservation buffer.⁵

Insolvent vs illiquid:

an insolvent bank's assets are worth less than its liabilities. An illiquid bank is solvent but can't raise cash fast enough.

Bank run:

many depositors pull money out at once for fear the bank will fail. Loans can't be sold quickly, so even a healthy bank can fall. Silicon Valley Bank failed after a run in March 2023.

Leverage:

assets divided by equity. The higher it is, the smaller the loss that wipes out the bank's equity.

Central banks

• Central bank tools (the US Federal Reserve): open market operations, the policy rate target, and interest on reserve balances. The Fed cut reserve requirements to 0% in March 2020.²
• Open market operations: buying government bonds adds reserves and pushes interest rates down. Selling them does the opposite.
• The federal funds rate is what banks charge each other for overnight loans of reserves. The Fed sets a target range, and changes flow through to mortgage, car-loan and savings rates.
• As lender of last resort, a central bank lends to banks in trouble, in the US through the discount window.
• Quantitative easing: large purchases of longer-term bonds when short-term rates are already near zero.
• FDIC deposit insurance covers $250,000 per depositor, per insured bank, per ownership category.² It stops most runs before they start.

Interest and loans

APR vs APY:

APR is the yearly rate without compounding. APY includes compounding, so it's higher whenever interest compounds more than once a year.

Amortization:

each fixed loan payment covers that period's interest first, and the rest pays down the principal. Early payments are mostly interest, later ones mostly principal.

Secured vs unsecured:

a secured loan is backed by collateral (a mortgage, a car loan). An unsecured loan (a credit card) isn't, so it charges more.

Fixed vs variable rate:

a fixed rate stays the same for the whole loan. A variable rate follows a benchmark rate.

Credit risk:

the chance a borrower won't repay (defaults). Riskier borrowers pay higher rates, and a credit score sums up a borrower's history.

Trading and investing^3,4

Stocks

Stock:

a share of ownership (equity) in a company. Common stock usually votes. Preferred stock usually doesn't, but its dividends are paid first.

Market capitalization:

share price times shares outstanding. The market's value of the whole company.

IPO:

a company's first sale of shares to the public. New shares are sold in the primary market. Investors then trade them with each other in the secondary market, on exchanges like the NYSE and Nasdaq.

Dividend:

a share of profits paid to shareholders. You must own the stock before the ex-dividend date to get the next one.

Stock split:

more shares at a lower price, same total value. After a 2-for-1 split, one $100 share becomes two $50 shares.

Buyback:

a company buys its own shares, so fewer are outstanding.

Bonds

Bond:

a loan to a government or company (debt). The face value (often $1,000) is repaid at maturity. The coupon rate sets the yearly interest as a percentage of face value.

Yield to maturity:

the yearly return if you buy at today's price and hold until maturity.

Premium vs discount:

a bond trades above face value when its coupon beats current rates, and below face value when it doesn't.

Zero-coupon bond:

pays no coupons. It's sold below face value and pays the full face value at maturity.

US Treasuries:

bills mature within a year, notes in 2 to 10 years, bonds in 20 or 30 years.

Credit rating:

BBB− (or Baa3) and above is investment grade. Below that is high yield ("junk"), which pays more for its higher risk of default.

Yield curve:

yields plotted by maturity. It usually slopes up. An inverted curve, with short rates above long rates, has often come before recessions.

• When interest rates rise, existing bond prices fall. Longer bonds fall more.

Orders and markets

• A market order fills right away at the best available price. A limit order fills only at your price or better. A stop order becomes a market order once the price hits your stop, often to limit losses (a stop-loss).
• The bid-ask spread is the gap between the highest bid and the lowest ask. It's wider in thinly traded markets.
• Liquidity means you can trade quickly near the last price. Volume is how many shares trade in a period. Higher volume usually means tighter spreads.
• A bull market is a long rise. A bear market is a fall of 20% or more from a recent high. A correction is a drop of about 10%.
• Market indices: the S&P 500 tracks 500 large US companies, weighted by market cap. The Dow Jones Industrial Average tracks 30, weighted by share price. The Nasdaq Composite covers stocks listed on the Nasdaq and leans toward tech.
• An index fund or ETF copies an index cheaply. ETFs trade all day on an exchange. Mutual funds are priced once a day at their net asset value.

Options and futures

Call option:

the right to buy at the strike price before expiration. A put is the right to sell. The buyer pays a premium, which is the most they can lose.

In the money:

a call whose stock trades above the strike, or a put whose stock trades below it.

American vs European:

an American option can be used any time before expiration. A European one only at expiration.

Futures:

an obligation to buy or sell later at a set price. Futures are standardized, traded on exchanges and settled daily. Forwards are private, custom versions.

Hedging vs speculating:

hedging uses derivatives to cut risk, like a farmer selling wheat futures to lock in a price. Speculating bets on where prices will go.

Risk, return and leverage

• Risk and return go together: higher expected returns come with more risk. Short-term government bills are treated as the risk-free rate.
• Volatility is how much returns swing, measured by standard deviation.
• Diversification reduces company-specific (unsystematic) risk, but not market-wide (systematic) risk.
• Beta measures how a stock moves with the market. \(\beta=1\) moves with it, \(\beta=1.5\) swings 50% more, and \(\beta<1\) swings less.
• Short selling: sell borrowed shares, hoping to buy them back cheaper. The possible loss has no limit, because the price can keep rising.
• Buying on margin means borrowing from your broker to buy more. Leverage magnifies gains and losses. If your equity falls too far, you get a margin call to add money.

Statics^1,2

Basics

Statics:

the study of bodies that aren't accelerating, usually at rest.

Scalar vs vector:

a scalar has only a size (mass, temperature). A vector has a size and a direction (force, velocity).

Force:

a push or pull, measured in newtons: \(1\text{ N}=1\text{ kg}\cdot\text{m/s}^2\).

Weight:

the force of gravity on a mass, \(W=mg\). A 10 kg box weighs about 98 N.

Rigid body:

an object that doesn't deform under load. Statics problems usually assume this.

Free-body diagram:

a sketch of one body on its own, showing every external force and support reaction acting on it. Start every problem with one.

Newton's third law:

every force has an equal and opposite reaction on the other body.

Forces and components

• Split a force \(F\) at angle \(\theta\) from the \(x\)-axis into \(F_x=F\cos\theta\) and \(F_y=F\sin\theta\).
• To add forces, add their \(x\) parts and their \(y\) parts separately, then combine them into the resultant.

Equilibrium and moments

• Equilibrium: \(\sum F=0\) and \(\sum M=0\). In 2D that's three equations (\(\sum F_x=0\), \(\sum F_y=0\), \(\sum M=0\)), so you can solve for at most three unknowns.
• Moment: \(M=F\cdot d\), with \(d\) the perpendicular distance from the point to the force's line of action. Counterclockwise is usually positive.
• Take moments about a point where unknown forces act. Those forces drop out of the equation.
• A couple (two equal, opposite, offset forces) creates a pure moment and no net force. Its moment is the same about every point.
• A two-force member has forces only at two points, so the forces must be equal, opposite and along the line joining those points.
• A structure is statically indeterminate when it has more unknown reactions than equilibrium equations.

Friction

• Static friction holds a body still and adjusts up to a maximum, \(f\le\mu_s N\). Kinetic friction acts once it slides, \(f_k=\mu_k N\). Usually \(\mu_k<\mu_s\), which is why it's harder to start a push than to keep going.
• Friction opposes the motion (or the motion that's about to happen). In the simple model, it doesn't depend on contact area.
• A block about to slip on an incline has \(\mu_s=\tan\theta\).

Trusses and pulleys

• In an ideal truss, members are two-force members joined by pins, so each one is either in tension (pulled) or compression (pushed).
• Method of joints: apply equilibrium at each pin, starting at a joint with no more than two unknowns. Method of sections: cut through no more than three members and take moments.
• Zero-force members: at an unloaded joint with only two members that aren't in line, both carry zero force. At an unloaded joint with three members, two of them in line, the third carries zero force.
• An ideal pulley only changes the direction of a rope. The tension is the same all along one rope.
• A single movable pulley halves the force needed, because two rope segments hold up the load.

Distributed loads, centroids and moment of inertia

Distributed load:

a load spread along a length, in N/m. Replace it with one force equal to the area under the load diagram, acting at that area's centroid.

Centroid:

the geometric center of an area.

Center of gravity:

the point where the total weight acts. For a uniform material, it's at the centroid.

Area moment of inertia:

how well a cross-section resists bending. Area far from the axis counts most, which is why I-beams put most of their material in the flanges.

• Because \(I\) grows with \(h^3\), doubling a beam's depth makes it 8 times stiffer in bending, while doubling its width only doubles it.

Dynamics¹

Describing motion

Distance vs displacement:

distance is the total path length (a scalar). Displacement is the straight-line change in position, with a direction (a vector).

Speed vs velocity:

speed is how fast. Velocity is how fast and in which direction.

Acceleration:

the rate of change of velocity, in m/s². Speeding up, slowing down and turning are all acceleration.

Free fall:

motion under gravity alone, with \(a=g\approx9.8\) m/s² downward (ignoring air).

Relative velocity:

the velocity of A as seen from B, \(v_{A/B}=v_A-v_B\).

Projectile motion

• Horizontal and vertical motion are independent. Horizontal velocity stays constant, and vertical velocity changes by \(g\) each second.
• At the top of its path, a projectile's vertical velocity is zero. Its horizontal velocity isn't.
• On level ground, the range is greatest at a 45° launch angle, and angles that add to 90° (30° and 60°) give the same range.

Forces and Newton's laws

• First law (inertia): a body keeps its velocity unless a net force acts on it. Second law: \(F_{net}=ma\). Third law: forces come in equal, opposite pairs acting on different bodies.
• Mass (kg) is the amount of matter and doesn't change. Weight (N) is \(mg\) and depends on gravity.
• The normal force pushes perpendicular to a surface. Tension pulls along a rope.
• On an incline, gravity splits into \(mg\sin\theta\) along the slope and \(mg\cos\theta\) into it.
• In circular motion, a net force toward the center is needed (centripetal force). It isn't a new force: tension, friction, gravity or the normal force supplies it.

Work, energy and power

Work:

force times distance along the force, in joules. A force perpendicular to the motion does no work.

Work-energy theorem:

the net work on a body equals its change in kinetic energy.

Conservation of energy:

without friction, \(KE+PE\) stays constant. Friction turns mechanical energy into heat.

Power:

the rate of doing work, in watts (\(1\text{ W}=1\text{ J/s}\)).

Efficiency:

useful energy out divided by energy in. It's always below 100%.

Momentum and collisions

• Momentum is conserved in every collision. Kinetic energy is conserved only in elastic ones.
• Doubling speed doubles momentum but quadruples kinetic energy.
• Impulse is force times time. Airbags and crumple zones stretch out the stopping time, which lowers the force.

Rotation and oscillation

Angular velocity:

\(\omega\), in rad/s. A point at radius \(r\) moves at \(v=r\omega\).

Torque:

the turning effect of a force, \(\tau=rF\sin\theta\).

Moment of inertia:

\(I=\sum mr^2\), a body's resistance to angular acceleration. Mass far from the axis counts most.

Angular momentum:

\(L=I\omega\), conserved without outside torque. A skater pulling in their arms lowers \(I\), so they spin faster.

Simple harmonic motion:

back-and-forth motion where the restoring force is proportional to the displacement, like a mass on a spring. The period doesn't depend on how far it swings.

Electrical circuits^1,3

Basics

Charge:

measured in coulombs (C). One electron carries \(-1.6\times10^{-19}\) C.

Current:

the rate charge flows, in amperes (\(1\text{ A}=1\text{ C/s}\)). Conventional current flows from + to −, opposite to the electrons.

Voltage:

the energy per unit charge between two points (\(1\text{ V}=1\text{ J/C}\)). It's what pushes current through a circuit.

Resistance:

how much a component opposes current, in ohms (Ω). A longer or thinner wire has more.

Short vs open circuit:

a short is a path with almost no resistance, so current gets very large. An open circuit has a break, so no current flows.

Ground:

the reference point the circuit calls 0 V.

Meters:

a voltmeter goes in parallel and ideally draws no current. An ammeter goes in series and ideally has no resistance.

Series and parallel

• In series the current is the same through each part, and the voltages add. In parallel the voltage is the same across each branch, and the currents add.
• Adding a resistor in parallel always lowers the total resistance. The total is smaller than the smallest branch.
• Two equal resistors in parallel give half of one: \(R\parallel R=\tfrac{R}{2}\).

Circuit laws and equivalents

• Kirchhoff's current law: the current into a node equals the current out. Voltage law: the voltages around any loop sum to zero.
• Thevenin equivalent: \(V_{th}\) is the open-circuit voltage, and \(R_{th}\) is the resistance seen with independent sources turned off.
• Norton equivalent: a current source \(I_N=V_{th}/R_{th}\) in parallel with \(R_{th}\). It behaves the same as the Thevenin version.
• Maximum power transfer: a load gets the most power when its resistance equals \(R_{th}\).

Capacitors and inductors

Capacitor:

two conductors separated by an insulator. It stores energy in an electric field. Its voltage can't jump instantly, and once DC settles it acts like an open circuit.

Inductor:

a coil of wire. It stores energy in a magnetic field. Its current can't jump instantly, and once DC settles it acts like a plain wire.

Time constant:

how fast an RC or RL circuit responds. After one \(\tau\), a charging capacitor reaches about 63% of its final voltage. After about \(5\tau\) it's essentially full.

• Capacitors combine the opposite way to resistors: in parallel they add, in series they combine like resistors in parallel. Inductors combine like resistors.

AC circuits

Alternating current:

current that reverses direction in a sine wave. Homes in North America get 120 V at 60 Hz, and most of Europe 230 V at 50 Hz.

Period and frequency:

\(T=\tfrac1f\). Angular frequency is \(\omega=2\pi f\).

RMS value:

the DC value that would deliver the same power. Mains voltages are quoted as RMS.

Impedance:

\(Z\), the AC version of resistance, in ohms. Reactance is the part from capacitors and inductors.

• An inductor's reactance grows with frequency, so it blocks high frequencies. A capacitor's shrinks with frequency, so it blocks low frequencies and DC.
• In an inductor, voltage leads current by 90°. In a capacitor, current leads voltage by 90°.

Thermodynamics¹

Basics

Temperature:

a measure of the average kinetic energy of the particles.

Heat:

energy that moves because of a temperature difference, always from hot to cold on its own. A body doesn't "contain" heat. It has internal energy.

Zeroth law:

if A and B are each in thermal equilibrium with C, they're in equilibrium with each other. That's what makes thermometers work.

System types:

an open system exchanges mass and energy with its surroundings, a closed system only energy, an isolated system neither.

Specific heat:

the energy to warm 1 kg by 1 K. Water's is high, about 4186 J/(kg·K), so it heats and cools slowly.

Latent heat:

the energy to change phase (melt or boil) without changing temperature.

• During a phase change, added heat changes the internal energy, not the temperature.
• Gas law calculations always use kelvin, never Celsius.

The first law and gas processes

• First law: energy is conserved. Heat added to a system either raises its internal energy or comes out as work.
• For an ideal gas, internal energy depends only on temperature.
• On a pressure-volume graph, the work done by a gas is the area under its path.

The second law, engines and refrigerators

Second law:

the entropy of an isolated system never decreases. Heat never flows from cold to hot on its own, and no engine turns heat fully into work.¹

Entropy:

a measure of disorder, or of how many ways the energy can be spread out.

Heat engine:

takes heat \(Q_H\) from a hot source, turns part into work, and dumps the rest \(Q_C\) to a cold sink.

Refrigerator and heat pump:

use work to move heat from cold to hot. A fridge is rated by the heat it removes, a heat pump by the heat it delivers.

Carnot cycle:

an ideal, reversible cycle of two isothermal and two adiabatic steps. No engine between the same two temperatures can beat its efficiency.

Third law:

a perfect crystal at absolute zero has zero entropy, and absolute zero can't actually be reached.

Heat transfer

Conduction:

heat passing through a material by direct contact, mainly in solids.

Convection:

heat carried by a moving fluid (liquid or gas), like warm air rising.

Radiation:

heat carried by electromagnetic waves. It needs no medium, which is how the Sun heats the Earth.

Thermal conductivity:

\(k\), how easily a material conducts heat. Metals are high (copper is about 400 W/(m·K)). Air and foam are low, so they insulate.

• Doubling a wall's thickness halves the heat conducted through it. Doubling an object's absolute temperature multiplies its radiated power by 16.

Cell biology¹

Basics

Cell theory:

all living things are made of cells, the cell is the basic unit of life, and every cell comes from an existing cell.

Prokaryote:

a cell with no nucleus and no membrane-bound organelles. Bacteria and archaea.

Eukaryote:

a cell with a nucleus and membrane-bound organelles. Animals, plants, fungi and protists.

Organelle:

a structure inside a cell with a specific job, like a tiny organ.

Cytoplasm:

everything inside the membrane except the nucleus. The fluid part is the cytosol.

Surface area to volume:

as a cell grows, its volume grows faster than its surface. Small cells can move nutrients and waste across their membrane fast enough, which limits cell size.

Organelles

• Plant cells have a cell wall, chloroplasts and a large central vacuole. Animal cells have none of these, but they have centrioles.
• Endosymbiotic theory: mitochondria and chloroplasts came from bacteria that were swallowed by an ancestral cell. They still have their own circular DNA, small ribosomes and a double membrane.
• A protein made for export travels: ribosome on rough ER → Golgi → vesicle → cell membrane.

Molecules of life

Carbohydrates:

sugars and their chains, used for energy and structure. Glucose is a simple sugar. Starch (plants) and glycogen (animals) store energy. Cellulose builds plant cell walls.

Lipids:

fats, oils, phospholipids and steroids. They store energy long term and make up membranes.

Proteins:

chains of amino acids (20 kinds) joined by peptide bonds. Their folded shape decides their job.

Nucleic acids:

DNA and RNA, built from nucleotides. They store and use genetic information.

Enzyme:

a protein that speeds up a reaction by lowering its activation energy, without being used up. The substrate binds at the active site. Too much heat or the wrong pH denatures it (ruins its shape).

ATP:

the cell's energy currency. Breaking it into ADP and phosphate releases energy for cell work.

Membranes and transport

• Fluid mosaic model: the membrane is a phospholipid bilayer, with water-loving heads facing out and water-fearing tails facing in, and proteins floating in it.
• Small nonpolar molecules (O₂, CO₂) pass straight through. Ions and large polar molecules (glucose) need protein channels or carriers.
• Passive transport needs no energy and moves substances down their concentration gradient: simple diffusion, facilitated diffusion (through proteins) and osmosis.
• Osmosis: water moves across a membrane toward the side with more solute.
• Active transport moves substances against their gradient using ATP. The sodium-potassium pump moves 3 Na⁺ out and 2 K⁺ in for each ATP.
• Endocytosis brings large particles in by wrapping them in membrane. Exocytosis releases them out.

Cellular respiration and photosynthesis

• Aerobic respiration yields about 30–32 ATP per glucose. Older textbooks say 36–38.
• Fermentation runs without oxygen and makes only the 2 ATP from glycolysis. Muscles make lactic acid. Yeast makes ethanol and CO₂.
• Photosynthesis has two stages. The light-dependent reactions in the thylakoid membranes split water, release O₂ and make ATP and NADPH. The Calvin cycle in the stroma uses them to fix CO₂ into sugar, with the enzyme rubisco.
• The O₂ released by photosynthesis comes from water, not from CO₂.
• Chlorophyll absorbs mostly red and blue light and reflects green, which is why leaves look green.

The cell cycle and cell division

Interphase:

most of a cell's life. G1 (growth), S (DNA is copied), G2 (getting ready to divide).

Chromatid:

one of the two identical copies of a chromosome after S phase, joined at the centromere.

Homologous chromosomes:

a matching pair, one from each parent, carrying the same genes.

Diploid vs haploid:

diploid cells (2n) have both sets of chromosomes. Haploid cells (n), the gametes, have one set.

Cancer:

cells dividing out of control after the cell cycle's checkpoints fail.

• Mitosis stages (PMAT): prophase (chromosomes condense, spindle forms), metaphase (chromosomes line up in the middle), anaphase (sister chromatids pulled apart), telophase (two nuclei form).
• Cytokinesis then splits the cytoplasm: a pinching furrow in animal cells, a new cell plate in plant cells.
• Meiosis creates variety three ways: crossing over in prophase I, independent assortment of chromosome pairs, and random fertilization.
• Nondisjunction is when chromosomes fail to separate, giving a gamete an extra or missing chromosome. An extra chromosome 21 causes Down syndrome.

Genetics¹

DNA, genes and chromosomes

Nucleotide:

the building block of DNA and RNA: a sugar, a phosphate and a nitrogen base.

DNA:

a double helix of two antiparallel strands. The sugar-phosphate backbone is on the outside, and the bases pair up in the middle.

Gene:

a stretch of DNA that codes for a product, usually a protein.

Chromosome:

one long DNA molecule wrapped around proteins called histones. The DNA-protein mix is chromatin.

Genome:

all the DNA in an organism.

• DNA base pairing: A–T and G–C. In RNA, U replaces T.
• A and G (purines) have two rings. C, T and U (pyrimidines) have one. A G–C pair has 3 hydrogen bonds and an A–T pair has 2, so DNA rich in G–C is harder to separate.
• Chargaff's rule: in double-stranded DNA, A = T and G = C.
• Humans have 23 pairs of chromosomes: 22 pairs of autosomes and one pair of sex chromosomes (usually XX in females, XY in males).

DNA replication

• Replication is semi-conservative: each new double helix keeps one old strand. It happens in the S phase.
• Helicase unzips the helix. Primase lays down a short RNA primer. DNA polymerase adds nucleotides, only in the 5′ to 3′ direction, and proofreads.
• The leading strand is built in one piece. The lagging strand is built in short Okazaki fragments, which DNA ligase joins.

From gene to protein

• Central dogma: DNA → (transcription) → mRNA → (translation) → protein.
• Transcription: RNA polymerase reads the template strand and builds a matching mRNA. In eukaryotes it happens in the nucleus.
• RNA processing (eukaryotes): the mRNA gets a cap and a poly-A tail, and its introns are cut out. The exons that remain are joined and leave the nucleus.
• Translation: a ribosome reads the mRNA three bases (one codon) at a time. Each tRNA has an anticodon that matches a codon and carries one amino acid.
• There are 64 codons. AUG is the start codon (methionine), and 3 codons are stops (UAA, UAG, UGA). The other 61 code for only 20 amino acids, so most amino acids have several codons.
• Gene regulation: every body cell has the same DNA, but different cells switch on different genes. In bacteria, the lac operon turns on lactose-digesting genes only when lactose is around. In eukaryotes, transcription factors bind near genes to turn them on or off.

Mutations

Substitution:

one base swapped for another. Silent if the amino acid stays the same, missense if it changes (sickle-cell disease), nonsense if it creates an early stop codon.

Frameshift:

an insertion or deletion of bases that isn't a multiple of 3. It shifts every codon after it, usually wrecking the protein.

Chromosome mutations:

a whole section deleted, duplicated, flipped (inversion) or moved to another chromosome (translocation).

Mutagen:

something that causes mutations, like UV light or certain chemicals.

• Mutations are the original source of all genetic variation. Only those in gametes are passed on to offspring.

Inheritance

Allele:

a version of a gene. You get one allele of each gene from each parent.

Homozygous vs heterozygous:

two identical alleles (AA or aa) vs two different ones (Aa).

Dominant vs recessive:

a dominant allele shows with just one copy. A recessive allele shows only with two copies.

Genotype vs phenotype:

the genotype is the alleles (AA, Aa, aa). The phenotype is the observed trait.

Law of segregation:

the two alleles of a gene separate, so each gamete gets one.

Law of independent assortment:

genes on different chromosomes are inherited independently. Genes close together on the same chromosome (linked genes) tend to be inherited together.

Incomplete dominance:

the heterozygote is a blend, like red × white snapdragons giving pink.

Codominance:

both alleles show fully, like type AB blood.

Polygenic trait:

controlled by many genes, so it varies smoothly, like height.

Test cross:

crossing an individual that shows the dominant trait with a homozygous recessive one, to find out if it's AA or Aa.

• Sex-linked genes sit on the X chromosome. X-linked recessive traits (hemophilia, red-green color blindness) show up more often in males, who have only one X.
• A father passes his X to all his daughters and to none of his sons. A carrier mother passes the allele to half her children on average.
• For independent events, multiply probabilities ("and"). For events that can't happen together, add them ("or").

Genetic technology

PCR:

the polymerase chain reaction copies a DNA segment many times. Each cycle heats the DNA to about 95 °C to separate it, cools it so primers attach, then warms it to about 72 °C so a heat-proof polymerase builds the copies. Each cycle doubles the DNA.

Gel electrophoresis:

sorts DNA fragments by size. DNA is negatively charged, so it moves toward the positive end, and smaller fragments travel farther.

Restriction enzyme:

cuts DNA at a specific sequence. DNA ligase glues pieces together.

Recombinant DNA:

DNA combined from different sources, like a human insulin gene put into a bacterial plasmid so bacteria make insulin.

CRISPR-Cas9:

a guide RNA leads the Cas9 enzyme to a matching DNA sequence, where it cuts so the gene can be edited.

Atomic structure¹

• Atomic number \(Z\) = protons. Mass number \(A\) = protons + neutrons. A neutral atom has \(Z\) electrons.
• Isotopes have the same \(Z\) but different numbers of neutrons. Ions have gained or lost electrons.
• Electron configuration fills 1s, 2s, 2p, 3s, 3p, 4s, 3d… Valence electrons are those in the outermost shell.
• Quantum numbers: \(n\) (shell), \(l\) (subshell), \(m_l\) (orbital), \(m_s\) (spin).
• Periodic trends: atomic radius shrinks across a period and grows down a group. Ionization energy and electronegativity do the opposite. Fluorine is the most electronegative element.

Stoichiometry¹

• Balance equations so each element has the same number of atoms on both sides. Then the coefficients give mole ratios.
• The limiting reagent runs out first and sets the theoretical yield.
• An empirical formula is the simplest whole-number ratio (CH₂O). A molecular formula is the actual count (C₆H₁₂O₆).