Course Review

Each place is worth ten times the one to its right: ones, tens, hundreds, thousands, ten-thousands, … Digits are grouped into periods of three (ones, thousands, millions) by commas.

To round to a place: look at the digit just to the right. If it is $5$ or more, round up; if $4$ or less, round down. Replace all digits to the right with zeros.

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• Parentheses / grouping symbols first.
• Exponents.
• Multiplication and Division, left to right (equal rank).
• Addition and Subtraction, left to right (equal rank).

In $b^n$, $b$ is the base and $n$ the exponent: $b^n=\underbrace{b\times b\times\cdots\times b}_{n\text{ times}}$. Any base to the first power is itself ($8^1=8$).

Perfect squares: $1,4,9,16,25,36,49,64,81,100,121,144$ (that is $1^2$ through $12^2$).

Integers are the whole numbers and their opposites: $\ldots,-3,-2,-1,0,1,2,3,\ldots$ On a number line, numbers increase to the right. So $-5<-2$ (farther left is smaller).

Absolute value $|a|$ is the distance from $0$, always $\ge 0$: $|-7|=7$ and $|7|=7$.

Adding: same signs $\to$ add and keep the sign; different signs $\to$ subtract and keep the sign of the larger absolute value.

Subtracting: add the opposite: $a-b=a+(-b)$.

Multiplying / Dividing: same signs $\to$ positive; different signs $\to$ negative.

A factor divides a number evenly; a multiple is what you get counting by that number. A prime has exactly two factors ($1$ and itself): $2,3,5,7,11,13,17,19,23,\ldots$ A composite has more than two factors. ($1$ is neither.)

Prime factorization writes a number as a product of primes (use a factor tree): $60=2^2\times3\times5$.

GCF (greatest common factor): the largest factor two numbers share --- multiply the primes they have in common. LCM (least common multiple): the smallest multiple they share --- take each prime to its highest power.

Useful identity: $\text{GCF}(a,b)\times\text{LCM}(a,b)=a\times b$.

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• 2: last digit even.
• 3: digit sum divisible by $3$.
• 4: last two digits divisible by $4$.
• 5: ends in $0$ or $5$.
• 6: divisible by $2$ and $3$.
• 9: digit sum divisible by $9$.
• 10: ends in $0$.

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Multiply or divide top and bottom by the same number for an equivalent fraction. Simplify by dividing both by their GCF. To compare, use a common denominator or cross-multiply: $\tfrac{a}{b}$ vs $\tfrac{c}{d}$ compare $a\cdot d$ vs $b\cdot c$.

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• Add / Subtract: need a common denominator, then add/subtract numerators: $\dfrac{a}{c}\pm\dfrac{b}{c}=\dfrac{a\pm b}{c}$.
• Multiply: straight across: $\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{ac}{bd}$.
• Divide: multiply by the reciprocal: $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}$.

Mixed $\to$ improper: $a\tfrac{b}{c}=\dfrac{a\cdot c+b}{c}$.   Improper $\to$ mixed: divide; quotient is the whole part, remainder over the divisor.

Places after the point: tenths, hundredths, thousandths, …

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• Add / Subtract: line up the decimal points, then compute.
• Multiply: ignore points, multiply, then place the point so the product has as many decimal places as both factors combined.
• Divide: move the divisor's point to make it whole; move the dividend's point the same number of places; then divide.

Fraction $\to$ decimal: divide numerator by denominator. Decimal $\to$ fraction: write over its place value, then simplify ($0.75=\tfrac{75}{100}=\tfrac{3}{4}$).

A decimal terminates if the simplified denominator has only factors of $2$ and/or $5$; otherwise it repeats. Repeating: $0.\overline{3}=\tfrac13$, shown with a bar over the repeating block.

A ratio compares two quantities: $a$ to $b$, $a{:}b$, or $\tfrac{a}{b}$. A rate compares different units (miles per hour). A unit rate has a denominator of $1$ --- divide to find “per one” ($$3$ for $2$ lb $=$ $$1.50$ per lb).

A proportion states two ratios are equal: $\dfrac{a}{b}=\dfrac{c}{d}$. Solve by cross-multiplying: $a\cdot d=b\cdot c$, then isolate the unknown.

Scale & similar figures: similar figures have equal corresponding angles and proportional sides. Set up matching sides as a proportion to find a missing length. A scale (e.g. $1\text{ in}:50\text{ mi}$) is a ratio between a drawing and reality.

Percent means “per $100$.” Percent $\to$ decimal: divide by $100$ (move point $2$ left): $45\%=0.45$. Decimal $\to$ percent: multiply by $100$. Percent $\to$ fraction: write over $100$ and simplify: $45\%=\tfrac{45}{100}=\tfrac{9}{20}$.

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• Percent of a number: $\text{part}=\text{percent(as decimal)}\times\text{whole}$.
• Percent equation: $\dfrac{\text{part}}{\text{whole}}=\dfrac{\%}{100}$.
• Percent change: $\dfrac{\text{new}-\text{old}}{\text{old}}\times100\%$ (increase if $+$, decrease if $-$).
• Tax / tip: add $\%$ of price. Discount: subtract $\%$ of price.
• Simple interest: $I=Prt$ ($P=$ principal, $r=$ rate as decimal, $t=$ years).

A rational number can be written as a fraction $\dfrac{a}{b}$ of integers with $b\ne0$. This includes integers, fractions, terminating decimals, and repeating decimals. Operations follow the fraction/decimal rules plus the integer sign rules.

Write the fraction in lowest terms and look at the denominator:

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• Only factors $2$ and/or $5$ $\Rightarrow$ terminates (e.g. $\tfrac{7}{20}=0.35$).
• Any other prime factor $\Rightarrow$ repeats (e.g. $\tfrac{5}{6}=0.8\overline{3}$).

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• Product: $a^m\cdot a^n=a^{m+n}$
• Quotient: $\dfrac{a^m}{a^n}=a^{m-n}$
• Power of a power: $(a^m)^n=a^{mn}$
• Power of a product: $(ab)^n=a^n b^n$
• Zero: $a^0=1$ ($a\ne0$)
• Negative: $a^{-n}=\dfrac{1}{a^n}$

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Write as $a\times10^n$ where $1\le a<10$ and $n$ is an integer. Large numbers use positive $n$; small numbers (less than $1$) use negative $n$.

$\sqrt{a}$ asks “what number squared gives $a$?” Squaring and square-rooting undo each other: $\sqrt{49}=7$ since $7^2=49$. Perfect-square roots are whole numbers; others are irrational (e.g. $\sqrt{2}\approx1.41$) and can be estimated between the two nearest perfect squares.

A variable is a letter standing for an unknown number. A term is a number, a variable, or their product (e.g. $3x$); the number is the coefficient. Evaluate an expression by substituting values for the variables and using PEMDAS.

Distributive: $a(b+c)=ab+ac$ (multiply the outside by each inside term).

Like terms have the same variable(s) to the same power; combine them by adding coefficients: $5x+3x=8x$, but $5x+3x^2$ cannot combine.

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• sum / more than $\to +$
• difference / less than $\to -$
• product / of / times $\to \times$
• quotient / per $\to \div$
• is / equals $\to =$

multicols “$7$ less than $x$” is $x-7$ (order flips).

An equation is a balance. Use inverse operations to isolate the variable, doing the same thing to both sides. Undo in reverse PEMDAS order: addition/subtraction first, then multiplication/division.

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• One-step: $x+5=12\Rightarrow x=7$.
• Two-step: $2x+3=11\Rightarrow 2x=8\Rightarrow x=4$.

Solve like equations, but the sign shows a range of solutions ($<,\le,>,\ge$). Key rule: when you multiply or divide both sides by a negative number, flip the inequality sign.

Graph on a number line: open circle for $<$ or $>$, closed circle for $\le$ or $\ge$.

The plane is formed by a horizontal $x$-axis and vertical $y$-axis meeting at the origin $(0,0)$. An ordered pair $(x,y)$ gives horizontal then vertical position (over, then up/down).

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• QI $(+,+)$   QII $(-,+)$
• QIII $(-,-)$   QIV $(+,-)$

Points on an axis are in no quadrant.

A rule like $y=2x+1$ pairs each $x$ with a $y$. Make a table, plot the $(x,y)$ pairs, and connect them --- a straight-line rule graphs as a line.

Measured in degrees. Acute $<90^\circ$; right $=90^\circ$; obtuse between $90^\circ$ and $180^\circ$; straight $=180^\circ$. Complementary angles sum to $90^\circ$; supplementary sum to $180^\circ$. Angles in a triangle sum to $180^\circ$.

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• Rectangle: $P=2l+2w$, $A=lw$
• Square: $P=4s$, $A=s^2$
• Triangle: $A=\tfrac12 bh$
• Parallelogram: $A=bh$
• Trapezoid: $A=\tfrac12(b_1+b_2)h$
• Circle: $C=2\pi r=\pi d$, $A=\pi r^2$

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For a right triangle with legs $a,b$ and hypotenuse $c$ (the side opposite the right angle):

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• Rect. prism: $V=lwh$
• Cube: $V=s^3$
• Prism: $V=(\text{base area})\times h$
• Cylinder: $V=\pi r^2 h$
• Rect. prism surface: $SA=2(lw+lh+wh)$
• Cylinder surface: $SA=2\pi r^2+2\pi rh$

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• Mean (average): $\dfrac{\text{sum of values}}{\text{number of values}}$.
• Median: middle value when ordered (average the two middle values if there is an even count).
• Mode: value that appears most often (can be none or several).
• Range: $\text{max}-\text{min}$.

Probability of an event: $P(E)=\dfrac{\text{favorable outcomes}}{\text{total outcomes}}$, a value from $0$ (impossible) to $1$ (certain). Complement: $P(\text{not }E)=1-P(E)$.

Counting principle: if one choice has $m$ options and another has $n$, together there are $m\times n$ outcomes. Independent events: $P(A\text{ and }B)=P(A)\times P(B)$.