1.6 Chapter 1 Study Guide

Definitions

• natural numbers ($\mathbb{N}$): The "counting" numbers, nonnegative whole numbers, aka $0, 1, 2, 3, ...$ forever.
• integers ($\mathbb{Z}$): All of the whole numbers, including negative numbers. These also go on forever but in both directions, $..., -3, -2, -1, 0, 1, 2, 3, ...$.
• rational numbers ($\mathbb{Q}$): All numbers that can be written as a ratio of two integers, so things like $\frac{1}{2}, -\frac{7}{5}, \frac{27}{4001},$ etc. These include integers, because we can write $4 = \frac{4}{1}$ if we want. Decimal representations of rational numbers either end, like $1.5$, or repeat forever, like $1.33333...$.
• real numbers ($\mathbb{R})$: All of the above numbers, together with irrational numbers, such as $\sqrt{2}, \sqrt{5}, \pi, e,$ etc. These numbers' decimal representations never end or have a repeating pattern.
• commutative: An operation is commutative if it doesn't matter what order you do it in. Addition and multiplication are commutative since $a+b = b+a$ and $ab = ba$, but subtraction and division aren't!
• factor: A number's factors are the numbers that can be multiplied to produce it. For example, factors of $12$ are $\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$.
• sum: the result of an addition.
• difference: the result of a subtraction.
• product: the result of a multiplication.
• quotient: the result of a division. The number you divide BY is the divisor. The number getting divided is the dividend.
• strictly less than / less than or equal to: symbols are $<$ and $\leq$ respectively. For example, $3 < \pi$.
• strictly greater than / greater than or equal to: symbols are $>$ and $\geq$ respectively. For example, if we say $x \geq 4$, that means $x$ could be equal to 4 or anything larger than that.
• numerator / denominator: The top of a fraction is the numerator. The bottom of a fraction is the denominator.
• reciprocal: The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$.
• compound fraction: A fraction that has fractions inside of its numerator or denominator or both.
• least common multiple (LCM): The smallest number that all of a set of numbers divide evenly into. Example: the LCM of $3, 4,$ and $6$ is $12$. This is used when getting a common denominator.
• percent: a proportion out of 100. Example: 20% is $20/100 = \frac{1}{5} = 0.2$.
• power / exponent: $a^k$ is "$a$ raised to the power $k$" and can be thought of as multiplying $a$ by itself a total of $k$ times. The base is $a$, and the exponent is $k$.
• radical / root: $\sqrt{a} = b$ means $b$ is the positive number such that $b^2 = a$. Similarly, $\sqrt[3]{a} = b$ means $b^3 = a$, and so on.
• scientific notation: a number is in scientific notation if it is written as X.XXXX times an appropriate power of 10. Examples: $1,234,000 = 1.234 \times 10^6$ and $0.00001234 = 1.234 \times 10^{-5}$. If the exponent is positive, it's a large number. If the exponent is negative, it's a tiny number.

Arithmetic With Negatives

• Subtraction is the same as adding a negative number, and adding negative numbers is the same as taking the difference (the sign of the answer matches the sign of the bigger number). For example, $2 - 3 = 2 + (-3) = -1$.
• Multiplication distributes over sums and differences, like $2(1+3) = 2\cdot 1 + 2 \cdot 3 = 2+6 = 8$, but you have to be careful when distributing a negative number. For example, $-3(a+b) = -3a - 3b$.
• Positive $\times$ positive = positive. Negative $\times$ negative = positive. Positive $\times$ negative = negative. The same applies to division!

Order of Operations (PEMDAS)

0. Parentheses first. Working from the innermost outward, completely take care of expressions inside parentheses.

1. Exponents (powers). If numbers are being raised to a power, compute those first.

2. Multiplication/Division. Working left to right, compute any multiplications and divisions.

3. Addition/Subtraction. Working left to right, compute any sums and differences.

Example:

$$2(3+(4-2)^2) - \frac{(3+1)}{4} = 2( 3+ (2)^2) - \frac{4}{4} = 2(3+4) - \frac{4}{4} = 2(7) - \frac{4}{4} = 14 - 1 = 13 \notag$$

Arithmetic With Fractions

• To multiply fractions, multiply across the top and multiply across the bottom: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ab}{cd}$.
• To divide fractions, multiply by the reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$.
• To add or subtract fractions, get a common denominator for all fractions, and then add or subtract (using parentheses if necessary).

Place Values in Decimals

When rounding to a certain place, look at the digit directly to the right. If it's less than 5, round down. If it's 5 and up, round up.

Arithmetic With Decimals

• Adding and subtracting: arrange vertically with decimal points lined up and add/subtract down columns as usual:

• Multiplying: ignore the decimal points and multiply the numbers, then count up the total number of decimal places and put the new point that far into the product. Example: for $1.5 \cdot 0.4$, first multiply $15 \cdot 4 = 60$, then each number had one decimal place, so put the point two places in to get $0.60$ as the answer.
• Powers of 10 trick: If multiplying by $10^k$, move the decimal point $k$ steps to the right, adding zeros into the spaces. Example: $4.2 \cdot 10^3 = 4200$. If dividing by $10^k$ (aka multiplying by $10^{-k}$), move the decimal point $k$ steps to the left, adding zeros if needed. Example: $4.2 \cdot 10^{-3} = 0.0042$.

Equivalent Fractions/Decimals to Know

Fraction	Decimal
$\frac{1}{2}$	$0.5$
$\frac{1}{3}$	$0.\overline{3}$, which means the 3s go on forever like $0.33333...$
$\frac{2}{3}$	$0.\overline{6}$, which due to rounding rules rounds to things like $0.6667$
$\frac{1}{4}$	$0.25$, like how 25 cents is a quarter of a dollar
$\frac{3}{4}$	$0.75$
$\frac{1}{5}$	$0.2$, which you can remember as $0.20$ because 20 is one-fifth of a 100
$\frac{1}{8}$	$0.125$
$\frac{3}{2}, \frac{5}{2}, \frac{7}{2}, ...$	$1.5 , 2.5, 3.5, ...$, which are just like finding mixed numbers from improper fractions

Percents

• To find XX% of a quantity, convert to a decimal by moving the point two steps to the left, getting 0.XX, and then multiply. Example: 10% of 30 would be $0.1 (30) = 3$.
• To convert from a ratio to a percent, just divide $\frac{\text{part}}{\text{whole}}$ and multiply by 100. Example: 2 out of 5 students come to office hours means $\frac{2}{5}\cdot 100 = 0.4 \cdot 100 = 40$% of students come to office hours.

Power Rules

$a^b a^c = a^{b+c}$	$\dfrac{a^b}{a^c} = a^b a^{-c} = a^{b-c}$	$(a^b)^c = a^{bc}$	$a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$
$a^0 = 1$	$a^1 = a$	$0^n = 0$	$1^n = 1$
$(ab)^c = a^c b^c$	$\textcolor{red}{(a+b)^c \neq a^c + b^c}$	$\text{negative}^{\text{even power}} = \text{positive}$	$\text{negative}^{\text{odd power}} = \text{negative}$

Root Rules

$\sqrt[n]{0} = 0$	$\sqrt[n]{1} = 1$	$( \sqrt[n]{a})^n = a$	$\sqrt[n]{a} = a^{\frac{1}{n}}$
$(\sqrt[n]{a})^m = \sqrt[n]{a^m} = a^{\frac{m}{n}}$	$\sqrt[n]{ab}= \sqrt[n]{a}\sqrt[n]{b}$	$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$	$\textcolor{red}{ \sqrt[n]{a+b} \neq \sqrt[n]{a} + \sqrt[n]{b} }$

Common Roots to Know

$\sqrt{4} = 2$	$\sqrt{400} = 20$
$\sqrt{9} = 3$	$\sqrt{3600} = 60$, etc.
$\sqrt{16} = 4$	$\sqrt{10000} = 100$, etc.
$\sqrt{25} = 5$	$\sqrt[3]{8} = 2$
$\sqrt{36} = 6$	$\sqrt[3]{27} = 3$
$\sqrt{49} = 7$	$\sqrt[3]{64} = 4$
$\sqrt{64} = 8$	$\sqrt[3]{125}= 5$
$\sqrt{81} = 9$	$\sqrt[3]{1000} = 10$
$\sqrt{100} = 10$	$\sqrt[4]{16} = 2$
$\sqrt{121} = 11$	$\sqrt[4]{81} = 3$
$\sqrt{144} = 12$	$\sqrt[4]{10000} = 10$

Comparing Fractions

• If two fractions have the same denominator, compare the numerators. The bigger numerator gives a larger fraction. Example: $\frac{2}{3} > \frac{1}{3}$.
• If they have the same numerator, compare the denominators. The bigger denominator gives a smaller fraction. Example: $\frac{1}{4} < \frac{1}{2}$.
• If they have different numerators and denominators, find a common denominator and then compare the new numerators. Example: $\frac{5}{4} = \frac{15}{12}$ is smaller than $\frac{4}{3} = \frac{16}{12}$.