Precalculus Study Notes: Sets of Numbers, Order of Operations, and Scientific Notation

Numbers, Data, and Problem Solving

Objectives

• Identify different sets of numbers.

• Apply order of operations.

• Convert standard notation into scientific notation and vice versa.

Sets of Numbers

Classification of Numbers

Numbers can be classified into several sets based on their properties. Understanding these sets is fundamental in precalculus and higher mathematics.

• Real Numbers (ℝ): The set of real numbers includes all rational and irrational numbers. Real numbers can be represented on the number line.

• Rational Numbers (ℚ): A rational number is any number that can be written as a ratio of two integers, that is, in the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. Rational numbers include all integers, as well as repeating and terminating decimals.

• Irrational Numbers: An irrational number cannot be written as a ratio of two integers. Its decimal expansion is non-repeating and non-terminating. Examples include $\sqrt{2}$ and $\pi$.

• Integers (ℤ): The set of integers includes all whole numbers and their negatives, as well as zero. That is, $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.

• Whole Numbers: Whole numbers include all natural numbers and zero. That is, $\{0, 1, 2, 3, ...\}$.

• Natural Numbers (ℕ): Also known as counting numbers, these are the numbers used for counting: $\{1, 2, 3, ...\}$.

Example: Classify the following real numbers

• 5 (Natural, Whole, Integer, Rational, Real)

• 1.2 (Rational, Real)

• 13/7 (Rational, Real)

• $\sqrt{15}$ (Irrational, Real)

• $-\sqrt{9}$ (Integer, Rational, Real)

Order of Operations

PEMDAS: Please Excuse My Dear Aunt Sally

To evaluate mathematical expressions correctly, follow the order of operations, often remembered by the acronym PEMDAS:

• P - Parentheses and grouping symbols: ( ), [ ], { }

• E - Exponents (and roots)

• M/D - Multiply and Divide (from left to right)

• A/S - Add and Subtract (from left to right)

Examples: Evaluate the following

• 1) $15 \div 6 + 3 - 2 \times 5$

• 2) $3(1 - 5)^2 - 4^2$

• 3) $\frac{10 - 6}{5 - 3} - 4 |2 - 7|$

Note: Always perform operations inside grouping symbols first, then exponents, followed by multiplication/division, and finally addition/subtraction.

Scientific Notation

Definition and Purpose

Scientific notation is a way to express very large or very small numbers in the form $c \times 10^n$, where $1 \leq |c| < 10$ and $n$ is an integer.

Examples

• The distance to the sun is 93,000,000 miles. In scientific notation: $9.3 \times 10^7$ miles.

• The size of a typical virus is 0.000005 cm. In scientific notation: $5 \times 10^{-6}$ cm.

Convert the following to scientific notation:

• 1) 45,000,000 → $4.5 \times 10^7$

• 2) 160 → $1.6 \times 10^2$

• 3) 0.00235 → $2.35 \times 10^{-3}$

• 4) 0.0000000091 → $9.1 \times 10^{-9}$

Operations with Scientific Notation

When multiplying or dividing numbers in scientific notation, use the following rules:

• Multiplication: Multiply the coefficients and add the exponents. $(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$

• Division: Divide the coefficients and subtract the exponents. $\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$

Examples: Evaluate and express in scientific notation

• 1) $\frac{3 \times 10^7}{2 \times 10^4} = 1.5 \times 10^3$

• 2) $(5 \times 10^3) \times (6 \times 10^4) = 30 \times 10^7 = 3.0 \times 10^8$

• 3) $\frac{4.6 \times 10^{-1}}{2 \times 10^2} = 2.3 \times 10^{-3}$

Summary Table: Sets of Numbers

Set	Symbol	Definition	Examples
Natural Numbers	ℕ	Counting numbers starting from 1	1, 2, 3, ...
Whole Numbers		Natural numbers and 0	0, 1, 2, 3, ...
Integers	ℤ	Whole numbers and their negatives	..., -2, -1, 0, 1, 2, ...
Rational Numbers	ℚ	Numbers expressible as $\frac{a}{b}$, $b \neq 0$	1/2, -3, 0.75, 5
Irrational Numbers		Non-repeating, non-terminating decimals	$\sqrt{2}$, $\pi$
Real Numbers	ℝ	All rational and irrational numbers	All above