Numbers and Quantities

The Decimal System and Numbers up to 100

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The Number Line

How to compare the size of numbers

What Are Number Pairs to 10?

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What Are Even and Odd Numbers

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The Place Value System

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Sets of Numbers

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Set Notation and Intervals

Number Patterns

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Exploring Number Patterns

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Number Sequences and Patterns

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What are arithmetic sequences?

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Square numbers, cubic numbers, rectangular numbers, and triangular numbers

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Recursive and Explicit Formulas

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What Are Series in Math?

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Arithmetic Series

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How to Calculate Amortizing Loans with Series Formulas

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Geometric Series Formulas

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Infinite Geometric Series and Convergence Intervals

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How Is Fixed-rate Mortgage Calciulated?

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How to Calculate Present Values Using Timelines

How to Calculate Future Values Using Timelines

Examples of Calculations with Series

What Is Decimal Notation?

Which Number is In the Tenths Place

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Which Number Is in the Hundredths Place?

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What Does Negative Numbers Mean?

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How to Calculate with Negative Numbers

What Are Prime Numbers?

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Prime Factorization Using the Division Method

Prime Factorization Using a Factor Tree

Scientific Notation

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Calculate with Scientific Notation?

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Scientific Notation and Standard Notation

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What Are Complex Numbers?

What Does the Complex Plane Mean?

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What Is Polar Form for Complex Numbers?

What Is the Norm and the Argument of a Complex Number?

What Is Euler’s Formula for Complex Numbers?

What Are Complex Conjugates?

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Addition and Subtraction of Complex Numbers

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How to Multiply Complex Numbers

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How Can You Simplify Complex Fractions

What Are the Roots of a Complex Number?

What Is de Moivre's Formula and How Do You Use It?

How to Find the Distance Between Two Complex Numbers

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How to Find the Midpoint of Two Complex Numbers

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How to Solve Linear Equations with Complex Numbers

How to Solve Quadratic Equations with Complex Numbers

How to Factorize Complex Polynomials

What Is the Fundamental Theorem of Algebra?

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Addition with Help from the Number Line

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Subtraction with Help from the Number Line

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Addition and Subtraction (The Place-value System)

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How to Add Using the Column Method

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How to Subtract Using the Column Method

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Multiplication as Repeated Addition

The Times Tables

Which Products Give the Answers in the Times Tables?

How to Multiply by 10

How to Multiply by 100

How to Multiply by 1000

How Long Multiplication Works

What Is Doubling in Math?

What Does Halving Mean in Math?

Division as Inverse Multiplication

What Are the Divisibility Rules?

How to Divide by 10

How to Divide by 100

How to Divide by 1000

How to Do Long Division

Examples of Long Division

What Are Powers in Math?

What Are the Power Rules?

What Are Powers of 10?

How Square Roots Work

Square Roots of Products and Fractions

Order of Operations (MDAS)

Parentheses and Order of Operations

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Order of Operations (PEMDAS)

Signs in Math (Positive or Negative)

How to Do Mental Math (Rounding)

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How to Do Mental Math (Estimates)

How to Round a Number

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How to Make Estimates (Mental Math)

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How to Calculate the Cost per Unit

What Are Fractions?

What Is the Value of a Fraction?

Proper Fraction, Improper Fraction and Mixed Fraction

How Do You Expand Fractions?

How Do You Simplify Fractions?

How to Simplify a Fraction by Canceling

Add and Subtract Fractions with Equal Denominators

Add and Subtract Fractions with Different Denominators

Find the Common Denominator by Multiplying Denominators

Find the Common Denominator by Expanding or Simplifying

Find the Common Denominator Using the Times Tables

How to Find the Common Denominator Using Factorization

How to Multiply Fractions

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How to Multiply Fractions by Cross-canceling

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How to Divide Fractions

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What Does Percentage Mean?

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How Can You Calculate Percentages?

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How to Find Relative Percent Change

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How to Calculate Growth Factor

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How to Use Growth Factors in Calculations

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How Do Percentage Points Work?

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Convert Percentages to Fractions and Decimal Numbers

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Convert Fractions and Decimal Numbers to Percentages

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How to Convert Between Decimal Numbers and Fractions

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Converting Fractions, Decimal Numbers and Percentages

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What are the units of measurement in the metric system?

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What are the metric units for length?

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What are the metric units for area?

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What are the metric units for volume?

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What are the metric units for weight?

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What Are the Prefixed for SI Units?

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Volumetric Mass Density (Metric)

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What Is Temperature and What Are Degrees Celsius?

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How Do You Calculate Metric Units of Measurement?

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Choosing Appropriate Metric Units of Measurement

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What Are the Units of Measurement in the Imperial System?

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What Are the Imperial Units for Length?

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What Are the Imperial Units for Area?

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What Are the Imperial Units for Volume?

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What Are the Imperial Units for Weight?

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How to Find Volumetric Mass Density (Imperial)

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What Is Temperature and What Are Degrees Fahrenheit?

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How Do You Calculate Imperial Units of Measurement?

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How Do You Convert Imperial to Metric?

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How Do You Convert Metric to Imperial?

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The Digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

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How Do Significant Figures Work?

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Measurement Uncertainty Mean

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What Does Relative Measurement Uncertainty Mean?

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What Is Digital and Analog Clock?

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Learning About the Clock (Seconds)

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Learning About the Clock (Minutes)

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Learning About the Clock (Hours)

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Learning About the Clock (Whole and Half Hours)

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Learning About the Clock (Quarters of an Hour)

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Learning About the Clock (Ten To and Ten Past)

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Learning About the Clock (Five To and Five Past)

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Mental Arithmetic with Time

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Calculations with the Clock

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What Is Day and Night?

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What Are Days and Weeks?

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What Are Months?

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What Are the Seasons of the Year?

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What Is the ratio Between Two Numbers?

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What Does Mixing Ratio Mean?

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How to Find Distance if You Have Velocity and Time

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What Is a Speed Diagram?

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Coins of the United States Dollar

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Banknotes of the United States Dollar

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How Do You Count Money

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How Math Is Used in Shopping

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How Currency Exchange Works

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How Can Personal Finance Best Be Managed?

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What Are the Different Types of Income?

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How Does Commission Salary Work?

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How Does Tax and Tax Basis Work?

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How Do Bonus Payments Work?

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How Value-added Tax is Calculated

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How Credit Card Interest Rates Work

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Krone value and consumer price index

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What Are Price Index and Consumer Price Index?

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What Are Real Wages in Economics?

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How Do Loans Work?

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What Is an Amortizing Loan?

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Fixed-rate Mortgage

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How to Budget Your Money?

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How to Use Spreadsheets for Accounting

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Interest over one period

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Interest Over Several Periods

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How to Compute Compound Interest on Savings

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What Do Revenue, Cost, Unit Cost and Profit Mean?

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How Do Price and Demand Affect Revenue?

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Marginal Revenue, Marginal Cost and Marginal Profit

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How to Find Cost-minimizing Input

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How to Find Profit-maximizing Input

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What is a vector?

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What Does Vector Coordinates Mean

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Position Vector

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How Does Vector Addition Work

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Multiplying Vector by Scalar

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Vector Between Two Points

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Length of a Vector

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Distance Between Two Points Using Vectors

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Parallel Vectors

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Scalar Product of Two Vectors (2D)

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How Can You Find the angle Between Two Vectors?

Two vectors making up an angle

The angle between two vectors is always an angle $α \in [0 °, 180 °]$ .

Formula

The Angle Between Two Vectors

\cos α = \frac{\vec{u} \cdot \vec{v}}{| \vec{u} | \cdot | \vec{v} |}, α \in [0 °, 180 °]

Example 1

Find the angle between the vectors $(4, 1)$ and $(2, - 3)$ .

You insert the numbers into the formula, which gives you

\begin{array}{l} \cos α & = \frac{(4, 1) \cdot (2, - 3)}{| (4, 1) | \cdot | (2, - 3) |} \\ = \frac{8 - 3}{\sqrt{4^{2} + 1^{2}} \cdot \sqrt{2^{2} + {(- 3)}^{2}}} \\ = \frac{5}{\sqrt{17} \cdot \sqrt{13}} \\ \approx \frac{5}{14.86} \\ \approx 0.336 . \end{array}

That means the angle is

α = \cos^{- 1} (0.336) \approx 70.3 ° .

Done

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Medians of a Triangle

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How to Parametrize a Line?

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The dot product in three dimensions

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How To Calculate the Area of a Parallelogram with Vectors?

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How to Calculate the Area of a Triangle with Vectors

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The volume of a rectangular prism, pyramid, and tetrahedron

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208

What does Parametrization mean?

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Parametrization of Lines in Space

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Plane

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Angle between two lines

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Angle between a line and a plane

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Angle between two planes

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Distance from a point to a line

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Distance between lines

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Distance from a point to a plane

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Distance between a line and a plane

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Distance between two planes

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219

What Is The Equation for a Sphere

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How to Find the Surface Area of a Spherical Segment

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221

The Intersection Between Planes and the Coordinate Axes

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How to Find the Intersection Between Line and Plane

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How to Find the Intersection Between Two Planes

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Intersection Between a Tangent Plane and a Sphere

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