The Science of Life Lecture Series

So, there has been a debate going around social media as of late.  It deals with the order of operations, and the fact that the vast majority of people seem to forget that each step of the order of operations ends with the phrase "from left to right".  That phrase is irrelevant most of the time, but when it's not irreverent, it's crucial. Damn you PEMDAS! So what is 9!/8!(3^2)?  Is it 81 or 1?  Or is it something else entirely?  Now I am going to tread lightly here so as to not enrage too many hearts. Before I answer this question, it would be a good idea to review the algebraic order of operations , or PEMDAS: Perform all operations inside parentheses using the remainder of the order of operations as your guide to the order.  Any parentheses within parentheses are to be performed from the inside out and from left to right. Perform any and all exponent from left to right. Perform all multiplication and division from left to right. Per...

When interpreting word problems as math equations, it is a good idea to keep in mind that certain phrases will always represent a certain mathematical operation.  Here is a list of some of these words for the basic operations. Addition Subtraction Multiplication Division Equals Exponent Parenthases Add Subtract Multiply Divide Is To the power of Quantity Sum Difference Product Quotient Yields Raised to the Increased by Decreased by Times As much Set as Plus Minus Double ($\times 2$) Halve ($\frac{x}{2}$) Equal to More Less Triple ($\times 3$) A third ($\frac{x}{3}$) Determined to be Combined Change Twice...

In the first section of this lecture series, I touched on positive integers as exponents.  Now I will delve further into the rules of exponents. Let's say that we're multiplying two exponents, $x^{5}$ and $x^{2}$, together.  Notice that $x^{2}=x \times x$ and $x^{5}=x \times x \times x \times x \times x$.  This means that if we multiply those two values together, we have $x^{2} \times x^{5}=(x \times x) \times (x \times x \times x \times x \times x)=x \times x \times x \times x \times x \times x \times x$.  This means that $x^{2} \times x^{5}=x^{7}$.  This means that when you're multiplying two exponents together, so long as the two bases are the same, all that needs to be done is adding the two exponents together, $x^{2} \times x^{5}=x^{5+2}=x^{7}$. We can use the same concept for division (quotients), $\frac{x^{5}}{x^{2}}$.  Notice that this yields $\frac{x^{5}}{x^{2}}=\frac{x \times x \times x...

Here, I will cover some of the basic operations and properties of real numbers. I say real numbers to differentiate from imaginary and complex numbers, both of which deal with the square root of -1, $\sqrt{-1}=i$. This course will not deal with the square root of -1, so all numbers will be real in this course. There are courses which deal with i, which typically have the word "Complex" in their name. The Absolute Value The absolute value (given by $\left | a \right |$) is the distance from the zero on the number line, regardless of the sign. This is to say $\left | -a |=\left | a |$.  The absolute value of -2 is 2, the absolute value of 2 is 2.  It does not matter what the initial value is, the sign after running it through the absolute value function will always, under every circumstance ever, be positive. Inequalities There are two types of inequality, less than and "greater than".  There are also two flavors of inequality, with and without the equal s...

In this first section from Chapter 1 , I'll cover some basics of Algebra, including basic terminology. Basic Terms: When a letter is used for a stand-in for a value which can take up any range of numbers, we call it a variable.  When a letter is used to represent a specific number (typically when we don't know what it is, but there can only be one), it is said to be a constant. Our Constants An example of this is the gross pay; the pay on your pay check before taxes is given by P G =H×W, where H is the hours worked and the W is your hourly wage.  Since hourly wage doesn't change from week to week, it is considered to be constant.  If you work part time, than your hours in any given week is considered a variable, because they do tend to change. An algebraic expression consists of a combination of constants, variables, and operation signs.  For example, H×W from the pay equation above is considered an expression.  When an equals sign is placed ...

Algebra is the mathematics of equalities and inequalities.  How much you'll spend at the grocery store, what you'll get on a paycheck after taxes, how much you'll owe or get back in taxes at the end of the year, how much money you'll have left at the end of the month after paying your bills; all of these problems and so many more are, by their very nature, algebraic problems. In this chapter, I'll cover some basics of Algebra and how to problem-solve with Algebra.  Problem solving can be applied to science, economics, and life.  After learning the algebraic problem solving techniques, you should try applying them to the specifics of your life. For those of you keeping track, here's the section breakdown: Section 1: Some Basics of Algebra Section 2: Operations and Properties of Real Numbers Section 3: Solving Equations Section 4: Introduction to Problem Solving Section 5: Formulas, Models, and Geometry Section 6: Properties of Exponents Sectio...