The Science of Life Lecture Series

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 describe some of the basics of algebraic manipulation.  First, we need to define a formula, which is any equation which uses letters to represent relationships between quantities.  An example of this is the area of a circle, $A=\pi r^{2}$.  You'll remember from Section 3 that we can solve for a variable.  The formula I used was the one relating masses of two object, their distance, a constant, and the Gravitational force. To solve a formula for a particular variable or constant, perform the following steps: Multiply both sides by any denominator there happens to be in order to cancel out any denominator and clear any fractions.  Combine like terms. Using the addition principle, get every term with the variable to be solved for on one side of the equals sign and every other term on the other side of the equals sign.  Combine like terms again if necessary, which may include factoring. Solve for the variable by using the multiplicatio...

Hello internet, and welcome to the Algebra Lecture Series from the Science of Life.  This entry is an introduction to algebraic problem solving.  In Algebra, there are five steps for problem solving, which I'll cover here. Familiarize yourself with the problem situation.  This is basically a situation of knowing what the context of the problem is.  Is the context taxes?  Grocery shopping?  Tipping?  Scaling up or down the ingredients of a cake to get a different size cake than what the recipe says the size will be?  Here are the sub-steps for this contexts step: Read the problem carefully.  Read it aloud if need be to understand the problem. List the information and state the question being asked.  Select variables to represent anything which is unknown and clearly state what the variables represent, and be descriptive about the statement. Obtain any relevant information and equations.  If you're painting a room, find the ...

In this third section of Chapter 1 , I will cover the basics of equation solving.  I've already touched on this in Physics , and will get to in in Chemistry; this is vital for all of math and science, including Economics. Equivalent Forms One of the most vital statements in Algebra is that two equations which have the same solutions are equations.  In all future endeavors, both science and in math, we look to solve for the variable that is unknown, and the final form of that equation doesn't look anything like the initial equation, but is in an equivalent form, so has all of the same solutions as the initial equations.  For any real numbers a, b, c, and d, $a+c=b+c$ is equivalent to $a=b$ and $a=b$ is equivalent to $a \times d=b \times d$.  What we can do from here is say that $a+c=b+c$ is equivalent to $a \times d=b \times d$. Like Terms In any equation, a "term" is any constant, variable, product of any number of constants and/or variables, or the qu...

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...