The Science of Life Lecture Series

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  In this entry, we move to the definition of an algebraic function. I feel it is a good idea to start with the definitions of domain and range.  The domain is defined as the set of all possible values for which the independent variable (the input, the x-value) can take up.  This is all possible values for the horizontal (x) axis.  The range is the set of all possible values for which the dependent variable (the output, the y-value) can take up.  This is all possible values for the vertical (y) axis. The blue horizontal axis (typically the x-axis) is the domain and the independent variable. The green vertical axis (typically the y-axis) is the range and the dependent variable. Now we move to the concept of a function.  A good definition of the function is any equation where each input value has exactly one output value.  The technical version of this is "any eq...

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...

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...

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...