The Science of Life Lecture Series

In science, there are many numbers which are either exceedingly large (for example, the number of water molecules in a liter of water) or exceedingly small (for example, the charge, in Coulombs, of a single electron).  This is the nature of the reality we live in.  This is why, in some cases, it is easier to read and right these extremely small and large values in a more compact notation.  Yes, this does somewhat reduce accuracy, but the loss of accuracy has very little baring on the accuracy of the final answer.  The most common compact notation in Social Sciences, Life Sciences, and Physical Sciences is called Scientific Notation . The notation for Scientific Notation is $N \times 10^{n}$, where N is a number which is at least 1 but less than 10 $[1 \le N < 10]$ and n is the magnitude of the number.  There are a couple steps to convert to scientific notation: Place a decimal point in between the first two significant digits of the number. Find the m...

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...

This is a natural continuation of Chapter 1 Section 2 .  When we make models, we need to make sure that, when we run the numbers, the dimensions of the solution naturally make sense.  The word "dimension" in physics means something more general than the definition we are used to here in every day life.  We typically think of dimensions as length, width, and height; we do live in a three dimensional world, after all.  The word dimension in science means the units that we are working with.  This could be Distance (like length of a box), or is could be time (duration of a trip), or it could be mass (I weigh 15 stone).  The dimensions of density are kilograms per cubic meter (kg/m 3 ). When these dimensions are used in mathematical equations, they are treated as if they are variables we've seen in Algebra ; they can be canceled and squared like any other value in mathematics.  For example, if we have a room that is 20ft.×30ft.×15ft, then we multiply...