Intro to Phys Sci I Mechanics Heat Instructor Carlos Perez Math review Even if this class does not have a heavy math component in order to understand physics concepts we will still need to use it Here s a refresher of the math tools we will use the most Make sure you understand what is written below you ll need it You can keep this document at hand for future use If you need additional help with this math review you can request o ce hours with the instructor or the LA They will be glad to help 1 Fractions Remember that a fraction a b has the number on top a called the numerator and the number on the bottom b called the denominator The denominator must be di erent from zero b cid 54 0 Some examples Any number can be written in fraction form by using 1 as the denominator This can always be done if needed For example Now this ought to remind you how to operate fractions 1 To multiply two fractions multiply the tops together and the bottoms together For example With numbers With numbers 2 To divide fractions ip the bottom fraction over and then multiply 2 5 7 3 9 14 2 1 2 5 7 3 9 14 2 1 2 5 9 14 3 7 1 2 6 35 9 1 14 2 9 28 9 14 2 3 To add or subtract fractions only like quantities i e fractions that have the same denominator can be added or subtracted You just simply operate the top numbers accordingly where the symbol represents addition subtraction For instance a c b c a b c 8 3 4 3 8 4 3 4 3 1 2 17 101 500 37 3 6 6 1 489 489 1 1 a b c d a c b d 5 7 3 2 6 1 15 14 47 3 3 1 6 47 3 47 3 282 3 94 a b g h a b g h a b h g a h b g Unlike quantities i e fractions that have di erent denominators must be rst converted to like quantities An easy way to convert the fractions to like quantities is to multiply together the two denominators of each fraction in case of an integer number apply the invisible denominator 1 Remember that the numerator will also change accordingly This method can be expressed algebraically The nal step is to simplify the resulting fraction Let s see how this works with two examples a b c d ad bc bd 3 4 7 2 3 2 7 4 4 2 34 8 5 3 5 6 5 6 5 3 3 2 15 6 17 2 4 2 5 3 2 3 17 4 5 2 2 Powers and roots Here is a quick reminder of how to deal with numbers and symbols raised to a certain power For the following explanations we will consider x to be any real number and we will call it the base We will also denote two other real numbers by m and n 1 Multiplication xm xn xm n Always make sure it s the same base number With numbers 32 33 35 243 2 Division xn xm n The number 1 xn can also be expressed as x n This means xm consistent with what we just state about products of the same base With numbers xm xn xm x n xm n xm n 27 23 27 2 3 27 3 27 3 24 16 3 The nth root of a number can be expressed by using exponents For example let p be a positive integer Maybe it will make a little more sense with actual numbers 4 Powers of powers xm n xm n Example Equivalently x1 p p x 81 3 3 8 2 23 2 8 2 64 23 2 23 2 26 64 5 Special case for any number x di erent from zero we have that x0 1 6 Even power of negative number x 2 x2 e g 3 2 32 9 This is true for any even power BE CAREFUL 32 9 because the even power does not a ect the negative sign if there are no parentheses This is true for any base and for any even power 7 Odd power of negative number x 3 x3 e g 2 3 23 8 This is true for any odd power and in this case it is the same if you use parentheses or not 2 In this course we will use words such as expression equation formula and function Some people might think they have similar meanings but each term is a di erent mathematical concept Let s take a quick look at these meanings and some examples 3 Terminology 3 1 Expressions An expression is a collection or combination of letters numbers and operators An expression does not involve the equal sign or any symbol for inequalities such as etc Examples of expressions are 17 5x y m n 3 23 3 x y It is also important to know how to give a worded description of expressions For instance the rst expression 17 5x y reads seventeen point ve times x plus pi times y 3 2 Equations An equation is a mathematical relation between two expressions The relation is that of equality symbolized by equal to This means that both sides have exactly the same value Examples 3x 2y 17 5x y 21x4 3x2 9x2 x 23 3 cid 96 mn2 4 5 4 5 As an example the last equation 3 cid 96 mn2 reads Three times cid 96 equals four fths of m times n squared To solve an equation means to nd the values of the unknown quantities x y m etc such that one or more equalities are satis ed For this course we will need to know how to solve certain types of equations Check the Solving Equations section for a review 3 3 Formulas A formula is a special type of equation Formulas or formulae express how one variable is related to other vari ables Because of this you need at least two di erent symbols to represent the variables in a formula Formulas usually have speci c meanings in math and science Let s see some examples To calculate the volume V of a sphere in terms of its radius r we can use the formula V r3 We can use 4 3 this formula to obtain the volume for di erent values of r 1 2 3 4 Where the symbol means we are approximating the resulting volume to one decimal place To calculate the area A of a triangle in terms of its base b and height h the formula is A description of this formula is the area is equal to one half of the base times the height 1 2 bh A worded r3 4 Radius r Volume V 3 V 20 6 V 1317 0 V 8 38 V 659 6 1 7 6 8 3 2 5 4 3 3 …