Copyright Sautter 2003 Modified by D. Morrish 2014

ALGEBRA & EQUATIONS the use of basic algebra requires only a few rules which are used over and over to solve equations. any number multiplied or divided by itself = 1 what is ever done to one side of an equation must be done to the other side. additions or subtractions which are enclosed in brackets are carried out first. when values in brackets are multiplied or divided by something, each value can be multiplied or divided separately before adding or subtracting the grouped terms. This does not apply to indices.

ALGEBRA & EQUATIONS if we add 10 to the left side we must add 10 to the right if we multiply the left side by 5 we must multiply the right by 5 rule 1 – a value divided by itself equals 1 rule 2 – operate on both sides equally

algebra & equations rule 3 – operation in brackets are done first the brackets terms (5 + 4) are added first the brackets terms (22 – 7) are subtracted first

algebra & equations rule 4 – values can be distributed through terms in brackets each term in the brackets must be multiplied by 4 all terms must be multiplied by every other term, then simplified

SOLVING ALGEBRAIC EQUATIONS solving an algebraic equation requires that the unknown variable be isolated on the left side of the equal sign in the numerator position and all other terms be placed on the right side of the equal sign.

Proportion When converting from one unit system to another we use proportional reasoning. e.g. How many joules is equivalent to 6.4 MeV? 1 eV = 1.6 X J 6.4 MeV =6.4 X 1.6 X J Or 1 eV = 1.6 X J 6.4 MeV = x J Therefore x / 1.6 X J = 6.4/1 So x = 6.4 X 1.6 X J 1 Which is the same as before

GRAPHS AND EQUATIONS Graphs can be considered as a picture of an equation showing an array of x and y values which were calculated from the equation. We will look at two different kinds of graphs, linear (straight line) and curved. Linear graphs are described by the general equation: y = mx + b Curved graphs are described by the general equation: Y = kx n Although graphs can be represented by many other equations, we will look at only these two basic relationships in detail

Y X THE dependant VARIABLE THE independant VARIABLE SLOPE VERTICAL INTERCEPT POINT b rise run SLOPE = RISE / RUN SLOPE =  Y /  X

Y X A constant A positive power other than 1 or zero The slope is always changing ( variable)

SLOPE ? CONSTANT SLOPE ? POSITIVE OR NEGATIVE? CONSTANT SLOPE ? POSITVE OR NEGATIVE ? SLOPE = 0 CONSTANT SLOPE ? POSITIVE OR NEGATIVE ? SLOPES & RATES TIME SLOPE = RISE / RUN SLOPE IS NEGATIVE SLOPE IS CONSTANT SLOPE IS NEGATIVE SLOPE IS VARIABLE SLOPE IS POSITIVE SLOPE IS VARIABLE SLOPE OF A TANGENT LINE TO A POINT = INSTANTANEOUS RATE GRAPH 1GRAPH 2 GRAPH 3GRAPH 4 DISPLACEMENTDISPLACEMENT DISPLACEMENTDISPLACEMENT DISPLACEMENTDISPLACEMENT DISPLACEMENTDISPLACEMENT

DISPLACEMENTDISPLACEMENT Time VELOCITYVELOCITY ACCELERATIONACCELERATION SS tt tt vv Slope of a tangent drawn to a point on a displacement vs time graph gives the instantaneous velocity at that point Slope of a tangent drawn to a point on a velocity vs time graph gives the instantaneous acceleration at that point

Y X X1X1 X2X2 AREA UNDER THE CURVE FROM X 1 TO X 2 Area =  Y  X (SUM OF THE BOXES) WIDTH OF EACH BOX =  X AREA MISSED - INCREASING THE NUMBER BOXES WILL REDUCE THIS ERROR! AS THE NUMBER OF BOXES INCREASES, THE ERROR DECREASES!

TRIGNOMETRY trignometric relationships are based on the right triangle (a triangle containing a 90 0 angle). the most fundamental concept is the pythagorean theorem (a 2 + b 2 = c 2 ) where a and b are the shorter sides (the legs) of the triangle and c is the longest side called the hypotenuse. ratios of the sides of the right triangle are given names such as sine, cosine and tangent. depending on the angle between a leg (one of the shorter sides) and the hypotenuse (the longest side), the ratio of sides for a particular angle always has the same value no matter what size the triangle.

C A RIGHT TRIANGLE  B C C C A 90 0    ++ = B

TRIG FUNCTIONS The ratio of the side opposite the angle and the hypotenuse is called the sine of the angle. The sine of 30 0 for example is always ½ no matter how large or small the triangle. This means that the opposite side is always half as long as the hypotenuse if the angle is (30 0 corresponds to 1/12 of a circle or one slice of a 12 slice pizza!) The ratio of the side adjacent to the angle and the hypotenuse is called the cosine. The cosine of 60 0 is always ½ which means this time the adjacent side is half as long as the hypotenuse. (60 0 represents 1/6 of a complete circle, one slice of a 6 slice pizza) The ratio of the side adjacent to the angle and the side opposite the angle is called the tangent. If the adjacent and the opposite sides are equal, the ratio (tangent value) is 1.0 and the angle is 45 0 ( 45 0 is 1/8 of a full circle)

 A B C Sin = A / C  Cos = B / C  Tan  = A / B A C B A B A RIGHT TRIANGLE C

Trig functions Right triangles may be drawn in any one of four quadrants. Quadrant I encompasses from 0 to 90 degrees (1/4 of a circle). It lies between the +x axis and the + y axis (between due east and due north). Quadrant II is the area between 90 and 180 degrees ( the next ¼ circle in the counterclockwise direction). It lies between the +y and the –x axis (between due north and due west). Quadrant III is the area between 180 and 270 degrees (the next ¼ circle in the counterclockwise direction). It lies between the –x and the –y axis (between due west and due south). Quadrant IV encompasses from 270 to 360 degrees ( the final ¼ circle). It lies between the –y and the +x axis (between due south and due east). The signs of the trig functions change depending upon in which quadrant the triangle is drawn.

y x radians  radians 3/2  radians 2  radians  Quadrant III Quadrant IV Quadrant I Quadrant II Sin Cos Tan      /2 radians 90 o 0 o 180 o 270 o 360 o

In science, we often encounter very large and very small numbers. Using scientific numbers makes working with these numbers easier

Scientific numbers use powers of 10

RULE 1 As the decimal is moved to the left The power of 10 increases one value for each decimal place moved Any number to the Zero power = 1

RULE 2 As the decimal is moved to the right The power of 10 decreases one value for each decimal place moved Any number to the Zero power = 1

RULE 3 When scientific numbers are multiplied The powers of 10 are added

RULE 4 When scientific numbers are divided The powers of 10 are subtracted

RULE 5 When scientific numbers are raised to powers The powers of 10 are multiplied

RULE 6 Roots of scientific numbers are treated as fractional powers. The powers of 10 are multiplied

RULE 7 When scientific numbers are added or subtracted The powers of 10 must be the same for each term. Powers of 10 are Different. Values Cannot be added ! Power are now the Same and values Can be added. Move the decimal And change the power Of 10

LOGARITHMS A logarithm (log) is a power of 10. If a number is written as 10 x then its log is x. For example 100 could be written as 10 2 therefore the log of 100 is 2. In chemistry calculations often small numbers are used like.0001 or The log of.0001 is therefore –4. For numbers that are not exact powers of 10 a calculator is used to find the log value. For example the log of is –2.46 as determined by the calculator. Logarithms do not always use powers of 10. Another common number used instead of 10 is 2.71, which is called base e. When the logarithm is the power of e it is called a natural log and the symbol used in ln rather than log.

LOGARITHMS Since logs are powers of 10 they are used just like the powers of 10 associated with scientific numbers. When log values are added, the numbers they represent are multiplied. When log values are subtracted, the numbers they represent are divided When logs are multiplied, the numbers they represent are raised to powers When logs are divided, the roots of numbers they represent are taken.