advertisement

Principles of Physics & Instrumentation of Medical Sonography Math Concepts Review Math Concepts Review Reciprocals Exponents Significant Figures Exponential Notation Units Unit Prefixes Unit Conversion Rules of Algebra Relationships Reciprocals The reciprocal or inverse of a number is ‘1’ divided by that number. Any number multiplied by its reciprocal equals one. The reciprocal of 250 is 1/250 or 0.004 250 x 1/250 = 1 What is the reciprocal of: 3? 111 ? .5 ? Exponents When we write 23, we mean 2 multiplied by itself 3 times 2 x 2 x 2 = 8. or The numeral 2 is the base The superscript numeral 3 is the exponent When written as an unknown variable, an is ‘a times a taken n times’. Two special cases are: a1=a and a0=1 Exponents - Multiplication Multiplying two numbers expressed as powers of the same base (an)(am) = an+m Exponents - Division Dividing one number raised to a power by another number of the same base - an = an-m am Knowing: 1 = a0 Then: 1 = a0 am am So… 1 = a-m am = a 0-m Exponents When the exponent is a fraction (a1/n), it is called a root. If a = 8 and n = 3, then… a1/n = 2 (the cube root of 8) Significant Figures A method of keeping track of numbers is to write only those figures that are significant. The zeros in 0.053 and 5,300 are not significant when using scientific notation, while the zero in 1.053 is. When zeros are “place keepers” for the decimal point, as in 5,300 and 0.053, they are not significant. Exponential Notation Writing out very small or large numbers is inefficient; so exponential notation is used. There are 3 separate types of numbers involved in exponential notation: 1. The first number in the notation is a whole number between and including 1 - 9, followed by a decimal point and any remaining significant numbers of the original number. 2. The second number represents the base which is 10. Expotential Notation 3. The final number is the exponent. The exponent is the “power” to which 10 is being raised. Its value can be positive or negative. If the exponent is positive, the resulting number > 0 If the exponent is negative, the resulting number < 0 If the exponent is 0, the resulting number will be the same as the first number of the equation Example I Express 50,000 in exponential notation: 1. If the number 1, then place the decimal point 1 place after the first number farthest to the left. If the number < ‘1’, then place the decimal point 1 place after the first non-zero number going from left to right. 5.0000 Express 50,000 in exponential notation 2. Place the multiplication sign (x) after that number. 5.0000 x 3. Count the number of places from the last number to the decimal point & note your direction (in this case there are 4 places to the left). Express 50,000 in exponential notation 4. The number of places will be the exponent; a left direction makes the exponent positive, whereas a right direction will make the exponent a negative. 5.0000 x 104 Since we are trying to be efficient, the 4 non-significant zeroes can be removed: 5 x 104 Example II Express 0.0001074 in exponential notation: 1. If the number 1, then place the decimal point 1 place after the first number farthest to the left. If the number < 1, then place the decimal point 1 place after the first non-zero number going from left to right. 1.074 2. Place the multiplication sign (x) after that number. 1.074 x 3. Count the number of places from the last significant number to the decimal point and note your direction (in this example there are 4 places to the left) 4. The number of places will be the exponent; a left direction makes the exponent positive, whereas a right direction will make the exponent negative. 1.074 x 10-4 This is as efficient as we can be; the zero is significant & cannot be removed Practice: Express these numbers in exponential notation a) 0.086010 b) 90,270,000,000,000,000 c) 0.0000000544 d) 360,000,000,000,001 Exponential Notation When performing addition or subtraction on exponential notation, both numbers must be expressed in the same power of 10. If they are not, convert the number(s) so that they are raised to the same power. Examples Addition: (a x 10n) + (b x 10n) = (a + b) x 10n Subtraction: (a x 10n) - (b x 10n) = (a - b) x 10n Examples Multiplication: (a x 10n) x (b x 10m) = (ab) x 10n+m Division: a x 10n b x 10m =a n-m X 10 b UNITS A unit is a predetermined quantity used as a standard of measurement. The unit tag gives a meaning to the number, it tells us what is being measured. Examples – inch, mile, gallon, square foot However, the medical and scientific world uses the Metric System S.I. (Systeme Internationale-French) Examples – centimeter, kilometer & liter Measurement units we will be using: Unit of Length: meter (m) Unit of Mass: gram (g) Unit of Volume: liter (l) Unit of Time: second (s) Each of the units of measurement can be made smaller or larger by adding a unit prefix Formulas we will be using: Area: Consists of two dimensions length (m) x width (m) = (m2) Volume: Consists of three dimensions Volume of a rectangular prism in cm: length (L) x width (W) x height (H) = cm3 Volume of a cylinder: V = r2 H Unit Prefixes Factor Prefix Powers of Ten Meaning Symbol giga 1,000,000,000 109 billion G mega 1,000,000 106 million M kilo 1,000 103 thousand k hecto 100 102 hundred h deca 10 101 ten da deci 1/10 (0.1) 10-1 tenth d centi 1/100 (0.01) 10-2 hundredth c milli 1/1,000 (0.001) 10-3 thousandth m micro 1/1,000,000 or (0.000001) 1/1,000,000,000 or (0.000000001) 10-6 millionth 10-9 billionth n nano nano micro milli centi deci UNIT Deca Hecto Kilo Mega Giga Unit Conversion To compare 2 numbers or to perform calculations, we must find the factor required to get the corresponding new unit. 1. Write out what you know from the math question. 2. Find the factor (in a table or memorize) 3. Multiply the beginning number by the factor and change the answer to the new unit tag micro 10-6 milli 10-3 centi deci Meter 10-2 10-1 100 Convert 25.4 millimeters to centimeters 1. Write what you start with: 25.4 mm. 2. Centi is to the right of the milli, we will change to larger units & get a smaller number 3. Determine the factor (10-3/10-2 = 10-1 = .1) 4. Multiply the beginning number by the factor & change the answer to the new unit tag: 25.4 mm X .1 = 2.54 cm micro 10-6 milli 10-3 centi deci 10-2 10-1 Second 100 Convert .4 seconds into microseconds 1. Write what you start with: .4 s 2. Microsecond is to the left of second, we will change to smaller units & get a larger number 3. Determine the factor:100/10-6=106 (1,000,000) 4. Multiply the beginning number by the factor and change the answer to the new unit tag 4 s X 1,000,000 = 400,000 s Practice: Convert 3.5 Megahertz (MHz) to Kilohertz (KHz) Rules of Algebra Solving for an unknown variable with known variables Whatever you do to one side of the equation, you must do on the other side. To solve for an unknown variable, it must be by itself, so we have to move all other variables to the other side of the equal sign. Recall: any symbol can be substituted for the letters in this equation Addition Solve for X, Y=15 & Z=30: X + Y = Z X + Y (-Y) = Z (- Y) X=Z–Y X = 30 – 15 X = 15 Subtraction Solve for X, Y=15 & Z=30: X – Y = Z X – Y (+Y) = Z (+ Y) X=Z+Y X = 30 + 15 X = 45 Division Solve for X, Z=12 & Y=3: XY= Z XY (Y) = Z (Y) X = ZY X = 12 3 X=4 Multiplication Solve for X, Z=12 & Y=3: X/Y = Z X/Y (x Y) = Z (x Y) X=ZxY X = 12 x 3 X = 36 Relationships: Unrelated: when you change one thing; the other factor does not change. Inversely Related: When you increase or decrease a variable; the other variable changes in the opposite direction Directly Related: When you increase or decrease a variable; the other variable changes in the same direction Inversely Related B A=C In the above equation, variables A & C are inversely related. This means that when the value of A is changed; the value of C must change in the opposite direction (and vice versa). To be inversely proportional the variables of interest must be: on opposite sides of the equal sign and on opposite sides of the divisor line. Directly Related A=B C In the above equation, variables A and B are directly related. This means when the value of A is changed; the value of B must change in the same direction (and vice versa). To be directly proportional the variables of interest must be: on opposite sides of the equal sign and on the same side of the divisor line. Direct Linear Proportionality Whenr A and B are directly related and whenever the value of A changes (increased or decreased), the value of B changes in the same direction by the same percentage. In a direct linear proportion, the variables of interest must be: on opposite sides of the equal sign and on the same side of the divisor line and when A is doubled, B doubles. If A is cut in half, B is also cut in half (or in the same percentage proportion) Direct Linear Proportionality Y = 3X; Substitute 1, 2 & 4 for X Y is direct linear proportional to X When you graph the 2 dependent variables (Y & X); the result is a straight line. Y o 12 10 8 o 6 4 2 o 1 2 3 4 5 X Direct Non-Linear Proportionality Whenever A and B are directly related and whenever the value of A changes (increased or decreased), the value of B changes in the same direction but NOT by the same percentage. In a direct non-linear proportion, the variables of interest must be: on opposite sides of the equal sign and on the same side of the divisor line and when A changes, B changes in the same direction but at a different percentage. Direct Non-Linear Proportionality Y = 3X2; Substitute 1, 2 & 4 for X Y is direct linear proportional to X When you graph the 2 dependent variables (Y & X); the result is NOT a straight line. Y 60 o 50 40 30 20 10 o 1 o 2 3 4 5 X