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TMAT 103 Chapter 1 Fundamental Concepts TMAT 103 §1.1 The Real Number System §1.1 – The Real Number System • Integers – Positive, Negative, Zero • Rationals • Irrationals • Reals – Real number line • Complex Numbers • Primes §1.1 – The Real Number System • Properties of Real Numbers (FYI) – – – – – – – – – Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Distributive Property of Multiplication over Addition Additive inverse Multiplicative inverse Additive identity Multiplicative identity §1.1 – The Real Number System • Signed Numbers – – – – – Absolute Value Adding 2 signed numbers Subtracting 2 signed numbers Multiplying 2 signed numbers Dividing 2 signed numbers §1.1 – The Real Number System • Examples – Calculate the following |–101| (- 1½) + (- 2¼) Bill, a diver, is 120 feet below the surface of the Pacific Ocean. Heather is directly above Bill in a balloon that is 260 feet above the Pacific Ocean. Find the distance between Bill and Heather. TMAT 103 §1.2 Zero and Order of Operations §1.2 – Zero and Order of Operations • Operations with 0 a0 a a0 a a0 0 If a b 0 then a 0 or b 0 0 b 0 (b 0) a 0 is meaningles s (a 0) 0 0 is indetermin ate §1.2 – Zero and Order of Operations • Examples – Calculate the following Find values of x that make the following meaningless: 3x – 7 2x + 1 Find values of x that make the following indeterminate: 2–x (2x – 7)(x – 2) . §1.2 – Zero and Order of Operations • Order of Operations – PEMDAS 1. Parenthesis 2. Exponents 3. Multiplications and Divisions in the order they appear left to right 4. Additions and Subtractions in the order they appear left to right §1.2 – Zero and Order of Operations • Examples – Calculate the following 5 3(7 2) 5 2 2(4 7) TMAT 103 §1.3 Scientific Notation and Powers of 10 §1.3 – Scientific Notation and Powers of 10 • Powers of 10 10 n , 10 n • Laws 10 m 10 n 10 m n 10 m mn 10 10 n (10 m ) n 10 mn 1 1 n n 10 n and 10 10 10 n 100 1 §1.3 – Scientific Notation and Powers of 10 • Scientific Notation – Changing a number from decimal form to scientific notation – Changing a number from scientific notation to decimal form §1.3 – Scientific Notation and Powers of 10 • Examples – Calculate the following Write the following in scientific notation 23700 17070000 .00325 Write the following in decimal form 7.23 x 106 6.2 x 10-3 TMAT 103 §1.4 Measurement §1.4 – Measurement • Measurement – Comparison of a quantity with a standard unit • In past, units not standard (1 pace, length of ear of corn, etc.) – Necessity dictated universally standard units • Approximate vs. exact – Accuracy (significant digits) – Precision §1.4 – Measurement • Accuracy (Significant Digits) Rules 1. 2. 3. 4. All non-zero digits are significant All zeros between significant digits are significant Tagged zeros are significant All numbers to the right of a significant digit AND a decimal point are significant 5. Non-tagged zeros to the right in a whole number are not significant 6. Zeros to the left in a measurement less than one are not significant §1.4 – Measurement • Examples – Calculate the following Find the accuracy (number of significant digits) of the following: 14.7 .000000000008 1404040 1404040.00030 §1.4 – Measurement • Precision – – – The smallest unit with which a measurement is made. In other words, the position of the rightmost significant digit. Ex: The precision of 239,000 miles is 1000 miles. Ex: The precision of 23.55 seconds is .01 seconds §1.4 – Measurement • Examples – Calculate the following Find the precision of each of the following: 1.0 m 360 V 350.000030 V §1.4 – Measurement • Precision and accuracy are different!!! – Ex: Determine which of the following measurements are more precise, and which is more accurate: 0.00032 feet 23540000 feet TMAT 103 §1.5 Operations with Measurements §1.5 – Operations with Measurements • Adding or subtracting measurements 1. Convert to the same units 2. Add or subtract 3. Round the result to the same precision as the least precise of the original measurements • Multiplying or dividing measurements 1. Convert to the same units 2. Multiply or divide 3. Round the result to the same number of significant digits as the original measurement with the least significant digits §1.5 – Operations with Measurements • Examples – Calculate the following Find the sum of: 178m, 33.7m and 100cm Find the product of: (.065m) and (.9282m) TMAT 103 §1.6 Algebraic Expressions §1.6 – Algebraic Expressions • Terminology – – – – – – – Variable Constant Term Numerical coefficient Monomial, binomial, trinomial, polynomial Degree of a monomial Degree of a polynomial §1.6 – Algebraic Expressions • Operations on Algebraic expressions – Adding expressions – Subtracting expressions – Evaluating expressions given the values of variables §1.6 – Algebraic Expressions • Examples – Calculate the following Find the degree of x2y Find the degree of x2y + w4 + a3b2 (4y + 11) + (11y – 2) (x2 + x + 17) – (3x – 4) TMAT 103 §1.7 Exponents and Radicals §1.7 – Exponents and Radicals • Laws of Exponents a m a n a mn am mn a an (a m ) n a mn (ab) n a nb n n an a n b b 1 a n n a a0 1 §1.7 – Exponents and Radicals • Examples – Simplify the following y6 y2 y6 y2 ( y 6 )2 3 13 3 x y 8 11 x y §1.7 – Exponents and Radicals • Radicals – Simplifying simple radicals • Ex: 36 – Simplifying radicals with the following property: ab a b • Ex: 18 TMAT 103 §1.8 Multiplication of Algebraic Expressions §1.8 – Multiplication of Algebraic Expressions • • • • Distributive Property FOIL Vertical multiplication Multiplication of general polynomials §1.8 – Multiplication of Algebraic Expressions • Examples – Calculate the following x2(y3 + z – 2) (x + 2)(x – 2) (3x2 + 4x – 1)(2y – 3z + 7) TMAT 103 §1.9 Division of Algebraic Expressions §1.9 – Division of Algebraic Expressions • Division by a monomial • Division by a polynomial §1.9 – Division of Algebraic Expressions • Examples – Calculate the following 14x2 – 10x 2x 6x4 + 4x3 + 2x2 – 11x + 1 (x – 2) 4y3 + 11y – 3 (2y + 1) TMAT 103 §1.10 Linear Equations §1.10 – Linear Equations • Four properties of equations 1. The same value can be added to both sides 2. The same value can be subtracted from both sides 3. The same non-zero value can be multiplied on both sides of the equation 4. The same non-zero value can divided on both sides of the equation §1.10 – Linear Equations • Examples – Calculate the following x – 4 = 12 4(2y – 3) – (3y + 7) = 6 ¼(½x + 8) = ½(x – 16) + 11 TMAT 103 §1.11 Formulas §1.11 – Formulas • Formula – equation, usually expressed in letters, that show the relationship between quantities • Solving a formula for a given letter §1.11 – Formulas • Examples – Calculate the following Solve f = ma for a Solve e = ƒx + for x Solve for R3: R1 R2 RB R1 R2 R3 TMAT 103 §1.12 Substitution of Data into Formulas §1.12 – Substitution of Data into Formulas • Using a formula to solve a problem where all but the unknown quantity is given 1. Solve for the unknown 2. Substitute all values with units 3. Solve §1.12 – Substitution of Data into Formulas • Examples – Calculate the following Solve f = ma for a when f = 3 and m = 17 Solve e = ƒx + for x when e = 11, ƒ = 3.5 and = .01 Solve for R3 when RB, R1, and R2 are all 11 R1 R2 RB R1 R2 R3 TMAT 103 §1.13 Applications involving Linear Equations §1.13 – Applications involving Linear Equations • Solving application problems 1. 2. 3. 4. 5. 6. Read the problem carefully If applicable, draw a picture Use a symbol to label the unknown quantity Write the equation that represents the problem Solve Check §1.13 – Applications involving Linear Equations • Examples – Calculate the following The difference between two numbers is 6. Their sum is 30. Find the 2 numbers. The perimeter of an isosceles triangle is 122cm. Its base is 4cm shorter than one of its equal sides. Find the lengths of the sides of the triangle. TMAT 103 §1.14 Ratio and Proportion §1.14 – Ratio and Proportion • Ratio: Quotient of 2 numbers or quantities • Proportion: Statement that 2 ratios are equal • If a c then ad bc b d §1.14 – Ratio and Proportion • Examples – Calculate the following Find x: 25 = 75 96 x The ratio of the length and the width of a rectangular field is 5:6. Find the dimensions of the field if its perimeter is 4400m. TMAT 103 §1.15 Variation §1.15 – Variation • Direct Variation – If 2 quantities, y and x, change and their ratio remains constant (y/x = k), the quantities vary directly, or y is directly proportional to x. In general, this relationship is written in the form y = kx, where k is the proportionality constant. – Example: m varies directly with n; m = 198 when n = 22. Find m when n = 35. §1.15 – Variation • Inverse Variation – If two quantities, y and x, change and their product remains constant (yx = k), the quantities vary inversely, or y is inversely proportional to x. In general, this relation is written y = k/x, where k is called the proportionality constant. – Example: d varies inversely with e; d = 4/5 when e = 9/16. Find d when e = 5/3. §1.15 – Variation • Joint Variation – One quantity varies jointly, with 2 or more quantities when it varies directly with the product of these quantities. In general, this relation is written y = kxz, where k is called the proportionality constant. – Example: y varies jointly with x and the square of z; y = 150 when x = 3 and z = 5. Find y when x = 12 and z = 8.