Real Numbers: Natural Numbers: N= f1; 2; 3; · · ·} Integers: Z= f0; −1; 1; −2; 2; −3; 3; · · ·} Note that every natural number is an integer. There are integers (negative numbers) that are not natural num- bers. 2 −1 13 Rational Numbers: Q=f ; ; 0; 3; − ; · · ·} 5 3 4 A rational number is a ratio of two integers. Note that 8 every integer is a rational number (e.g. 8 = ), but 1 there are rational numbers that are not integers (frac- 2 tions whose denominator is not 1. E.g. ). 3 When expressed as a decimal, every rational number ei- ther terminate after some point, or continue indefinitely in a repeated pattern. E.g. 3 = 0:6 5 79 = 2:393939 ··· = 2:39 33 1 Irrational numbers are numbers thatp cannot be ex- pressed as ratios of two integers. e.g. 2, π. When ex- pressed as decimals, irrational numbers don't terminate and don't repeat. E.g. π = 3:14159 ··· Note that rational numbers are never irrational and vice versa. Putting together rational and irrational numbers give us the set of Real Numbers, R. I.e., real numbers are the collection of all the rational and irrational numbers. Please note that the terms natural, rational, irra- tional, when used in the above case, are not adjectives, but nouns. The set of Real Numbers is a specific collec- tion of numbers that have particular properties in math- ematics. The word real is not used to describe the nature of the numbers and does not imply that the real numbers are any more or less real (used as a description) than any other numbers. 2 An inequality involves one of four inequality symbols: < less than > greater than ≤ less than or equal to ≥ greater than or equal to 3 < 5 is a true statement 5 < 5 is a false statement 5 ≤ 5 is a true statement We can graph an inequality statement like x < 10 on a number line. We use open parentheses ( ) for < or >, and use bracket [ ] for ≤ or ≥. 3 The absolute value of a number is the distance the number is from zero. Algebraically, 8 < x if x ≥ 0 jxj = : −x if x < 0 To add two negative numbers, add the absolute value of the two numbers, and turn the result into negative. E.g. −4 + −3 = −(4 + 3) = −7 To add a positive number with a negative number. First determine which number has a larger absolute value. If the negative number has a larger absolute value, the re- sult is negative, otherwise the result is positive. Subtract the smaller absolute value from the larger absolute value and assign a sign as given above. E.g. −5 + 3 = −(5 − 3) = −2 To subtract a negative number is to add the positive: E.g. 3 − −4 = 3 + 4 = 7 To add a negative is to subtract the positive: 4 + −6 = 4 − 6 = −2 The product of two negative numbers is positive. The product of a negative number with a positive number is negative. Note: j6 − 3j 6= 6 + 3 The absolute value does not change any operation inside it, the absolute value only changes a negative number to its opposite when the final result inside the absolute value, after all the operations are done, is a negative 4 number. 5 Exponents: Any number to a positive integer power means that num- ber times itself that many times. xn = x · x · x ··· x | {z } n times E.g. x3 = x · x · x E.g. 34 = 3 · 3 · 3 · 3 = 81 E.g. 25 = 2 · 2 · 2 · 2 · 2 = 32. In evaluating an expression, we need to always follow the Order of Operations: Parenthesis, acting as a grouping symbol, always has the highest priority. Exponents, or raising to a power, is the next highest priority. Multiplication and division should be done after expo- nents, they have the same priority. If an expression has both multiplication and division written, it should be evaluated from left to right. Addition and subtraction has the same, lowest priority. E.g. 4 + 32 − (2 + 3) + 12 − 2 · 5 = 4 + 32 − 5 + 12 − 2 · 5 = 4 + 9 − 5 + 12 − 2 · 5 = 4 + 9 − 5 + 12 − 10 = 13 − 5 + 12 − 10 6 = 8 + 12 − 10 = 20 − 10 = 10 7 Simplifying Algebraic Expressions: A variable is a letter used to represent a number. A variable is used in one of two purposes: 1. It acts as a pronoun for an unknown quantity, for ex- ample, we may not know the age of John, but we know that John is 5 years older than Mary, we can let x rep- resent the age of John, and y represent the age of Mary, we can then write x = y + 5. 2. A variable also can be used as a generic expression to represent a collection of numbers (or other objects) that share a particular characteristic when we need to write formulas about the numbers: E.g. For any real number x, y, we have: x + y = y + x. A constant is a number whose value is fixed. The num- ber 3, for example, is a constant. A constant can also be represented by a letter (or any other symbol) as long as it is understood that the number represented by the letter has a fixed value. An expression is a collection of variables and constants with mathematical operations: E.g. 3x + 5 p E.g. x + 4y − x Each expression can have a value if each variable of the expression is assigned a value. For example, in the ex- pression: 5x + 3y − 4z, if x = 4, y = 1, and z = 2, then the expression has the value: 8 5(4) + 3(1) − 4(2) = 20 + 3 − 8 = 15. To find the value of the expression given the values of the variables is to evaluate the expression. 9 If we have the equal sign (=) between two expressions, we have an equation. An equation can be true or false. E.g. 3 + 2 = 9 E.g. 4 − 2 = 2 E.g. x + 3 = 5 The first equation is always false, the second equation is always true (called an identity), and the third equation is sometimes true (if x = 2). To find the value(s) of the variable that would make the equation a true equation is to solve the equation. Please understand the difference between an expression and an equation. An expression does not need to be solved. An expression may be evaluated if all the values of the variables are known or assigned. An equation, on the other hand, can be solved. An equation is either true or false. 10 A term in an expression is separated by addition or sub- traction from other terms. E.g. 4x − 3xy + 2z has three terms, they are 4x, −3xy, and 2z. Note that we say the first term is 4x, not 4 and x, and we say that the second term is −3xy, because the sign in front of a term is part of the term. The constant (the number) in a term is called the coefficient of the term. In the example, the coefficient of the first term is 4, the coefficient of the second term is −3, and the coefficient of the third term is 2. If a term does not have a number explicitely written, the coefficient is 1. E.g. x + 3y. In this expression with two terms, the coefficient of the first term is 1. E.g. 4x2 −y+12xz2. In this expression with three terms, the coefficient of the second term is −1. Two terms are like terms if they involve the same vari- ables to the same power: E.g. 4x2y, 6x2y are like terms, they both have variables x and y, and the power of x are both 2, and the power of y are both 1. E.g. 3x, 4xy are unlike terms, since the first term does not have y. E.g. 4x2, −8x are unlike terms, because the first term has x to the 2 power, and second term has x to the first power. If the terms are alike, we can combine like terms by 11 adding their coefficients. E.g. 3x + 5x = 8x E.g. 4x2 + 3x − 2x2 = 2x2 + 3x E.g. x + 4x = 5x. Notice in the third example that there is the invisible coefficient of 1 in the first term, and that's why the sim- plified expression is 5x. When simplifying an expression, we are trying to write an expression to an equivalent expression that is easier to read or evaluate. Two expressions are equivalent if they would have the same value for the same values of the variables. In the above 3 examples, the expression on the right is equiva- lent to the expression on the left. 12 Properties of the Real Numbers: For any real numbers x, y, z, Commutative property of Addition: x + y = y + x Commutative property of Multiplication: xy = yx Associative property of Addition: (x + y) + z = x + (y + z) Associative property of Multiplication: (xy)z = x(yz) Distributive property x(y + z) = xy + xz (y + z)x = yx + zx = xy + xz x(y − z) = xy − xz (y − z)x = yx − zx = xy − xz Notice that when we use the distributive property with a negative number outside of the parentheses, the negative sign will also need to be distributed. E.g. 4x − 3(x − 5) = 4x − 3x + 15 = x + 15. Notice that the negative sign in front of 3 is also dis- tributed to the x (making it −3x) and to the −5, making it +15.