Natural Numbers are the numbers 1, 2, 3, 4,... . The ellipsis ... implies that the numbers continue forever. Our number system is a base-10 system, there are 10 digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that we use to form all the natural numbers. The (relative) location of the digits in a number determines its value. Example: The number 4, 235 has a value of 4 × 1000 + 2 × 200 + 3 × 10 + 5 × 1 The digit 5 is in the units place and has a value of 5 × 1 The digit 3 is in the tens place and has a value of 3 × 10 The digit 2 is in the hundreds place and has a value of 2 × 100 The digit 4 is in the thousands place and has a value of 4 × 1000 An equation is a mathematical relationship that contains an equal sign. Example: 3 + 4 = 2 + 5 is an equation. This is equation is a true equation. Example: 12 + 5 = 13 is an equation. This equation is a false equation. An inequality is a mathematical relationship that contains an inequality sign. There are four inequality signs: < less than > greater than ≤ less than or equal to ≥ greater than or equal to Example: 3 > 2 is an inequality. This inequality is true. Example: 4 ≤ 0 is an inequality. This inequality is false. Example: 12 ≥ 12 is an inequality. This inequality is true. Example: 6 < 6 is an inequality. This inequality is false. Notice that in order for the inequality < to be true, the number on the left must be strightly less than the number on the right. Similarly, in order for the inequality > to be true, the number on the left must be strightly greater than the number on the right. On the other hand, in order for the inequality ≤ to be true, the number on the left can be less than or equal to the number on the right. Similarly, in order for the inequality ≥ to be true, the number on the left can be greater than or equal to the number on the right. We use the number line to represent the numbers graphically. We can graph a particular number by marking a dot on that number. Example: Graph the number 5 on the real number line: 0 1 2 3 4 5 6 7 Addition is commutative, meaning that we can add two numbers in different order and it will not affect the result: Example: 2 + 5 = 5 + 2 The result of an addition is the sum. Subtraction is not commutative, meaning that you cannot change the order of subtrating two numbers without changing the result. Example: 7 − 3 6= 3 − 7 Multiplication is also communtative. For example 3 × 4 = 4 × 3 In algebra, we often use a dot, ·, to mean multiplication, so 3 · 5 means 3 times 5. We may also use the parenthesis to mean multiplication. Example (7)(8) means 7 times 8. Any number multiplied to 0 is 0. Example: 3 · 0 = 0 Any number multiplied to 1 is the original number. Example: 6 · 1 = 6 Distributive Property: If we have numbers a, b, c, the following is always true: a(b + c) = ab + ac a(b − c) = ab − ac Example: 3(6 + 2) = 3(6) + 3(2) Example: 7(9 − 4) = 7(9) − 7(4) Often in mathematics, we want to multiply the same number by itself multiple number of times. For example, we may want to multiply the number 2 by itself seven times. Instead of writing 2 · 2 · 2 · 2 · 2 · 2 · 2, we use exponents to simplify the writing. We write the above as: 2 · 2 · 2 · 2 · 2 · 2 · 2 = 27 = 128 In the exponential form 27, the number 2 is the base, the number 7 is the exponent. The expression means 2 times itself 7 times. Example: 38 = 3 · 3 · 3 · 3 = 81 Example: 53 = 5 · 5 · 5 = 125 In algebra, we use the horizontal bar to represent division. 20 Example = 20 ÷ 5 = 4 5 0 divided by any number is 0 0 Example = 0 4 0 Example = 0 7 Any number divided by 1 is itself: 12 Example = 12 1 3 Example = 3 1 Any number divided by itself is 1: 9 Example = 1 9 15 Example = 1 15 Division by 0 is undefined. 8 Example is undefined. 0 The square root of a number is a number whose square is the givenn number.We √ use the radical symbol, , to represent the (positive) square root of a number. √ Example 16 = 4 √ Example 25 = 5 Order of Operation: In order to evaluate an expression, we need to follow the order of operation according to the priority of the operations: Grouping symbols, including the parenthesis ( ), brackets [ ], braces {}, radicals √ , and the fraction bar −, always has the highest priority. Exponents, or raising to a power, is the next highest priority. Multiplication and division should be done after exponents, 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. If an expression has both addition and subtraction, the operations should be evaluated from left to right. Example: 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 = 8 + 12 − 10 = 20 − 10 = 10 Example 3 + 3 5 + p19 − (7 − 4) − 22 + 2 √ 3 + 3 = 5 + 19 − 3 − 22 + 2 √ 3 + 3 = 5 + 16 − 22 + 2 √ 6 = 5 + 16 − 22 + 2 6 = 5 + 4 − 22 + 2 = 5 + 4 − 22 + 3 = 5 + 4 − 4 + 3 = 9 − 4 + 3 = 5 + 3 = 8