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Understanding Integers and Number Line Concept

Learn about integers, absolute value, number line graphing, inequalities, addition, subtraction, and multiplication with real numbers. Understand the relationship between integers and their properties.

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Understanding Integers and Number Line Concept

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  1. CHAPTER 5 Number Theory and the Real Number System

  2. 5.2 • The Integers; Order of Operations

  3. Objectives • Define the integers. • Graph integers on a number line. • Use symbols < and >. • Find the absolute value of an integer. • Perform operations with integers. • Use the order of operations agreement.

  4. Define the Integers • The set consisting of the natural numbers, 0, and the negatives of the natural numbers is called the set of integers. • Notice the term positive integers is another name for the natural numbers. The positive integers can be written in two ways: • Use a “+” sign. For example, +4 is “positive four”. • Do not write any sign. For example, 4 is also “positive four”.

  5. The Number Line • The number line is a graph we use to visualize the set of integers, as well as sets of other numbers. • Notice, zero is neither positive nor negative.

  6. Example: Graphing Integers on a Number Line • Graph: • 3 • 4 • 0 • Solution: Place a dot at the correct location for each integer.

  7. Use the Symbols < and > • Looking at the graph, 4 and 1 are graphed below. • Observe that 4 is to the left of 1 on the number line. This means that -4 is less than -1. • Also observe that 1 is to the right of 4 on the number line. This means that 1 is greater then 4.

  8. Use the Symbols < and > • The symbols < and > are called inequality symbols. • These symbols always point to the lesser of the two real numbers when the inequality statement is true.

  9. Example: Using the Symbols < and > • Insert either < or > in the shaded area between the integers to make each statement true: • 4 3 • 1 5 • 5 2 • 0 3

  10. Example: Using the Symbols < and > continued • 4 < 3 (negative 4 is less than 3) because 4 is to the left of 3 on the number line. • 1 > 5 (negative 1 is greater than negative 5) because 1 is to the right of 5 on the number line. • 5 < 2 ( negative 5 is less than negative 2) because 5 is to the left of 2 on the number line. • 0 > 3 (zero is greater than negative 3) because 0 is to the right of 3 on the number line.

  11. Use the Symbols < and > • The symbols < and > may be combined with an equal sign, as shown in the following table:

  12. Absolute Value • The absolute value of an integer a, denoted by |a|, is the distance from 0 to a on the number line. • Because absolute value describes a distance, it is never negative.

  13. Example: Finding Absolute Value • Find the absolute value: • |3| b. |5| c. |0| • Solution: • | 3| = 3 because 3 is 3 units away from 0. • |5| = 5 because 5 is 5 units away from 0. • |0| = 0 because 0 is 0 units away from itself.

  14. Addition of Integers • Rule • If the integers have the same sign, • Add their absolute values. • The sign of the sum is the • same sign of the two numbers. • If the integers have different signs, • Subtract the smaller absolute • value from the larger absolute • value. • The sign of the sum is the same as the sign of the number with the larger absolute value. Examples

  15. Study Tip • A good analogy for adding integers is temperatures above and below zero on the thermometer. Think of a thermometer as a number line standing straight up. For example,

  16. Additive Inverses • Additive inverses have the same absolute value, but lie on opposite sides of zero on the number line. • When we add additive inverses, the sum is equal to zero. • For example: • 18 + (18) = 0 • (7) + 7 = 0 • In general, the sum of any integer and its additive inverse is 0: a+ (a) = 0

  17. Subtraction of Integers • For all integers a and b, • a – b = a + (b). • In words, to subtract b from a, add the additive inverse of b to a. The result of subtraction is called the difference.

  18. Example: Subtracting of Integers • Subtract: • a. 17 – (–11) b. –18 – (–5) c. –18 – 5

  19. Multiplication of Integers • The result of multiplying two or more numbers is called the product of the numbers. • Think of multiplication as repeated addition or subtraction that starts at 0. For example,

  20. Multiplication of Integers: Rules • Rule • The product of two integers with different signs is found by multiplying their absolute • values. The product is negative. • The product of two integers with the same signs is found by multiplying their absolute • values. The product is positive. • The product of 0 and any integer is 0: Examples • 7(5) = 35 • (6)(11) = 66 • 17(0) = 0

  21. Multiplication of Integers: Rules • Rule • If no number is 0, a product with an odd number of negative factors is found by • multiplying absolute values. The product is negative. • If no number is 0, a product with an even number of negative factors is found by • multiplying absolute values. The product is positive. Examples

  22. Exponential Notation • Because exponents indicate repeated multiplication, rules for multiplying can be used to evaluate exponential expressions.

  23. Example: Evaluating Exponential Notation • Evaluate: a. (6)2 b. 62 c. (5)3 d. (2)4 • Solution:

  24. Division of Integers • The result of dividing the integer a by the nonzero integer b is called the quotient of numbers. • We write this quotient as or a / b. This means that 4(3) = 12.

  25. Division of Integers Rules • Rule • 1. The quotient of two integers with different signs is found by dividing their absolute values. The quotient is negative. • 2. The quotient of two integers with the same sign is found by dividing their absolute values. The quotient is positive. • 3. Zero divided by any nonzero integer is zero. • 4. Division by 0 is undefined. Examples

  26. Order of Operations • Perform all operations within grouping symbols. • Evaluate all exponential expressions. • Do all the multiplications and divisions in the order in which they occur, working from left to right. • Finally, do all additions and subtractions in the order in which they occur, working from left to right.

  27. Example: Using the Order of Operations • Simplify 62 – 24 ÷ 22 · 3 + 1. • Solution: There are no grouping symbols. Thus, we begin by evaluating exponential expressions. • 62 – 24 ÷ 22 · 3 + 1 = 36 – 24 ÷ 4 · 3 + 1 • = 36 – 6 · 3 + 1 • = 36 – 18 + 1 • = 18 + 1 • = 19

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