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Math 300 Basic College Mathematics Chapter 10 Signed Numbers Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 1 of 16 10.1 Signed numbers Integers consist of whole numbers and their opposites. (see diagram below) Neither positive nor negative (zero) Negative Integers –4 –3 –2 –1 Positive Integers 0 1 2 3 4 The natural numbers to the right of zero are called positive integers. The opposites of the natural numbers to the left of zero on the number line are called negative integers. Zero is neither positive nor negative. Therefore, the integers to the left of zero are opposites of the integers to the right of zero and zero serves as its own opposite. Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 2 of 16 10.1 Signed numbers (cont) Integers and the Real World Integers correspond to many real-world problems and situations. Some common uses for negative integers are as follows: Time: Before an event Temperature: Degrees below zero Money: Amount lost, spent, owed, or withdrawn Elevation: Depth below sea level Travel: Motion in the backward (reverse) direction Otherwise, the integer is probably represented as a positive number. Example: Represent each quantity by a signed number. 1. A scuba diver is swimming 35 feet below the surface of the water in the Gulf of Mexico. 2. The Minnesota Viking football team lost 15 yards on a play. Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 3 of 16 10.1 Signed numbers (cont) Graphing Signed Numbers To graph signed numbers on a number line means to make a dot on the number line and label the dot with its corresponding number. Example: Graph the signed numbers in the list on a number line. 1. 4, 0, 2, 3 –4 –3 –2 1 4 –1 0 Math 300 M-G 4e Chapter 10; Rev: Mar 2011 1 2 3 4 Page 4 of 16 10.1 Signed numbers (cont) Order Real numbers are named in order on the number line. Things you will need to know to compare them: a. positive numbers > negative numbers. b. negative numbers < positive numbers. c. As you travel right on the number line, the numbers always get larger. (i.e.: -6 is greater than -9, because -6 is farther to the right) -9 -6 Using < and > symbols: -3 0 -6 > -9 and -9 < -6 Example: Use either < or > or for to write a true sentence. Picture where each number lies on a number line. 1. 0 –7 2. 6 –6 3. –3 4. –10 –4 –14 Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 5 of 16 10.1 Signed numbers (cont) Absolute value The absolute value of a number is its distance from zero on a number line. The symbol x is used to represent the absolute value of a number x. The absolute value of a number is always positive. If there is a negative sign on the outside of the absolute value symbol, then the answer would be negative (though the absolute value part is positive). Example: Find the value. 1. 7 2. 0 3. 8 4. 10 3 5. 8 Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 6 of 16 10.1 Signed numbers (cont) The opposite of a number When two numbers are the same distance from zero on a number line but are on opposite sides of zero, they are called opposites. The phrase “the opposite of” is written in symbol as “ – ”. If a is a number, then – (a) = – a. If a is a number, then – (– a) = a. Example: Find the opposite of each number. Question Equivalent to: Opposite of: 1. 8 2. 3. 4 9 43 4. 14 Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 7 of 16 10.2 Adding Signed Numbers ADDING 2 NEGATIVE INTEGERS Add the numbers and make the answer negative. Example #1: 7 + (10) = 17 {Think: 7 + 10 = 17, now write the answer with a negative sign as 17.} ADDING THE NUMBER ZERO TO ANY NUMBER When adding any number (positive or negative) to zero, your answer will be that number. Example #2: 8 + 0 = 8 Example #3: 4+0=4 ADDING A POSITIVE AND A NEGATIVE INTEGER Subtract the numbers and take the sign of the larger number. Example #4: 11 + (5) = 6 {Think: 11 – 5 = 6 and since 11 is the larger number, the answer will be positive.} 1 + (6) = 5 {Think: 6 – 1 = 5 and since 6 is the larger number, the answer will be negative.} Example #5: Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 8 of 16 10.2 Adding Signed Numbers (cont) ADDING OPPOSITES Any number added to its opposite is equal to zero. Example #6: 20 + (20) = 0 Example #7: 12 + 12 = 0 ADDING SEVERAL INTEGERS BOTH POSTIVE & NEGATIVE First, add all the positive numbers together. Then, add all the negative numbers together. Finally, add the results. Example #8: (15) + (37) + 25 + 42 + (59) + (14) Step 1: Add all of the positive numbers. 25 + 42 = 67 Step 2: Add all of the negative numbers. 15 + (37) + (59) + (14) = 125 Step 3: Add the results. 67 + (125) = 58 So, (15) + (37) + 25 + 42 + (59) + (14) = 58 Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 9 of 16 10.3 Subtracting Signed Numbers The first thing you will need to know in order to subtract integers is finding the opposite or additive inverse of a number. To find the opposite, or additive inverse, you would change only the sign of that number. Examples: Find the opposite of each number a. 34 b. 8.3 c. 0 Answers: a. The opposite of 34 is 34 b. The opposite of 8.3 is 8.3 c. The opposite of 0 is 0 Note: The opposite or additive inverse of a positive number is negative. The opposite or additive inverse of a negative number is positive. Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 10 of 16 10.3 Subtracting Signed Numbers (cont) Now to subtract signed numbers, you must add its opposite or additive inverse of the number that is being subtracted. To subtract signed numbers, change the minus sign to a plus sign and change the number behind the minus sign to its opposite. Examples: Subtract each of the following. a. 2 6 **In this problem, the 6 is positive.** b. 4 9 **In this problem, the 9 is negative.** Answers: a. 2 6 4 **After taking the opposite of 6 from the original problem, it becomes negative.** b. 4 9 5 **After taking the opposite of 9 from the original problem, it becomes positive.** Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 11 of 16 10.4 Multiplying and Dividing Signed Numbers Multiplying Signed Numbers Positive Positive = Positive Example: 5 5 25 Positive Negative = Negative Example: 3 4 12 Negative Positive = Negative Example: 2 5 10 Negative Negative = Positive Example: 2 4 8 Multiplying by Zero Any number (positive or negative) multiplied by zero is equal to zero. Example: 0 9 0 15 0 0 Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 12 of 16 10.4 Multiplying and Dividing Signed Numbers (cont) Dividing Signed Numbers Positive Positive = Positive Example: Positive 14 2 7 Negative = Negative Example: 6 2 3 Negative Positive = Negative Example: 24 8 3 Negative Negative = Positive 15 Example: 3 5 Division by Zero Zero divided by any number (positive or negative) is equal to zero. (Hint: When the numerator is zero, the answer is zero!) Examples: 0 21 0 0 0 2 Any number divided by zero is undefined. (Hint: When the denominator is zero, the answer is undefined!) Examples: 3 0 undefined Math 300 M-G 4e Chapter 10; Rev: Mar 2011 6 undefined 0 Page 13 of 16 10.5 Order of Operations Parentheses Exponents / Roots Multiplication / Division (Left to Right) Addition / Subtraction (Left to Right) Example: Simplify. 1. 2. 9 27 3 8 4 3 11 7 3. 4. 79 12 42 23 Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 14 of 16 Math 300 Chapter 10 Glossary Negative numbers - numbers less than 0 Positive numbers - numbers greater than 0 Signed numbers - the collection of numbers that included positive numbers, negative numbers and zero Integers - consists of zero, the natural numbers, and the opposites of the natural numbers. Absolute Value - the distance the number is from zero Opposite - if two numbers are the same distance from zero on the number line then they are opposites. Properties Adding two numbers with the same sign 1. add their absolute value 2. use the common sign as the sign of the sum Adding two numbers with different signs 1. find the larger absolute value minus the smaller absolute value 2. use the sign of the larger absolute value as the sign of the sum Subtracting two numbers: If a and b are numbers, then a – b = a + ( -b) The product of two numbers having the same sign is positive (+)(+) = + (-)(-) = + The product of two numbers having different signs is negative (+)(-) = - (-)(+) = - Math 300 M-G 4e Chapter 10; Rev: Mar 2011 Page 15 of 16 Math 300 Chapter 10 Glossary Hints The absolute value of a number is always positive. If you think of < and > as arrowheads, they always point to the smaller value. When subtracting a negative number, think of it as adding a positive, 5 – ( -2) = 5 + 2 When multiplying and dividing, if you have an even number of negatives, the result is positive. If you have an odd number of negatives, the result is negative. Be aware of the difference: (-5) 52 = -(5)(5) = -25 Math 300 M-G 4e Chapter 10; Rev: Mar 2011 = (-5)(-5) = 25. However, - Page 16 of 16