2.1 Combining Opposite Numbers Number Types By now, you probably know what fractions and whole numbers are. The set of Whole Numbers includes all positive numbers and zero, without fractions or decimals. The set of Whole Numbers includes the Natural Numbers and zero. The set of Natural Numbers is sometimes referred to as the set of counting numbers, because they are all the numbers you would use to count whole things (fingers, dogs, rocks, trees, and so on). Another group of numbers is the ______________________. The Integers are all the Whole Numbers and their negatives. Again, no decimals or fractions, just {...-4, -3, -2, -1, 0, 1, 2, 3, 4, ...}. Whole numbers are known as a subset of integers (they are within the set of Integers). The last set of numbers this book will address are rational numbers. A rational number is any number that can be written as a _____________________________. This includes terminating decimals (decimals that end, like 0.5) and repeating decimals (decimals that repeat in a pattern, 0.5858585858..., also written as: _____________________). All whole numbers and integers can also be written as fractions. For example, __________________. Opposite numbers are the same number with opposite signs. All opposite numbers are called additive inverses. All additive inverses have a sum of zero. If you were to place the number 'a' and its opposite, '-a', on a number line, you would see they are the same distance away from zero. Both 3 and -3 are 3 units away from zero, making them additive inverses and opposites. The distance away from zero is also called the __________________________ of a number. Absolute value is always positive because distance is always positive. Absolute value is marked with vertical lines on either side of the number. |−12|=12 |4.5|=4.5 Dance Moves Juan really wants to impress the girls at the next school dance, so he decides to take dance lessons. He notices that dancing is just a series of steps; move two steps forward, two steps back, one to the left, one to the right, etc. At the school dance, Juan creates his own series of moves. Here is a list of the steps: ● 3 steps to the left ● 2 steps to the right ● 4 steps to the left ● 3 steps to the right Then he repeats. If his starting position is zero on the number line, where does he end up after one cycle? Use the number line and Juan's feet to help you towards your answer. Remember this! ● ________________numbers are positive numbers without fractional or decimal parts, including zero. ● _________________are whole numbers that are positive and ____________________. Whole numbers are a part of the set of integers. ● ______________________numbers are all numbers that can be written as a fraction, including whole numbers, integers, fractions, terminating, decimals, and repeating decimals. ● The absolute value of a number is its _____________________from zero on the number line. Distance is always positive, therefore, so is the absolute value. ● Two numbers are opposites of each other, or additive inverses, if their sum is zero. 2.2 Modeling the Addition of Integers The answer for an addition problem can be ________________or _________________. The answer can also be zero if the integers are opposites. When adding integers, start at the first integer on the number line. If the second integer is positive, move that many units to the right and where you land is the answer. If the second integer is negative, move that many units to the left and where you land is the answer. Discussion Question Start at 0. Describe two dance steps that will bring you back to where you started. The commutative property of addition: For any two numbers, the order in which you add them does not matter; a+b=b+a Since the order in which you add the three numbers does not matter, so do the combination that is easiest for you. So, rather than do 9+(−3)+2, you could do 9+2+(−3) In this case, parentheses are used around the negative number to separate it from the plus sign, so they do not change this problem in any way. Remember this! When adding integers, start at the first integer (on the number line) and move to the right or left, depending on the sign of the second integer to find the answer. Subtracting Integers Subtraction is defined as taking away a value from another. As you saw in the previous lessons, adding a negative number is subtraction. On a number line, adding an integer meant moving units in the direction of the sign of that number. So, adding -4 would mean moving 4 units to the left. With subtraction, it is the opposite. Subtracting -4 would mean moving 4 units to the right. Therefore, subtraction can also be defined as adding the opposite. 2−(−4)=2+4 When doing subtraction problems, change the problem to adding the opposite before starting. KEEP, CHANGE, CHANGE! Diving Depths Fatima is learning how to scuba dive. She is experimenting with different depths as she swims in the ocean. If zero is the surface of the water, she starts 5 feet below that and then dives another 12 feet further. How far below the surface is she? When subtracting integers, remember to add the opposite. To help you organize subtraction with the different cases, see the table below. Discussion Question Why is subtracting a negative integer the same as adding a positive integer? Banking Gina has a savings account with $23.64 in it. She makes a withdrawal of $15.67 and then a deposit of $6.78. How much money is in her account? Remember this! ● When subtracting integers, fractions, or decimals, always change the problem to adding the opposite of the second number. Then, approach like addition. ● When adding, if the signs of the numbers are the same, add the values and then give the answer the same sign as the numbers you added. ● When adding, if the signs of the numbers are different, subtract the values, and then give the answer the sign of the larger number Properties of Addition In this lesson, you will combine multiple addition and subtraction expressions and solve. To help solve these types of problems, there are two properties that can be used. The first is the Commutative Property of Addition, which states that the order in which you add two rational numbers ____________________. So, for any two rational numbers a and b, a+b=b+a The second property is the Associative Property of Addition. This property states that for any three rational numbers, you may pick the two that you want to add together first. You do not have to add them in the order in which they appear in the problem. So, in this case, for any three rational numbers; a, b and c, a+b+c=(a+b)+c=a+(b+c) Notice the grouping symbols (parentheses) here. They indicate that it does not matter which two numbers you add together first. Pick the two that are easiest for you, then add the third. Other grouping symbols are braces, { }, brackets, [ ], and even absolute value, | |. This property also works for more than three rational numbers. Switch the Order How many different ways can you add 7+4+(−6)+(−3)? Multi-Step Multiplication and Division Properties of Multiplication Just like addition, multiplication has a commutative and associative property. The Commutative Property of Multiplication says that the order in which you multiply any two rational numbers, order does not matter. So, for any two rational numbers a and b, a×b=b×a ab=ba a⋅b=b⋅a Notice the three different ways to write the multiplication of two numbers. In addition, you could put parentheses around a and b, (a)(b)=(b)(a) It is important to note that if there is no sign between two variables (letters) or a variable and number, you can assume that the two numbers are being ______________________. The Associative Property of Multiplication states that for any three rational numbers, you may pick the two that you want to multiply together first. You do not have to multiply them in the order in which they appear in the problem. So, in this case, for any three rational numbers: a, b and c, a×b×c=a×(b×c)=(a×b)×c abc=a(bc)=(ab)c a⋅b⋅c=a⋅(b⋅c)=(a⋅b)⋅c Just like with the Associative Property of Addition, the Associative Property of Multiplication can use any of the grouping symbols, including absolute value. This property also works for more than three rational numbers. Switch the Order How many different ways can you multiply -8⋅2⋅-5⋅2? The Commutative and Associative Properties work for multiplication like they do for addition. There are two more properties to explore here: Identity and Inverse. Recall that for addition the additive identity was ________because any number you add to _________ is still that number. What number can you multiply by any number and the answer is still that number? a⋅1=a The Identity Property of Multiplication states that for any number, a, when multiplied by 1, the answer is still a. The Inverse Property of Multiplication is similar. What number times a is equal to 1? You can figure this out by working backwards from the identity property. Remember how when dividing by a fraction, you can flip the fraction and multiply by the reciprocal instead? Well, for any number a, you can multiply by its reciprocal to obtain 1. ________________________ It is important to note that the sign of the reciprocal of a number is the same as the sign of the number. Reciprocals are not opposites. Recall that the Associative and Commutative Properties do not work with division. For the last question in the interactive, you need to perform the indicated operations in order from left to right. Additionally, this interactive used multiplying by an inverse interchangeably with dividing by the number. Familiarize yourself with using both multiplying by fractions and dividing by a whole number. Depending on the problem, one might be easier than the other.