CHAPTER 2 NOTES
2-1 Adding Rational Numbers
In the real number system, for all numbers, “x”, there exists an opposite, “-x” such that when they are added together, you will always get zero.
X + (-X) = 0
(Inverse Property of Addition, sometimes called the property of opposites)
Example:
These opposite numbers are called additive inverses. Inverses are things that “undo” each other, so the positive and negative of the same number “undo” each other and you will always get zero when they are combined together.
The Additive Identity is zero. It’s the number that can be added, without changing the identity of the number. What’s the only thing you can add without changing the number? Zero! Thus zero is the additive identity.
X + 0 = X and 0 + X = X
(Identity Property of Addition, sometimes called the addition property of zero)
Example:
Rules:
When adding numbers with the same sign (such as two positive numbers or two negative numbers) add the numbers and keep the sign.
Think of a number line. If you start at a positive 6 and add another positive 7, you end on the number line at a positive 13.
Example:
Likewise, if you start on a number line at a negative 3 and add another negative 5, you go in the negative direction and end at a negative 8
Example:
If you are adding one matrix to another matrix of the same dimension, simply add the corresponding elements and simplify.
Example:
2-2 Subtracting Rational Numbers
When subtracting rational numbers, you can think of subtraction as adding the opposite, since the negative is the inverse of subtraction. If you do this, then you simply follow the rules for addition from the last section.
Example: -3 – (-6)
If you need a more visual representation, you can use tiles. Tiles are blocks that are used to help see positives and negatives. If you need to take away (subtract) negatives, but don’t have any, you get negatives by adding in “Zero Pairs” which contain a positive and a negative. Then you will have negatives to take away.
Example: -2 – (-7)
Overall the general rule when subtracting numbers is to add the opposite. However, an answer is still written with only one sign. If a number is negative, we do not write + (-4), simply we write -4.
2-3 Multiplying and Dividing Rational Numbers
Multiplicative property of zero:
Anything multiplied by 0 is 0.
Example: 5 * 0 = 0 and (-5) * 0 = 0
Multiplicative Property of 1 (Also called the multiplication identity)
Anything multiplied by one is itself.
Example: 3 * 1 = 3 and (-3) * 1 + (-3)
Multiplication Property of (-1)
Anything multiplied by a negative one is itself with the opposite sign.
Example: 4 * (-1) = (-4) and (-4) * (-1) = 4
Inverse Property of Multiplication (also called property of reciprocals)
Recall the inverse of addition was subtraction (or adding the opposite) because it “undoes” the addition. Here we are looking for what “undoes” the multiplication, but still by using multiplication. We call this number the reciprocal. When you multiply by the reciprocal, you get one.
The reciprocal of any number is itself as a fraction, flipped over.
Example: The reciprocal of ½ is 2/1 and the reciprocal of 5 is 1/5. When you multiply a number and it’s reciprocal, you should always get one.
General Rule for Multiplying and Dividing with negative numbers
If you have an odd number of negatives, the answer is negative, if you have an even number of negatives, the answer is positive.
2-4 The Distributive Property
The distributive property is similar to when you were a kid and brought treats for your class. You had to make sure everyone got the same number (otherwise the kids would fight) The number they got was the number to be distributed, and the kids were what they were distributed to.
In mathematical terms it would look like this:
Cupcakes to be given out (kid 1 + kid 2 + kid 3 + kid 4 + …)
The number of cupcakes you would need is the number you want to give each kid TIMES the number of kids.
Algebraically, it looks like this:
5(x + 3) = 5*x + 5 * 3 = 5x + 15
When simplifying, remember you can only combine LIKE terms. In other words, only terms with the same variable and the same exponent can be combined. Then you combine them by adding the coefficients. (The number in front of the term)
Examples:
2/3(6y + 9) -3(5x – 5)
2-5 Properties of Numbers
Commutative Property – works with addition and subtraction. It states the order of the terms does not matter
Example: a + b = b + a and ab = ba
Associative Property – works with addition and subtraction. It states that you can group any two items together and the value of the expression will not be changed, as long as the order of the terms does not change
Example: a + (b + c) = (a + b) + c and (ab)c= (a(bc)
Identity of Addition – States that adding zero to any real number does not change the number.
Example: a + 0 = a and 0 + a = a
Identity of Multiplication – States that multiplying any real number by one does not change the number.
Example: b * (1) = b and (1) * b = b
Inverse Property of Addition (Property of Opposites) – States that adding any number and its inverse will result in zero.
Example: w + (-w) = 0 and (-w) + w = 0
Inverse Property of Multiplication (Property of Reciprocals) – States that multiplying any number by its reciprocal will result in one.
Example: h * (1/h) = 1 and (1/h) * h = 1
Symmetric Property – States that if two expressions are equivalent, it does not matter which side of the equal sign they are on.
Example: if x = 3, then 3 = x is also true.
Distributive Property – states that each item in parenthesis must be multiplied by the same number on the outside.
Example: 5(x + 3) = 5*x + 5 * 3 = 5x + 15
Multiplication Property of Zero – states that anything multiplied by zero results in zero
Multiplication Property of (-1) – States that anything multiplied by (-1) is itself with the opposite sign.
Identifying properties is necessary to understand and justify Algebraic processes.
Deductive Reasoning – the reasoning you use when you make a conclusion based on FACT. For example, when it pours outside, you know for a fact that the ground will be wet.
Inductive Reasoning – the reasoning you use when you make a conclusion based on previous experiences, but not necessarily fact. For example, when it pours outside, businesses on Main Street get busy – (people are coming in out of the rain) but this can not be proven – there may not be any people on Main Street when it is pouring outside – your conclusion is just based on past observations.
2-6 Theoretical and Experimental Probability
The theoretical probability of an event, P(event), tells you how likely an event is to occur. It is found by taking the number of favorable outcomes over the total number of outcomes.
Example: If there are 12 red marbles in a jar of 50 marbles, the probability I draw a red marble on any given draw is 12/50 because there are 12 favorable outcomes, and 50 total outcomes.
Theoretically, this is what we expect, however if we actually draw 50 times (conduct an experiment) we might not get exactly 12 red marbles. This type of probability is called Experimental (because we actually conducted an experiment to get the results)
The Complement of an event is the probability that an event does NOT occur. In the example above, the complement of drawing a red marble would be the probability that we did NOT draw a red marble. This would be 38/50 since there are 38 non-red marbles out of the 50 total.
The odds of an event happening are the favorable to the non-favorable. In the example above again, the odds of drawing a red marble would be 12/38 since there are 12 red (favorable) and 38 non-red (non-favorable). Notice that the combined value is the total.
2-7 Probability of Compound Events
If two events are independent, the probability of each of them happening is the PRODUCT of the probabilities of each.
Example: The probability of rolling a 2 on a dice and then drawing an Ace out of a deck of cards would be 1/6 * 4/52 since the two events are unrelated.
1/6 * 4/52 = 1/78
If two events are DEPENDENT on each other, the probability is the PRODUCT of the first event and the second event after the first event has occurred.
Example: The probability of drawing an Ace then a Queen out of a deck of cards, without replacing the first would be 4/52 * 4/51 since there would now only be 51 cards from which to draw the Queen.
Independent vs. Dependent Events
The difference is found in the problem itself. Read carefully to determine if the problem is asking for “With Replacement” or “Without Replacement”
Random events – If an event is Random, then every event has an equal chance of being selected.
In the real number system, for all numbers, “x”, there exists an opposite, “-x” such that when they are added together, you will always get zero.
X + (-X) = 0
(Inverse Property of Addition, sometimes called the property of opposites)
Example:
These opposite numbers are called additive inverses. Inverses are things that “undo” each other, so the positive and negative of the same number “undo” each other and you will always get zero when they are combined together.
The Additive Identity is zero. It’s the number that can be added, without changing the identity of the number. What’s the only thing you can add without changing the number? Zero! Thus zero is the additive identity.
X + 0 = X and 0 + X = X
(Identity Property of Addition, sometimes called the addition property of zero)
Example:
Rules:
When adding numbers with the same sign (such as two positive numbers or two negative numbers) add the numbers and keep the sign.
Think of a number line. If you start at a positive 6 and add another positive 7, you end on the number line at a positive 13.
Example:
Likewise, if you start on a number line at a negative 3 and add another negative 5, you go in the negative direction and end at a negative 8
Example:
If you are adding one matrix to another matrix of the same dimension, simply add the corresponding elements and simplify.
Example:
2-2 Subtracting Rational Numbers
When subtracting rational numbers, you can think of subtraction as adding the opposite, since the negative is the inverse of subtraction. If you do this, then you simply follow the rules for addition from the last section.
Example: -3 – (-6)
If you need a more visual representation, you can use tiles. Tiles are blocks that are used to help see positives and negatives. If you need to take away (subtract) negatives, but don’t have any, you get negatives by adding in “Zero Pairs” which contain a positive and a negative. Then you will have negatives to take away.
Example: -2 – (-7)
Overall the general rule when subtracting numbers is to add the opposite. However, an answer is still written with only one sign. If a number is negative, we do not write + (-4), simply we write -4.
2-3 Multiplying and Dividing Rational Numbers
Multiplicative property of zero:
Anything multiplied by 0 is 0.
Example: 5 * 0 = 0 and (-5) * 0 = 0
Multiplicative Property of 1 (Also called the multiplication identity)
Anything multiplied by one is itself.
Example: 3 * 1 = 3 and (-3) * 1 + (-3)
Multiplication Property of (-1)
Anything multiplied by a negative one is itself with the opposite sign.
Example: 4 * (-1) = (-4) and (-4) * (-1) = 4
Inverse Property of Multiplication (also called property of reciprocals)
Recall the inverse of addition was subtraction (or adding the opposite) because it “undoes” the addition. Here we are looking for what “undoes” the multiplication, but still by using multiplication. We call this number the reciprocal. When you multiply by the reciprocal, you get one.
The reciprocal of any number is itself as a fraction, flipped over.
Example: The reciprocal of ½ is 2/1 and the reciprocal of 5 is 1/5. When you multiply a number and it’s reciprocal, you should always get one.
General Rule for Multiplying and Dividing with negative numbers
If you have an odd number of negatives, the answer is negative, if you have an even number of negatives, the answer is positive.
2-4 The Distributive Property
The distributive property is similar to when you were a kid and brought treats for your class. You had to make sure everyone got the same number (otherwise the kids would fight) The number they got was the number to be distributed, and the kids were what they were distributed to.
In mathematical terms it would look like this:
Cupcakes to be given out (kid 1 + kid 2 + kid 3 + kid 4 + …)
The number of cupcakes you would need is the number you want to give each kid TIMES the number of kids.
Algebraically, it looks like this:
5(x + 3) = 5*x + 5 * 3 = 5x + 15
When simplifying, remember you can only combine LIKE terms. In other words, only terms with the same variable and the same exponent can be combined. Then you combine them by adding the coefficients. (The number in front of the term)
Examples:
2/3(6y + 9) -3(5x – 5)
2-5 Properties of Numbers
Commutative Property – works with addition and subtraction. It states the order of the terms does not matter
Example: a + b = b + a and ab = ba
Associative Property – works with addition and subtraction. It states that you can group any two items together and the value of the expression will not be changed, as long as the order of the terms does not change
Example: a + (b + c) = (a + b) + c and (ab)c= (a(bc)
Identity of Addition – States that adding zero to any real number does not change the number.
Example: a + 0 = a and 0 + a = a
Identity of Multiplication – States that multiplying any real number by one does not change the number.
Example: b * (1) = b and (1) * b = b
Inverse Property of Addition (Property of Opposites) – States that adding any number and its inverse will result in zero.
Example: w + (-w) = 0 and (-w) + w = 0
Inverse Property of Multiplication (Property of Reciprocals) – States that multiplying any number by its reciprocal will result in one.
Example: h * (1/h) = 1 and (1/h) * h = 1
Symmetric Property – States that if two expressions are equivalent, it does not matter which side of the equal sign they are on.
Example: if x = 3, then 3 = x is also true.
Distributive Property – states that each item in parenthesis must be multiplied by the same number on the outside.
Example: 5(x + 3) = 5*x + 5 * 3 = 5x + 15
Multiplication Property of Zero – states that anything multiplied by zero results in zero
Multiplication Property of (-1) – States that anything multiplied by (-1) is itself with the opposite sign.
Identifying properties is necessary to understand and justify Algebraic processes.
Deductive Reasoning – the reasoning you use when you make a conclusion based on FACT. For example, when it pours outside, you know for a fact that the ground will be wet.
Inductive Reasoning – the reasoning you use when you make a conclusion based on previous experiences, but not necessarily fact. For example, when it pours outside, businesses on Main Street get busy – (people are coming in out of the rain) but this can not be proven – there may not be any people on Main Street when it is pouring outside – your conclusion is just based on past observations.
2-6 Theoretical and Experimental Probability
The theoretical probability of an event, P(event), tells you how likely an event is to occur. It is found by taking the number of favorable outcomes over the total number of outcomes.
Example: If there are 12 red marbles in a jar of 50 marbles, the probability I draw a red marble on any given draw is 12/50 because there are 12 favorable outcomes, and 50 total outcomes.
Theoretically, this is what we expect, however if we actually draw 50 times (conduct an experiment) we might not get exactly 12 red marbles. This type of probability is called Experimental (because we actually conducted an experiment to get the results)
The Complement of an event is the probability that an event does NOT occur. In the example above, the complement of drawing a red marble would be the probability that we did NOT draw a red marble. This would be 38/50 since there are 38 non-red marbles out of the 50 total.
The odds of an event happening are the favorable to the non-favorable. In the example above again, the odds of drawing a red marble would be 12/38 since there are 12 red (favorable) and 38 non-red (non-favorable). Notice that the combined value is the total.
2-7 Probability of Compound Events
If two events are independent, the probability of each of them happening is the PRODUCT of the probabilities of each.
Example: The probability of rolling a 2 on a dice and then drawing an Ace out of a deck of cards would be 1/6 * 4/52 since the two events are unrelated.
1/6 * 4/52 = 1/78
If two events are DEPENDENT on each other, the probability is the PRODUCT of the first event and the second event after the first event has occurred.
Example: The probability of drawing an Ace then a Queen out of a deck of cards, without replacing the first would be 4/52 * 4/51 since there would now only be 51 cards from which to draw the Queen.
Independent vs. Dependent Events
The difference is found in the problem itself. Read carefully to determine if the problem is asking for “With Replacement” or “Without Replacement”
Random events – If an event is Random, then every event has an equal chance of being selected.