Why do we classify numbers? Why do we give them names like integers, irrational numbers, or negative numbers? For the same reason we classify anything. We want to make sure that everyone has an understanding of what specific numbers are called and what they mean.
After all, there is a difference between 25 and negative 32 and 4 to the 6th power. In this Mometrix video, we will provide an overview of numbers and their classifications. Numbers are our way of keeping order. We count the amount of money we have.
We measure distance. We use percentages to indicate a sale. Numbers are an integral part of our everyday existence, whether they are whole numbers, rational numbers, or the first type of numbers we're going to look at, real numbers. A real number is any value of a continuous quantity that can represent distance on a number line.
Essentially, it's any number you can think of. 50 is a real number. 1 billion is a very large real number.
Real numbers encompass three classifications of numbers, which we'll talk about in a little bit. Whole numbers, rational numbers, and irrational numbers are all real numbers. Imaginary numbers are not real numbers. They are complex numbers that are written as a real number multiplied by an imaginary unit i.
For instance, the square root of negative 1 calculates as the imaginary number i, and the square root of negative 25 is 5i. Even though imaginary numbers aren't real numbers, they do have value. Electricians use imaginary numbers when working with currents and voltage.
Imaginary numbers are also used in complex calculus computations. So, just because these numbers are called imaginary doesn't mean they aren't useful. Whole numbers are numbers that we count with. 1, 2, 3, 4, and 5 are all whole numbers.
So are negative 17 and 0. Whole numbers do not have fractions or decimals. All whole numbers are called integers. Integers can be positive or negative whole numbers. All integers and whole numbers are part of a bigger group called rational numbers.
This group also includes fractions and decimals. That means that 3 fifths and 7.25 are rational numbers. Rational numbers can also be positive or negative. Rational numbers have opposites, which are called irrational numbers. These numbers can't be written as a simple fraction.
Pi is the most famous irrational number. We have a close approximation of how to calculate pi, but it's just a close approximation. Pi is renowned for going on and on forever. That's why it's an irrational number. You can't easily write it.
as a fraction. Natural numbers are those that are positive integers, although there is some debate as to whether natural numbers start at 0 or 1. Negative numbers are, well, exactly that. They are the numbers below 0. There are several other number classifications as well. Numbers are divided into even and odd numbers. If you can divide a number by 2, that number is even.
So 24 36 and 74 are all even numbers because if you divide them by 2 you get 12, 18, and 37. Even numbers always end with 0, 2, 4, 6, or 8. Odd numbers can't be divided by 2 and leave a whole number. Any odd number divided by 2 will leave a fraction. So 17 divided by 2 is 8.5.
23 divided by 2 is 11.5. All odd numbers will end in 1, 3, 5, 7, or 9. Numerators and denominators form fractions, which are compromised of two integers. The number on top is the numerator.
The number on the bottom is the denominator. The numerator, the top number, shows how many parts we have. The denominator, the bottom number, shows how many parts make a whole. Let's say you have 6 apples and 3 of the apples get eaten. The number of apples you have left over would be displayed as 3 sixths.
You would then divide 3, the top number, into 6, the bottom number, to determine the percentage of remaining apples. In this case, the number is 50%. So, that's our look at numbers and their classifications. From whole numbers to irrational numbers, we need to know what to call numbers so we can know what they mean. Hi, welcome to this video on rational and irrational numbers.
Rational and irrational numbers comprise the real number system. This Venn diagram shows a visual representation of how real numbers are classified. The natural numbers comprise the smallest subset, which is also known as the set of counting numbers. These are all positive, non-decimal values, starting at 1. All numbers are the natural numbers. plus the value of 0. The integer set of numbers includes whole numbers in all negative non-decimal values.
Rational numbers include all the sets seen here, in addition to the fractional values in between. An easy way to remember this is that the word ratio is in the name of this classification. All numbers included in the rational number set can be written as a ratio of integers.
If A And b are integers. Rational numbers can be written as a over b, as long as b does not equal 0. Clearly, the set of integers can be written as ratios because any integer divided by 1 results in the original integer. As illustrated here, integers can be expressed as fractions in infinite numbers of ways. The integer 3 can be represented as the fractions 3 over 1, 6 over 2, and even negative 24 over negative 8. The integer negative 5 can be represented as the following fractions.
And the integer 0 can be represented as the fractions. As a side note, these are not the only fractions that result in these integers. They're just a few of the many examples that exist.
Fractions can also be written as decimals. For example, point 1 is equivalent to 1 over 10 because the 1 is in the tenths decimal place. 0.13 is equivalent to 13 one hundredths because the 3 is in the hundredths decimal place and the 1 is in the tenths decimal place. Point 237 is equivalent to 237 over 1000 because the 7 is in the thousandths decimal place and so on. These decimals can be written as fractions, so they are considered rational.
Other decimals have repeating patterns. These are also considered rational because they can be expressed as a fraction based on the following proof. The repeating decimal 2.17 represents the digits 2.17171717 and so on. So let's try this as a practice problem. Let x equal 2.17 repeating.
which represents the hundredths place. So let's multiply both sides of the equation by 100. 100 times x, 2.17 repeating times 100. This results in 100x. which is equal to 217.17 repeating.
We moved the decimal over two spots because we multiplied by 100. Now let's subtract the original equation from this one, meaning minus x and minus 2.17 repeating. So we have 99x is equal to 215. Notice that the repeating portion of the decimal is now eliminated. Solving for x results in dividing both sides by 99. x is equal to 215 over 99. This is a fractional representation of x equal to 2.17 repeating.
This proof shows that repeating decimals are also considered rational because they can be written as a fraction of integers. If you plug this into your calculator, you will get something close to, probably rounded, 2.17 repeating. It is important to note that not all decimals are repeating.
Some decimals have an infinite number of non-repeating digits, and therefore cannot be expressed as a fraction of integers. These types of real numbers are classified as irrational. While there are an infinite number of irrational numbers in the real number system, most commonly used in mathematics are the square roots of non-perfect squares, like square root of 2, for example, and the constants pi and e. The notation for irrational numbers allows for efficiency in math applications.
For geometry, you may recall that pi is equal to about 3.141594 infinity. This is derived from the circumference of any circle and its diameter. Because the decimal value is non-repeating and infinite, we use an approximate value in math applications. Business applications regarding continuously compounded interest employ the irrational value of e, which has an approximate value of 2.718 for infinity. As you know, there are times when we have to algebraically adjust how a number or an equation appears in order to proceed with our math work.
We can use the greatest common factor and the least common multiple to do this. The greatest common factor is the largest number that is a factor of two or more numbers, and the least common multiple is the smallest number that is a multiple of two or more numbers. To see how these concepts are useful, let's look at adding fractions. Before we can add fractions, we have to make sure the denominators are the same. by creating an equivalent fraction.
In this example, the least common multiple of 3 and 6 must be determined. In other words, what is the smallest number that both 3 and 6 can divide into evenly? With a little thought, we realize that 6 is the least common multiple, because 6 divided by 3 is 2, and 6 divided by 6 is 1. The fraction, 2 over 3, is then adjusted to the equivalent fraction, 4 over 6, by multiplying both the numerator and denominator by 2. Now the two fractions with common denominators can be added for a final value of 5 over 6. In the context of adding or subtracting fractions, the least common multiple is referred to as the least common denominator.
In general, you need to determine a number larger than or equal to two or more numbers to find their least common multiple. It's important to note that there is more than one way to determine the least common multiple. One way is to simply list all the multiples of the values in question and select the smallest shared value, as seen here. This illustrates that the least common multiple of 8, 4, and 6 is 24, because it is the smallest number that 8, 4, and 6 can all divide into evenly. Another common method involves the prime factorization of each value.
Remember, a prime number is only divisible by 1 and itself. Once the prime factors are determined, list the shared factors once, and then multiply them by the other remaining prime factors. The result is the least common multiple.
The least common multiple can also be found by common or repeated division. This method is sometimes considered faster and more efficient than listing multiples and finding prime factors. Here's an example of finding the least common multiple of 3, 6, and 9 using this method. Divide the numbers by the factors of any of the three numbers.
6 has a factor of 2, so let's use 2. 9 and 3 cannot be divided by 2, so we'll just rewrite 9 and 3 here. Repeat this process until all of the numbers are reduced to 1. Then, multiply all of the factors together to get the least common multiple. Now that methods for finding least common multiples have been introduced, we'll need to change our mindset to finding the greatest common factor of two or more numbers.
We will be identifying a value smaller than or equal to the numbers being considered. In other words, ask yourself, what is the largest value that divides both of these numbers? Understanding this concept is essential for dividing and factoring polynomials.
Prime factorization can also be used to determine the greatest common factor. However, rather than multiplying all the prime factors like we did for the least common multiple, we'll multiply only the prime factors that the numbers share. The resulting product is the greatest common factor.
Let's wrap up with a couple of true or false review questions. Number 1. The least common multiple of 45 and 60 is 15. The answer is false. The greatest common factor of 45 and 60 is 15, but the least common multiple is 180. Number two, the least common multiple is a number greater than or equal to the numbers being considered. The answer is true. The least common multiple is greater than or equal to the numbers being considered, while the greatest common factor is equal to or less than the numbers being considered.
Thanks for watching, happy studying! Hey guys, today we're going to take a look at the mathematical operations addition, subtraction, multiplication, and division. These four operations serve as the fundamental building blocks for all math, so it is crucial to have a solid understanding to build upon.
Let's dive in. We use addition and subtraction to solve many real-world situations. Addition and subtraction are simply the mathematical terms used to describe combining and taking away.
When we add, we are combining or increasing. When we subtract, we are taking away or decreasing. As a reminder, the symbol we use for addition is a plus sign, the answer to an addition problem is called the sum, the symbol we use for subtraction is a minus sign, and the answer to a subtraction problem is called the difference.
Essentially, addition and subtraction are opposite operations. One adds value and the other deducts value. One strategy for visualizing these two operations is to use a number line. We will use a number line to illustrate the following examples.
Let's imagine a situation that involves the sale of popcorn. For this scenario, let's assume that you are trying to raise money by selling bags of popcorn, and you start with 20 bags. When your first customer arrives, they wish to purchase 4 bags of popcorn. This means that your remaining number of bags will decrease.
We can represent this situation with a simple equation that involves subtraction. We started with 20 bags, and we decreased by 4, or subtracted 4. Our subtraction equation is written as 20 minus 4 equals 16. On a number line, we can represent this deduction by starting at 20 and then moving backwards 4 units in the negative direction. Each jump backwards represents subtraction by 1. Now let's say you started with 20 bags of popcorn and ended up with 6 bags left at the end of the day.
You need to replenish your stock in order to keep up your sales, so you make 4 more bags of popcorn. How many bags of popcorn do you now have available to sell? For this scenario, since we are looking at an increase of bags, we will use addition. This situation can be described using the equation 6 plus 4 equals 10. You had 6 bags initially, and then combined that amount with 4 more bags.
Now you have 10 bags in all. On a number line, addition is represented by jumps to the right in the positive direction. Each jump to the right represents the addition of one unit. So in this example, we would be starting at 6 and jumping 4 units to the right. We can see that we land on 10. It is important to notice that when using addition, the order of the values does not matter.
For example, 10 plus 30 is the same as 30 plus 10. The placement or arrangement of the values has no effect on the outcome. Both arrangements would equal 40. However, the same is not true for subtraction. Does 30 take away 10 mean the same thing as 10 take away 30?
Clearly not. We can see that the order matters when dealing with a situation involving subtraction. The technical term for this quality is known as the commutative property.
Essentially, the property is true for operations where the values can move around, commute, and the outcome of the expression or equation will not change. The commutative property applies to addition, but not to subtraction. Another operation that also shares the commutative property is multiplication. Let's discuss multiplication together with division as we did for addition and subtraction.
Multiplication and division are similar to addition and subtraction in that they perform opposite functions. The function of multiplication is to represent multiple groups of a certain value, whereas division is designed to show the separating or subdividing of a value into smaller groups. As a reminder, the symbol we use for multiplication is a time sign. The answer to a multiplication problem is called the product. The symbol we use for division is a division sign.
And the answer to a division problem is called the quotient. Multiplication is essentially a convenient and time efficient way to show what's called repeated addition. For example, if you need to fill 30 bags of popcorn and each bag requires 60 kernels, it could take hours to count up how many kernels you need in total by just using addition.
A faster and more efficient way to make this calculation would be to use repeated addition. Instead of counting each seed independently, we would group them up and add the groups together. The calculation would then become 30 groups of 60. This grouping, for the purpose of repeated addition, is the multiplication process at its core.
30 groups of 60 is written as 30 times 60, which is 1,800. So, 1,800 kernels are required to fill up 30 bags of popcorn. Both addition and multiplication are commutative, because the order does not affect the answer. 30 groups of 60 gets us the same result as 60 groups of 30. Our last operation, division, can be considered multiplications opposite. When we use division, we are essentially splitting up a large group into smaller subgroups.
For our popcorn example, we can use division to answer the following question. How many bags of popcorn can I make using 1,800 kernels if each bag requires 60 seeds? This situation requires us to divide the large value, 1,800, into groups of 60. Each smaller subgroup will now represent a bag of popcorn. 1,800 divided into groups of 60 is represented as 1,800 divided by 60. In this case, the answer is 30. So 30 bags of popcorn can be made with our 1,800 kernels. As you can see, division is not commutative, because the order of the values plays a crucial role in determining the answer.
1,800 divided by 60 is not the same thing as 60 divided by 1,800. Now, I know what you are thinking. What does this phrase actually mean?
Quite a bit actually, because that saying provides the key to remembering an important math concept. The order of operations. The order of operations is one of the more critical mathematical concepts you'll learn because it dictates how we calculate problems. It gives us a template so that everyone solves math problems the same way.
Let's start off with a simple question. What is an operation? An operation is a mathematical action. Addition, subtraction, multiplication, division, and calculating the root are all examples of a mathematical operation.
Let's take a look at this problem. Looks easy, right? Well, it wouldn't be so easy if we didn't understand the order in which the math operation occurs. If we didn't have rules to determine what calculations we should make first, we'd come up with different answers.
Should you start by subtracting 4 minus 6 and then multiplying by 7? No, the order of operations tells us how to solve a math problem. And this brings us back to Aunt Sally. Operations have a specific order, and this is what Please Excuse My Dear Aunt Sally helps us to understand.
It's an acronym PEMDAS, or PEMDAS, that tells us in which order we should solve a mathematical problem. So first is PLEASE, which stands for parentheses. So we solve everything inside of the parentheses first.
Then e, excuse, which is for exponents. We solve that after we solve everything in a parentheses. Multiplication, which is the my, and this happens from left to right.
And then division, which is the dear, which also happens left to right. And then we have addition and subtraction, which also happens from left to right. And this is aunt and Sally.
Okay, so now that we know the order of operations, let's apply it to our problem that we have here and solve. So if, let's kind of go down our list. We don't have parentheses and we don't have exponents, but we do have multiplication. So we do that before we do any addition and subtraction. So let's go ahead and multiply 7 times 4. That gives us 28, and now we're subtracting 6. Which gives us 22. Now let's look at another problem.
Without the operations, you could calculate this as 7 plus 7, which is equal to 14, times 3, which is equal to 42. And this would be wrong. Remember, you multiply before you add. Therefore, the equation should look like this. So, when we do problems like this, we can use parentheses to group together our numbers that are going to take place first.
So, in this case, it's 7 times 3. And when we do that, we get 21, and we have plus 7 left over. When we add those together, we get 28. And that's our answer. Let's look at some more complex problems.
The order of operations dictates how to solve this problem. Remember, you multiply exponents first. Here is the wrong way to solve the problem. Why is that wrong?
Because you violated the order of operations. You do not multiply first. You perform an operation on the exponent first.
This is how it should be done. See, solving the equation in the right order provides the correct answer. Let's try out one more problem. This one is a little bit more challenging, but it perfectly illustrates the order of operations.
Remember the order. What do we do first? The number inside the parentheses.
So, 8 times 6 is equal to 48, then we subtract 15 and that gives us 33. Here is how the problem looks now. So, our next step is multiplication and division. So, let's perform all of our multiplication and division problems and then see what we have left. Now we finish with addition and subtraction.
So here's what we have. And our answer is 37. There is an exception. If an equation only has one expression, you don't have to follow the order of operations.
Here are some examples of single expressions. 10 plus 10. Well, there are no other operations, so you just need to go ahead and add them together and you get 20. Same thing with subtraction, multiplication, or division. All of those are single expressions.
Before we dive in, let's review the basic parts of a fraction. Remember, a fraction simply represents a part of a whole. It has a numerator and a denominator, which tells us what the part is and what the whole is.
Let's look at the fraction 3 fourths as an example. We can see that the 3 is our numerator and the 4 is our denominator. So the fraction 3 fourths is really saying 3 parts out of 4 parts total.
It can also be helpful to visualize 3 fourths as simply 1 fourth plus 1 fourth plus 1 fourth. It is very important to remember that a denominator of 4 does not represent the value of 4. A denominator of 4 represents the value of 1 that is divided up into 4 equal parts, or 4 fourths. The type of fraction we're working with here represents a value less than one whole.
3 fourths is not quite one. If we had 4 fourths, that would be equivalent to one, but we only have 3 out of 4 parts. We see and use fractions that are less than one all the time in our daily lives, whether it's for things like recipes or keeping track of time.
Recipes often call for amounts such as 1 half teaspoon of salt, and we often keep track of time in terms of quarter hours. like a quarter past three for 315. Though we observe this type of fraction very frequently in our daily lives, it is not the only type of fraction. Consider the following scenario. You're ordering pizza for a big celebration.
There will be lots of hungry guests at this celebration, so you order three pizzas. Each pizza is cut into six slices. This means that each pizza has six equal parts, and as a fraction, six would be considered our whole, or our denominator. If your first guest eats two slices, we would represent this as the fraction 2 sixths. Two parts out of six parts total.
But what if that first guest was really hungry and grabbed seven slices? Again, each pizza was cut into six equal slices, so six remains as our whole, or denominator. But this time our part is seven.
In this scenario, our numerator is larger than our denominator, seven over six. Fractions with a numerator larger than their denominator are referred to as improper fractions. Essentially, improper fractions equal a value that is more than 1. One whole pizza would be represented by 6 over 6, or 6 sixths. 7 over 6 represents 7 sixths, which is more than one pizza.
This could be visualized as 1 over 6 plus 1 over 6 plus 1 over 6 plus 1 over 6 plus 1 over 6 plus 1 over 6 plus 1 over 6 plus 1 over 6. equals 7 over 6. It can also be written in another form called a mixed number. An improper fraction and a mixed number will represent the same amount, but simply be written in a different form. For example, the improper fraction 7 over 6 could also be written as the mixed number 1 and 1 sixth.
Mixed numbers and improper fractions share the same amount, but as a mixed number, the parts are collected and consolidated into as many groups of one whole as possible. For example, 4 over 4 would be grouped together as 1. 7 over 7 would also be grouped together as 1. Any value where the numerator is equivalent to the denominator would be expressed simply as 1. In our pizza example, the guest took 7 slices from a group of pizzas that were sliced into sixths. We said this could be represented as the improper fraction 7 over 6, or visualized as 1 over 6 plus 1 over 6 plus 1 over 6 plus 1 over 6 plus 1 over 6 plus 1 over 6. plus 1 over 6. As a mixed number, we would group 6 of these sixths in order to form 6 over 6, or one whole. By grouping 6 over 6 together, we can see that 1 over 6 is left over. We would write our mixed number as 1 and 1 sixth.
Let's try a few more examples. Let's write the following improper fractions as mixed numbers. 4 thirds can be visualized as 1 over 3 plus 1 over 3 plus 1 over 3 plus 1 over 3. We know that 3 over 3 is equal to So let's group three of these thirds together. We are now left with one and one third as our mixed number.
Three halves can be visualized as one half plus one half plus one half. We then know that 2 halves, 2 over 2, makes one whole, and we're left with one half left over. So 3 halves as a mixed number is 1 and 1 half. Let's try one more example, 7 fourths. 7 fourths is the same as 1 fourth plus 1 fourth.
plus 1 fourth, plus 1 fourth, plus 1 fourth, plus 1 fourth, plus 1 fourth. Now we know that 4 fourths are grouped as one whole. So these 4 fourths are pulled over to equal 1, and we're left with 1, 2, 3 fourths. 7 fourths, written as a mixed number, is 1 and 3 fourths.
This process will take place in reverse in order to convert between a mixed number to an improper fraction. For example, If we started with a mixed number 1 and 3 fourths, and we wanted to convert it to an equivalent improper fraction, we would take a look at the whole number, in this case it is 1. This whole number is really representing the denominator that's in the fraction. In this case, it's 4, so the 1 is equal to 4 over 4. When we combine these 4 fourths with the 3 fourths, we end up with 7 fourths total, or 7 over 4. Welcome to this video on rates, and more specifically, unit rates.
This math concept is practical and useful, so we hope that you come away with a solid understanding and confidence in interpreting the rates that are all around you. Let's get started. Rates are spoken about and used every day in many aspects of life.
The concept is pretty straightforward, as rates are nothing more than ratios of values that represent different units of measure. The values that are being compared are called the terms of the rate. Let's consider the treasure trove of rates that may be in your grocery cart the next time you go shopping.
In the examples listed here, the rate that you agree to pay is simply the cost related to the quantity of the product. Let's say we have a 12-ounce box of pasta, which costs $1.49. The rate is then $1.49 per 12 ounces. If we have a pound of deli meat, and the cost is $9.99, our rate would then be $9.99 per pound. Lastly, let's say we have an 8-pack of 20-ounce bottles of soda, and the cost is $5.98.
Our rate, then, is $5.98 per pack. If we look at this last example a little closer, we'll see that there is room to break down the rate even further. The example provides the price of an 8-pack, but what if I want to determine the cost of one 20-ounce bottle? The ratio of $5.98 to 8 bottles provides that information. This quick calculation tells me that each 20 ounce bottle costs approximately 75 cents.
This breakdown to determine the cost per bottle may be helpful to determine whether I buy the 8 pack of one type of soda or the individual 20 ounce bottles of another brand on sale for 50 cents each. The process of breaking down the cost to a smaller unit reveals the unit rate of the product. This is helpful to make informed decisions at the store, as the volumes of product in various packaging are often different. By comparing unit rates, savvy customers are able to make price comparisons based on common units of the product, regardless of packaging and advertised sale prices. Let's break down the soda cost per bottle further to determine the cost per ounce.
If one 20-ounce bottle costs roughly 75 cents, then dividing that cost by 20 ounces reveals the cost per ounce. So 75 cents per ounce divided by 20 ounces gives you roughly 4 cents per ounce. As you can see, breaking down costs to the smallest unit reveals the cost savings of the sale.
Of course, saving a few cents per ounce of soda may not be the deciding factor of your purchase. Other factors come into play when consumers are shopping, like brand loyalty and personal preference. However, comparing unit costs provides an objective way of making consumer choices based on the price. The important thing to remember when analyzing unit rates is that the units must be the same. Let's consider another example to illustrate this point.
Suppose you are on a road trip in Wyoming and on the first day you covered 300 miles in four hours of mostly highway driving. You can quickly determine your average rate of speed as miles per hour with the following calculation. Coincidentally, your friend is traveling in Germany, where the standard unit of measure is in kilometers. And she reports that she covered approximately 513 kilometers in four hours on her first day of the road trip.
Her average speed would be calculated as... Clearly, this is comparing apples to oranges in the sense that the underlying units are not the same. A conversion of either miles to kilometers or kilometers to miles must be made to make a fair comparison of average speed. Keep in mind that a kilometer is a shorter unit of distance than a mile.
One mile is equal to approximately 1.609 kilometers. To convert your average speed of 75 miles per hour to kilometers, simply multiply 75 by 1.609. On the other hand, you could convert your friend's reported km per hour to miles per hour.
One kilometer is equal to approximately.6215 of one mile. Multiply this conversion factor by your friend's daily average speed to convert to miles per hour. The way that you convert does not matter as long as you compare the average speeds of the same unit.
Both conversions show that your friend in Europe traveled at a faster rate on the first day of her trip. Hi and welcome to this video on ratios. Ratios are used all the time in many aspects of our lives. In this video, we'll review what ratios represent and how they should be interpreted.
As mentioned, ratios are frequently used, but sometimes not fully understood. Simply put, ratios are an efficient way to compare numeric values of different categories. For example, let's say that you have a room of 20 people, comprised of 12 women and 8 men. The two categories are men and women, so your ratios would look like this.
The ratio of women to men is 12 to 8, and the ratio of men to women is 8 to 12. Note. that ratios can be simplified by dividing both values by the common factor of 4, which will not change the meaning of the ratio. The simplified ratio still represents the relationship of the number of women to men, 3 to 2, and the number of men to women, 2 to 3. There are two ways that ratios can be written. This example uses a colon, but you could also use a division bar to form a fraction.
12 over 8 simplifies to 3 over 2. It is very important to note that the order of the categories matters when building a ratio. The first value is technically referred to as the antecedent, and the second value is referred to as the consequent. The antecedent is always compared to the consequent.
Interpreting the ratio in context can sometimes be tricky, but becomes easier with practice. Let's look at some examples of ratios you might come across in day-to-day life. Cooking. You are making a batch of your mom's delicious salad dressing. The recipe calls for many ingredients, including 2 cups of extra virgin olive oil and 3 cloves of chopped garlic.
In this case, the ratio of oil to cloves is 2 to 3, or 2 over 3, and the ratio of garlic to oil is 3 to 2, or 3 over 2. In cooking, you might hear this referred to as 2 parts oil, 3 parts garlic. This is because the units here are different. Cloves vs. cups.
Shopping. Grocery store displays allow us to compare the value of products based on their unit cost. Many times, this is simply a quick ratio of cost to unit, with a unit being ounce, pound, etc. Suppose you're scanning the cereal aisle and narrow your choice down to your two favorites. You decide to be practical, and base your decision purely on cost.
Both have similar prices, but the volume of the packaging is different. Calculating a quick ratio of cost to unit for each box will reveal the better value. Brand A costs $5.79 and has a volume of 20.35 ounces. Dividing the price by the volume gives us the cost of the cereal per ounce, 28 cents. Brand B costs $6.39 and has a volume of 24.15 ounces.
Dividing the price by the volume gives us 26 cents per ounce. Despite being a higher cost per package, choice B is a better value by unit price. This example was fairly straightforward because the volume of the cereal packages were both measured in ounces. Many times a unit conversion must be made before a unit price can be determined.
For example, if we're trying to compare the unit cost of a pound of fresh broccoli to a package of frozen broccoli measured in ounces, we would need to know that 1 pound is equal to 16 ounces before we could calculate the unit cost ratio. In this case, the frozen broccoli would be the better value based on the smaller unit price per ounce. A linear equation can be expressed in many different ways, but no matter which form you use, it just represents a straight line. The standard form of a linear equation is written as ax plus by equals c, where a, b, and c are constants, and x and y represent variables. This form of the equation is very useful for some purposes in math.
For example, a line can be quickly graphed when it is in this form by finding the x and y intercepts. There are also methods of solving systems of equations that require each equation in the system to be written in this form. Rearranging the standard form equation into slope-intercept form, y equals mx plus b, reveals other key features of the line, namely the slope and the y-intercept.
The slope of a line describes the slant, or steepness, and the y-intercept is the point on the graph where the line crosses the y-axis. Notationally, slope is represented by m, and the y-intercept is represented by b. Because the y-intercept is an actual point on the coordinate plane, It is represented as an ordered pair, 0, b. Sometimes you will be asked to rearrange an equation from one form to another. Here's an example.
2x plus 3y equals 12. Remember, the standard form of a linear equation is ax plus by equals c. So this is currently in standard form. If we want to change it to slope-intercept form, we are going to need to rearrange it so that y is by itself on our left side. Our first step is to subtract 2x from both sides.
3y equals 12 minus 2x is what we have now. Then we're going to divide everything by 3, which gives us y equals 4 minus 2 thirds x. This is almost right, but if we look again at slope-intercept form, we see that we need the x term to be in front.
Thankfully, we can think of this as 4 plus negative 2 thirds x, and use the commutative property of addition to swap their places. This gives us y equals negative two-thirds x plus 4. The resulting equation is more informative about the line than the original equation in standard form. The coefficient of x negative two-thirds is the slope.
A negative slope tells us that the line slants downward from left to right. The y-intercept of 4 tells us that the line crosses the y-axis at the point. Now that we have seen how to convert a standard form equation into slope-intercept form, let's practice recognizing the key features of slope and the y-intercept with a few examples.
For these examples, we want to name the slope, describe the slant of the line, and name the y-intercept as an ordered pair. y equals 2x plus 3. The slope of this equation is m equals 2, the coefficient of the x variable. A positive slope indicates that the line slants upward from left to right. The y-intercept is b equals 3, which indicates that the line crosses the y-axis at the point. Let's try another one.
y equals 3 fifths x minus 2 thirds. The slope of this equation is m equals 3 fifths. Because the slope is positive, the line slants upward from left to right. The y-intercept is negative two-thirds, which indicates that the line crosses the y-axis at the point. Let's try one more.
y equals negative 5x minus 2. The slope is m equals negative 5. Negative slope means that the line slants downward from left to right. The line crosses the y-axis at the point. As you can imagine, knowing where the line crosses the y-axis and the slope of the line will make the line very easy to graph.
Now let's take a look at how slope provides you with instructions to graph from the y-intercept. Any value of slope can be looked at as a fraction, where the numerator indicates where to move along the y-axis, and the denominator indicates where to move along the x-axis. Movement along the y-axis is typically referred to as the rise.
A positive rise value would instruct a move up the y-axis, while a negative rise would indicate a move down the y-axis. Likewise, a positive run value would mean a shift to the right, and a negative run would mean a shift to the left. Here are a few examples to practice identifying the rise and run indicated by a given slope.
M equals 5. This slope is not written as a fraction, but any whole number can be rewritten as a fraction over 1. m equals 5 over 1. The numerator, 5, is the rise. The positive value means up 5. The denominator is the run. Positive value means right 1. Let's try another one. m equals negative 2 thirds.
When you have a negative slope, you can consider either the numerator or the denominator to be negative. Not both. For this example, let's consider the numerator, the rise, to be the negative value. Negative value means down 2. The denominator 3 is the run.
Positive value means right 3. Here's one last example. m equals negative 3. First we need to rewrite the whole number as a fraction. Negative 3 over 1. Let the numerator be the negative value.
Rise is negative 3. Down 3. Run is 1. Write 1. Now onto some graphing. People sometimes find it helpful to use the notation of slope-intercept form to get started. Begin at b and move according to m. Let's graph the linear equation in slope-intercept form.
y equals two-thirds x minus 2. Step 1. Begin at b. Plot the y-intercept 0, negative 2. First point. Step 2. Move by m equals 2 thirds.
Rise equals up 2. Run equals right 3. Plot the second point at 3, 0. Step 3. Repeat the rise equals up 2 and run equals right 3. Plot the third point at 6, 2. Step 4. Draw a straight line through the three points. Got the hang of it? Let's try one more.
y equals negative 2x minus 3. Step 1. Begin at B. Plot the y-intercept first point. Step 2. Move by m equals negative 2 over 1. Let the rise be negative. Rise equals down 2. Run equals right 1. Plot the second point at.
Step 3. Repeat the rise equals down 2 and run equals right 1. Plot the third point at 2, negative 7. Step 4. Draw a straight line through the three points. Note the slant downward from left to right due to the negative slope in this equation. Now that we have had some review of the key features of linear equations, we have the tools to explore the point-slope form.
This form is of special use if we know one point that is on the line and the slope. The general form of this arrangement is y minus y1. equals m times x minus x1, where m equals the slope and x1, y1 is another point that is known on the line. Using this template, let's practice identifying the slope and the point from the following examples of point-slope form. y minus 5 equals 3 times x minus 2. This is a straightforward example.
First, identify the slope as the coefficient outside the parentheses, m equals 3. When naming the point on the line, Note that in the general form, the x coordinate is being subtracted from x, and the y coordinate is being subtracted from y. So the ordered pair of the point will be. Here's another one. y plus 3 equals negative 1 half times x minus 4. The slope in this equation is m equals negative 1 half. How is this equation different from the general form y minus y1 equals m times x minus x1?
You may have noticed that the y value of the point 3 is being added. To identify the point that is on this line, the equation must look like the general form, which subtracts the coordinates of the point. Therefore, the point can be seen more clearly if the equation is written as y minus negative 3 equals negative one-half times x minus 4. Subtracting a negative value is the same as addition.
Now we can see that this line travels through the point 4, negative 3, and has a slope of m equals negative one half. Let's look at one more example. y plus 12 equals negative 3 times x plus 5. By now you can quickly see that the slope of this equation is m equals negative 3. This equation also does not match the general form, but it can be rewritten as follows.
y minus negative 12 equals negative 3 times x minus negative 5. This adjustment reveals that the point that is on the line is negative 5, negative 12. Once you feel comfortable with identifying the slope and the point from this form, you can graph the line as we did before. y minus 5 equals 1 half times x minus 2. Step 1, identify the slope m equals 1 half. Step 2, identify the point on the line 2, 5. Some students find it helpful to switch the sign of the given formula to determine the coordinates of the point.
Step 3, identify the point on the line 2, 5. Step 3. Plot the point as the first point. Step 4. Move by m equals one half. Rise equals up one. Run equals right 2. Plot the second point at.
Step 5. Repeat the rise equals up 1 and run equals right 2. Plot the third point at. Step 6. Draw a straight line through the three points. Alright, we've covered a lot of ground in this video regarding the different ways linear equations can be written.
While the structure of the equations looks different, they all represent a line. The use of each depends on what you are given or what you are asked to do. Hi and welcome to this video about polygons.
In this video we will explore four things. One, what polygons are. Two, the different parts of a polygon.
Three, ways to classify polygons. And four, how to determine the number of diagonals in a polygon. The term polygon is derived from the Greek words polis meaning many and gania meaning angle.
So polygons have many angles. First, let's explore how polygons are constructed. Consider the geometric point represented by a dot.
The point is zero-dimensional. It has no length, width, height, nothing. Now let's consider two connected points.
This is called a line. or a line segment. A line segment is one-dimensional. It has length, but no width or height. When multiple line segments are connected end-to-end, polygons such as this can be formed.
Polygons are two-dimensional. They have no thickness, like this. In order to be a polygon, the shape must be closed.
In other words, every endpoint must be connected to another endpoint. This figure is comprised of connected segments, but the result is not a polygon. This shape is closed, but not made up of any connected segments, so it is also not a polygon.
A polygon is defined as a two-dimensional closed shape composed of two-dimensional closed segments. composed of line segments. The sides of polygons are called edges and the angles created where the edges intersect are called vertices. Polygons also have as many edges as vertices. Polygons are named by the number of edges they have.
This polygon has three edges, three vertices, and is called a triangle. The triangle has the smallest number of edges and vertices of any polygon. It is impossible to create a two-sided polygon. Some common polygons are quadrilaterals, which have four sides, pentagons, which have five sides, hexagons, six sides, heptagons, seven sides, octagons, eight sides, nonagons, nine sides, decagons, ten sides, and dodecagons, which have twelve sides.
No polygons with any number of edges have names, the general n-gon is typically used for all other polygons where n represents the number of sides. For instance, a 30-sided polygon is called a tricontagon, but it's often simply called a 30-gon. Polygons can be regular or irregular. Regular polygons have congruent edges and congruent vertices.
For example, this is a regular quadrilateral. The edges are the same length And the vertices have the same measure. This is an irregular quadrilateral. The vertices have the same measure, but the edges have different lengths. This is an irregular quadrilateral.
The edges are the same length, but the vertices have different measures. This is an irregular quadrilateral. Neither the edges nor the vertices have the same measure. Polygons can also be convex or concave. When one or more vertices of a polygon measures more than 180 degrees, the result is a concave polygon.
For example, this is a convex hexagon. All the vertices measure less than 180 degrees. This is a concave hexagon. One vertex measures more than 180 degrees. Now remember, only polygons with four or more sides can be concave because it's not possible for a triangle to contain an angle measuring more than 180 degrees, which is a straight line.
Concave polygons cannot be regular because all the vertices will never be the same measure. Polygons also contain diagonals. Diagonals are line segments joining two vertices that are not next to each other. As you can see here, this irregular convex pentagon has five diagonals. This is an irregular concave pentagon.
It also has five diagonals, even though the concavity causes diagonals to lie outside the polygon. Triangles do not have diagonals, because there is no way to connect two vertices with segments that are not edges. Now that you know the basics of polygons, let's use the diagrams to figure out a formula for finding the number of diagonals in any polygon. In this case, the number of diagonals connecting to each vertex is 2, which is 3 less than the number of vertices.
From any vertex, diagonals cannot connect to the vertex itself or to the vertices that they are one away from because it would be edges. That makes three vertices from every vertex that aren't included. To generalize this, we'll use n minus 3, where n is the number of vertices of the polygon.
Now each vertex has the same number of diagonals connecting to it. So in this case, we can see that the total number of diagonal connections to vertices is 5 times 5 minus 3, which is equal to 10. In general, we can say the total number of diagonal connections is n times n minus 3. When we figure out this total though, we are counting each diagonal twice because diagonals have two endpoints. In order to figure out the number of unique diagonals we need to divide our total by 2. In our Pentagon this looks like 5, which corresponds to our diagram. Therefore for any size polygon our equation can be written as this n times parentheses n minus 3 all over 2 or divided by 2. Conceptually, this can be remembered as...
Using our formula, we can determine the number of unique diagonals in, for example, a 17-gon. which is equal to 119 unique diagonals. We can also see algebraically that triangles have no diagonals. We can also figure out how many edges or vertices a polygon has by the number of unique diagonals. Suppose a polygon has 44 unique diagonals.
How many edges does the polygon have? Alright, so here we are at answer n, the number of edges the polygon has is either equal to 11 or negative 8 because what we want to do is we want to make sure that we end with the result of 0 which means that n somewhere has to be 11 or negative 8 in order to get a small multiplying 0 by another answer, resulting in 0. Since n can't be negative, because we can't have negative edges, it can't be negative 8. So we know that our answer is 11. The number of edges is 11, making it an 11-gon. It's been said that a picture is worth a thousand words.
This is true of the many different ways that data can be transformed into visual representations for easier interpretation. Data in and of itself can be difficult to explain, but when it is carefully organized, patterns can be revealed or it may become apparent that no patterns exist. Either way, both results are meaningful.
Advances in technology enable us to collect and track data like never before. An interpretation of data is simplified with the use of graphs. Data is collected for many reasons, ranging from simple consumer surveys to track preferences to election polling to predict the winner of a political office. Graphs are used to make that data easier to digest or comprehend. Before we get to the graphs, you should know that there are different types of data that can be collected.
The first one we'll talk about is qualitative data. This data would result from a survey that groups respondents by category. For example, you can survey students in your school and ask them to report their eye color, gender, area code, and favorite movie type. These categories provide information for your study.
In this case, maybe you're wondering whether there is a relationship between the number of blue-eyed sci-fi fans living in a particular area code. Well, you may be able to detect a pattern in the counts, or percentages, of qualitative data by category. Graphs that are used to depict qualitative data are pie charts and bar graphs. A pie chart looks exactly like what you might expect.
A pie, a circle representing 100% of a particular variable of interest, is divided into categories. Segments of the pie represent a percentage of the whole. Pie charts can also show counts.
In this example, 6 people, or 30% of the respondents, favored romantic movies. Bar charts also show counts, or percentages, by category. but in a different way.
It looks like a city in the distance with skyscrapers at varying heights. This bar chart shows categories on the horizontal axis with associated counts on the vertical axis. Next, we have pictographs. As the name suggests, a pictograph provides visual context. Visual cues and images are used to represent the data, and keys are provided to tell you what each image or symbol represents.
In this survey of student fruit preference, Apple's took the lead with 20 students reporting them as their favorite. Another type of data that can be collected is known as quantitative. Unlike qualitative data, you can do math with quantitative data because it is numeric in nature.
For example, what if you're interested in the average AP statistics score of the graduating class of your high school? Gathering reliable data may be the trickiest part of your study, but a histogram can relay trends in the results with precision. A histogram resembles a bar graph, with the exception of spaces between the bars. There's a range of numeric values displayed on the horizontal axis, and the counts of students are displayed on the vertical axis.
The pattern of scores may appear somewhat symmetric, which is typically referred to as normally distributed. This would suggest that most students scored in the middle of the range, and that there were some high achievers and others who were not so successful. A histogram could also show a skew to the right or left. which may reveal information about overall student understanding. Scatter plots show data points for two quantitative variables on the same graph.
The pattern revealed could show an association between the two variables. The data is analyzed with respect to direction, form, and strength. Direction can be positive if a rising pattern is seen in the data, or negative if a falling pattern is seen.
The form of the scatter plot identifies whether the data appears somewhat linear, curved, or cone-like in shape. Strength associates how tightly the points are gathered. An example of a strong positive, somewhat linear pattern could be revealed in a scatter plot of shoe size versus height of male high school students.
Because typical male students continue to grow during the high school years, the data for all male students would likely show underclassmen data points at the lower end of the scatter plot and upperclassmen represented at the higher end. Data points that do not fit a recognizable pattern are referred to as outliers. Finally, a scatter plot with no pattern at all is also of interest to prove no association between the two variables. Regression analysis can be performed on scatter plot data that has been characterized as strong linear in shape and either positive or negative in direction. This regression line serves as a predictive model only for estimating other points of interest within the range of the data set.
Line graphs are very useful to recognize how patterns and data sets change over time. Unlike a linear regression model that runs through a set of points, a line graph connects the data points. The graph of mobile phone sales over time shows an overall increasing pattern. However, there was a noticeable dip in the sales between 2011 and 2012. Graphing multiple variables together in a line graph can also be very informative.
For example, suppose temperature and rainfall amounts in a particular region were graphed simultaneously. As can be seen, contrasting weather patterns are revealed. Finally, stem and leaf charts contain actual data values. The pattern revealed is similar to that of a bar chart that has been rotated clockwise 90 degrees. The stem of the chart lists the digits of an indicated place value vertically.
In this example, the stem represents the tens of a data set of ages. The leaves branch outward with the digits of the ones place for each age in the data set. A key should always be part of the graph so the data can be interpreted correctly. This data set shows a pattern that seems to be skewed left towards the younger ages. There are 19 people under 20 and only 7 people over 50. Thanks for watching and happy studying!