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Video Transcript (0:05) So what's a number? And what does it mean to say there are different types of numbers? Well, to answer these questions we're going to have to start at the beginning with the basics. (0:18) The most basic thing you can do in mathematics is count - 1 2 3 4 5. These numbers - these counting numbers - are the simplest number type that we have. They're called the natural numbers. And the natural numbers are great for representing all sorts of stuff, but they're not perfect there's lots of real-life stuff you can't do with only natural numbers so we need more stuff (0:45) The first thing that we can do to improve the natural numbers - to add something else - is to add the number zero. This system of the natural numbers together with 0, is called the whole numbers. (0:57) This is better, but there's still more that's missing. If all we have are just the whole numbers how do we represent something like debt? How do we describe the elevation of something that's underground? The next thing we can add to this system is the negative numbers which represent these kinds of situations. This system of the negative numbers together with the whole numbers is called the integers. (1:27) This is great. It's much better, but it's still not perfect. If I eat half of a pizza how do we represent that with only the integers? We can't. We need something more. So, the next thing we add is all of the fractions. This system of the integers together with all of the fractions is known as the rational numbers. (1:55) Let's talk about that for just a second. (1:59) The definition of the rational numbers is any number that you can write as a ratio of two integers. This includes numbers like -2/3 but it also includes numbers like five because we can write that as 5/1. It will also include any decimal number that either terminates like 2.5 or repeats like point three repeating (which you might recognize as the decimal for ⅓). (2:30) To summarize, we can tell if something is a rational number if it’s an integer, a fraction, a decimal that stops or a decimal that repeats. (2:44) Now it might be tempting to think that this is the top -this is everything- but even now we're still missing some stuff. What about decimals that don't terminate and don’t repeat. Stuff like pi or the square root of 2. We call these numbers, the ones that can't be written as a fraction, the irrational numbers. (3:07) And if we put those two groups together the rationals and the irrationals we get the real numbers. (3:17) And you might think that's it that's the top - that's all of the numbers - what could we possibly be missing? (3:25) Well one thing that we can't do with just the real numbers is to take the square root of -1. So mathematicians came up with a new number “i” which is defined to be the square root of -1 and we call these numbers that contain “i” the imaginary numbers. So, 2i, 3i, negative a million “i” are all imaginary. (3:55) And when we combine these with real numbers we get complex numbers like 2 + 3i or 10 - 2i. (4:13) One of the most important things to understand about these number types is that they form a hierarchy. What does it mean to be a hierarchy? Here’s an example. Take me. I live in an apartment, Which is in my apartment complex which is in Houston which is in Texas which if I combine it with the other states makes up the United States which if I combined it with the other countries makes up the Earth. (4:45) So if I’m in my apartment, I’m also on earth. I’m also in Houston. It’s the same way with numbers. (5:00) We made these new number types by taking the old ones and adding new stuff so if I have a question like what number type is 5? Well it's a natural number because it's a counting number but it's also a whole number and it's also an integer and it's also a rational number and it's also a real number and it's even also a complex number, but it’s not an irrational number and it’s not an imaginary number.
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The ISEE will directly and indirectly test your knowledge of number types. For example, a word problem may include rules where its values may only be integers or whole numbers. Real Numbers (Numbers that Exist) Natural/Counting Numbers Natural numbers include positive whole numbers. They do not include negative numbers, decimals or fractions. They also do not include zero. They could also be called Counting numbers, so you can think of these as the numbers we use to talk about things in the real world, like talking about how many people there are in the room. Correct ✓ and ✘ Incorrect Examples ✓ 1, 2, 3, 4 ✘ -2, 0, 2.5 Whole Numbers You can think of whole numbers and natural numbers as the very first numbers you learned when you were first learning about number lines. includes positive numbers includes zero does not include negative numbers does not include decimals or fractions Correct ✓ and ✘ Incorrect Examples ✓ 0, 1, 2, 3, 4 ✘ -4, -1.5, 3.7 Integers Eventually, you learned that negative numbers exist on the left side of 0 on number lines. includes positive numbers includes zero includes negative numbers does not include decimals or fractions Correct ✓ and ✘ Incorrect Examples ✓ -2, -1, 0, 1, 2 ✘ -4.3, -2.9, ½ Rational Numbers Afterwards, you learned that it was possible to have a fraction of a number, and that these exist in between the integers that you had previously learned about. any number that can be represented as a fraction Correct ✓ and ✘ Incorrect Examples ✓ -3, 4.25, 0.004 ✘ √2, √3, π Irrational Numbers While you can approximate where irrational numbers might fall on a number line, you will never find the exact location like you can for the rest of the real numbers. any number that cannot be represented as a fraction Correct ✓ and ✘ Incorrect Examples ✓ √2, √3, π ✘ 3.14, 22/7 While all natural numbers are integers, not all integers are natural numbers, as by definition negative numbers are not natural numbers. The same can be said for numbers in different categories, i.e. 2 is a rational, integer, and natural number, -2 is a rational number and an integer, whereas -0.25 is only a rational number. In that way, some numbers will fall under the umbrella of multiple categories, while others may fit only one. However, falling under the smaller umbrellas does not automatically make numbers fall under the larger ones, such as how not all rational numbers are real numbers. Pi or π is not considered a rational number. As you may have noticed, 3.14 and 22/7 are listed as rational numbers, and it is very likely that you have used these numbers to represent pi in the past. However, they are not the true value of pi, which is derived from the ratio of a circle’s circumference to its diameter. These numbers are simply useful approximations. In reality, there is no fraction or ratio of non-zero integers that would properly equal pi. In most cases, you are allowed to conveniently use approximations of pi in your math problems because it is not necessary to calculate using digits past the thousandths place. Imaginary Numbers (Numbers That Don’t Exist) Imaginary numbers are numbers that don’t exist: the square root (or even root) of a negative number squaring a negative number results in a positive number, thus the square root of a negative number does not exist generally represented as i = √-1 Correct ✓ and ✘ Incorrect Examples ✓ √-2, 4√-25 ✘ √7, √19 Let’s examine the idea of non-existent numbers by looking at a typical square root problem. To solve for x, we take the square root of both sides: Now take a look at an alternate problem. If we try to solve this equation the same way as the previous one, we run into a problem: Squaring a positive or negative number will always result in a positive number. Thus, we cannot find the square root of a negative number, because the square root of a negative number does not exist. This is where imaginary numbers come in. Imaginary numbers (represented as i ) are theorized such that i2 = -1. Therefore, imaginary numbers are generally represented as: Returning to problem 2, we can now solve for x using imaginary numbers: One more example with Exponentiation In this case, we see that the square root of a larger negative number is a multiple of i. It is also important to know the values of i using exponentiation. Given that i is equal to the square root of -1, the value of in is: This cycle of exponentiation will continue indefinitely. Complex Numbers Complex numbers are “complex” because they include a real number a and an imaginary number bi. represented as the binomial a + bi act like binomials in calculations Correct ✓ and ✘ Incorrect Examples ✓ (4 + 3i), (3 + 2i), (1 - 4i), and (-3 + 2.5i) , 12 ✘ x3 – 3x, x + 15 (4 + 3i), (3 + 2i), (1 - 4i), and (-3 + 2.5i) are all complex because they contain a real component and an imaginary component. On the other hand (x3 - 3x) and (x + 15) are not complex numbers as they are simply algebraic expressions! It is important to note that a number such as 12 actually is a complex number, since we could rewrite 12 as 12 + 0i! This means that any ordinary number you can think of is complex. Complex numbers encompass every type of number that can be expressed as a + bi. It may help you to conceptualize complex numbers by graphing them on a cartesian plot. A cartesian plane has two axes, the x-axis and the y-axis, which are perpendicular to each other. In this case, the x-axis represents real numbers, and the y-axis represents imaginary numbers. Instead of x-coordinates and y-coordinates, we use the real component and the imaginary component of each complex number to graph the coordinates. Adding and Subtracting Complex Numbers The graph above will make it easier to understand why adding and subtracting complex numbers is actually quite straightforward. If you want to compute (4 + 3i) + (3 + 2i) all you have to do is combine like terms. That is: Returning to the cartesian plot, adding or subtracting a complex number is simply about shifting coordinates along the real axis (x) and the imaginary axis (y). Let’s try another one: Multiplying and Dividing Complex Numbers Multiplying complex numbers is just like multiplying binomials. For example: Dividing complex numbers is also done in a similar manner to binomials. How would you reduce the following fraction? The fraction above actually must be reduced since i cannot exist in the denominator of a fraction (since it is still technically a square root!). We fix this by multiplying the numerator and denominator by the conjugate of the denominator, which is simply the denominator with its sign flipped. If the denominator is a + bi, then the conjugate is simply a - bi. This results in the simplification we see above. The next example will leave the answer in incomplete form, since we are simply illustrating how you can cancel identical complex numbers from the numerators and denominators. Try using the same method as above to simplify this answer further! We can eliminate (-3 + 2i) from the numerator and denominator, thus removing like terms. Number Types Practice Problems List three examples of rational numbers: List three examples of natural numbers: Why are some rational numbers not integers? What is the main difference between whole numbers and natural numbers? If a is a negative integer, then 2a can be a natural number. True or False? Variables a, b, and c are rational numbers and a < b < c. If b = ¼, which of the following is NOT a possible combination for the values (a, c)? (0.2, 3) (-¼, ⅛) (1/10, 0.26) (-0.99, 1) Answers to Practice Problems Three rational number examples: 5, ½, ⅔ Three natural numbers: 17, 15, 18 Some rational numbers are simply the ratio of two integers, like ⅔. This is not an integer since its decimal representation is 0.666… but this is a rational number. Whole numbers include 0 and natural numbers do not include 0. Remember this because 0 looks like a (w)hole! False (-¼, ⅛) Complex Numbers Practcie Problems Answers to Practice Problems -13 + 9i 11 + 16i -18i + 12 7 + 34i (-63-36i) / (38-44i) A (and also B, since all numbers are complex!)
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