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Numbers This is a number line.
It represents all of the positive and negative numbers. A mark is made at each integer (whole number). Positive numbers extend to the right of zero (greater than zero), and negative numbers extend to the left (less than zero.) By putting points on the line, we can represent various things.
An open circle is used here because we are showing “less than, but not equal to, –1.” A filled circle is used here because we are showing “greater than, and equal to, 3.” We can also use it for adding and subtracting easily.
And we can see that switching the order of the numbers we are adding does not change the answer. We can also see that adding a negative number is the same thing as subtracting a positive number (in other words, 5 + (–3) is the same as 5 – 3).
Arithmetic o What are equations? Equations show that two things are equal. So for example, let’s use the equations: 4=9–5 10 = 4 + 6 The equals sign means that whatever is on one side is equal in value to whatever is on the other side. This also means that each side is interchangeable with the other side. Because there is a 4 in each equation, we can replace the 4 in the second equation with 9 – 5, because we know that it is interchangeable with 4, and it will remain true. 10 = (9 – 5) + 6
2 Because the equals sign means that the two sides are the same as each other, when we change one side, we must change the other side the same way to keep them the same. Let’s use the following equation to show how this is done. 5=9+x We want to find out what value x has, or what x is equal to. So we need x to be by itself on one side of the equals sign, and whatever is on the other side will be what x is equal to. To get x by itself, we need to move the 9 to the other side, so we will subtract 9 from both sides (to keep the sides the same.) 5–9=9–9+x –4 = x So x is equal to –4. o
Addition and Subtraction Addition is just a way of counting up faster. So if I want to count 16 more than 14, I can count sixteen numbers past 14, or I can add 16 + 14, which is faster. 16 + 14 = 30, so I know that if I counted sixteen numbers past 14, I would get to 30. Subtraction is addition’s opposite. You can think of it as counting down faster, but it will be helpful in a few minutes to see it as the opposite of addition. Simply put, every arithmetic operation has an opposite, and subtraction is addition’s opposite because it is undoing what addition does. 30 – 14 = 16, and 30 – 16 = 14.
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Multiplication and Division Multiplication is a way of adding faster. So it’s the next step up in counting speed, in a way. Multiplication is only useable for counting the same amount multiple times. If I want to count to 12 five times, I could either do it by counting (the slowest way), I could add 12 + 12 + 12 + 12 + 12 (the next fastest way), or I could simply multiply 12 * 5 (the fastest way yet.) 12 * 5 = 60, so if I counted to 12, then counted 12 more another four times, I’d eventually reach 60. Note for multiplying: pay attention to signs. +*+ = +, –*– = +, and +*– = – (also, –*+ = –, because it does not matter what order two numbers are multiplied in.) If the signs are the same, the answer will be positive; if they are different, the answer is negative. Division is multiplication’s opposite. It undoes what multiplication does. 60/5 = 12, and 60/12 = 5. Note that multiplication may be indicated by a variety of symbols. All of these mean you need to multiply: xy x*y xy (x)(y) x(y) xy
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Exponents and Radicals Exponents are a way of multiplying faster. This can be thought of as being the fastest method of counting, but it is useable only when you are multiplying the same number multiple times. So if I need to multiply by 4, and then multiply that answer by 4 again, and then multiply that answer by 4 again, I can instead use 43. This is the same thing as 4*4*4. And 43 = 64.
3 Radicals are the opposite of exponents, and they undo what exponents do. √16 = 4 (this reads “the square root of 16 equals 4.”) While it will rarely be important to you in lower levels of math, it is important to note that √16 also equals –4. When you take a square root, the answer will always be positive and negative, which is to say that there are always two correct answers to a square root problem, a positive and a negative of the same number. The same is true for any time you are taking an even numbered root of a number – square root, fourth root, sixth root, etc. – because this goes back to multiplying signs. If you are taking an odd numbered root – cube (third) root, fifth root, etc. – there is only one valid answer, and it is either positive or negative, but not both. This might seem confusing, but some examples will show the reason why. √16 = 4*4, and √16 = –4*–4 too. So 42 and (–4)2 are both equal to 16. 3 √27 = 3*3*3. But (–3)3 equals –27, because –3*–3*–3 = 9 * –3 = –27 4 √16 = 2*2*2*2, and 4√16 = –2*–2*–2*–2 too, because –2*–2*–2*–2 = 4*–2*–2 = –8*–2 = 16 o
Order of operations The order of operations is the correct order to solve an equation that has multiple steps. The reason for following the order of operations is that solving an equation out of order will often give you a different answer, and having this standard ensures that everyone does math the same way, and gets the same answers. The order of operations is PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Let’s use the following equation as an example: (5 + 32) / (28/2*2) The first thing we need to do is solve whatever’s inside the parentheses first. Our first set of parentheses is (5+32). Within those parentheses, we have an addition and an exponent. The order of operations says to solve the exponent first. 32 = 3*3 = 9. Now we have (5+9) in the parentheses, and 5+9 = 14. Now let’s rewrite the equation, replacing the first set of parentheses with 14. 14 / (28/2*2) Our next set of parentheses is (28/2*2). First, multiply: 2*2 = 4. Then, divide: 28/4 = 7. Now, rewrite the equation, replacing the second set of parentheses with 7. 14 / 7 14/7 = 2, so our answer is 2. Now let’s see what would happen if we solved the problem out of order, just moving from left to right. (5 + 32) / (28/2*2) = (82) / (28/2*2) = 64 / (28/2*2) = 64 / (14*2) = 64/28 = 2.285714…… This is why the order of operations matters.
Fractions o What do fractions really mean? Fractions are simply a way of showing division problems. 3/4 is the same thing as 3 divided by 4. Most commonly, fractions are used to show pieces of one whole. So ordinarily, you won’t see a fraction like 5/3, you will more commonly see 1 2/3, which means the same thing. (1 is the same as 3/3, or 3 thirds of one. Then there are 2 thirds left over, so 5/3 = 1 2/3, and the fraction is only used to show that extra amount that is less than one whole.) The number on top of the fraction is known as the numerator, and the number on the bottom is known as the denominator.
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Adding, subtracting fractions To add fractions, both fractions being added must have the same denominator. Let’s use an example: 2/3 + 3/5 = ? It’s hard to think about how to add these two numbers together, because they don’t represent the same thing. One represents thirds, or “one whole divided into three equal pieces.” The other represents fifths, or “one whole divided into five equal pieces.” We can’t count them up unless they’re the same thing, so we need to change the denominators to be the same. The easy way to do this is to multiply them together. 3*5 = 15, so that will be our new denominator. To change 2/3 into ?/15, we must multiply both the 2 and the 3 by 5 (because that’s the amount we need to multiply 3 by to get it to be 15.) 2*5/3*5 = 10/15 Now we need to multiply both parts of 3/5 by 3 so that it has a 15 in the denominator too. 3*3/5*3 = 9/15. Our problem now looks like this: 10/15 + 9/15 (because 2/3 = 10/15, and 3/5 = 9/15) 10+9=19, so the answer is: 10/15 + 9/15 = 19/15, or 1 4/15.
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Multiplying, dividing fractions Multiplying two fractions together is simple – you simply multiply the numerators together, and multiply the denominators together. 3/5 * 5/6 = 3*5/5*6 3*5/5*6 = 15/30, or 1/2. Dividing fractions is exactly like multiplying them, but with one extra step: Before multiplying them together, flip one of them upside down. 1/2 / 5/6 = 1/2 * 6/5 1/2 * 6/5 = 1*6/2*5 1*6/2*5 = 6/10, or 3/5.
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Simplifying fractions Improper fractions are fractions with a numerator that is larger than their denominator, such as 18/4. To simplify an improper fraction, simply think about what 18/4 represents.
When you look at it this way, it is obvious that 18/4 = 4 1/2. Each 4/4 is equal to one whole, so add 4/4’s until you are close to 18. 4/4 + 4/4 + 4/4 + 4/4 = 16/4, and 2/4 more makes 18/4. Since the extra 2/4 is the same as 1/2, the answer we get is 4 1/2.
5 Complex fractions are fractions where the denominator is also a fraction of its own. Simplifying them is as easy as dividing the fraction on top by the fraction on the bottom.
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Decimals Decimals are a way of representing parts of a whole, just like fractions, but decimals represent the same thing in regular numbers which makes them easier to use in a lot of situations. For instance, 3 1/5 = 3.2. To convert a fraction to decimals, just divide the numerator by the denominator. To convert decimals to a fraction (this will be a little more confusing), put the decimal in the numerator, and the denominator will be 1 followed by the same number of zeros as there are digits in the numerator. Here’s an example: 5/16 = 5 divided by 16 = 0.3125 0.3125 = 3125/10000 = 5/16 (after you simplify it)
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Percentages Percentages are simply fractions where the denominator is always 100. The best way to remember this is by the name: percent literally means per 100, because cent comes from the Latin word for 100. (Think 100 cents in a dollar.) 70% = 70/100 37% = 37/100 Because the denominator is 100, you can also easily represent percentages as decimals, by simply moving the decimal point two places to the left. 46.3% = .463 99.9% = .999
Mean, median, and mode o Mean – The mean is the average of a set of numbers. You calculate the mean by adding all of the numbers in the set, then dividing that answer by the number of numbers in the original set. Example: Find the mean of the numbers 5, 2, 8, 9, 10, 9, 14, 3, 7, 13 5 + 2 + 8 + 9 + 10 + 9 + 14 + 3 + 7 + 13 = 80 There are 10 numbers in the set, so 80/10 = 8. o Median – The median is the number in the middle of the set when it is in numerical order. The first thing to do is put all of the numbers in order, then find which number is in the middle. If there are two numbers in the middle, take the average of those two numbers. Example: Find the median of the numbers 5, 2, 8, 9, 10, 9, 14, 3, 7, 13 First, put them in order and see what numbers are in the middle: 2, 3, 5, 7, 8, 9, 9, 10, 13, 14 There are two numbers in the middle: 8 and 9. Let’s take the average: 8 + 9 = 17 17/2 = 8.5 The median is 8.5.
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Mode – The mode is simply the number that appears most often in a set of numbers. It is possible for there to be more than one mode, if two numbers appear the same number of times, but more often than the other numbers. It is also possible for there to be no mode, if all numbers appear the same number of times. Example: Find the mode of the numbers 5, 2, 8, 9, 10, 9, 14, 3, 7, 13 The number 9 shows up twice and all other numbers show up once. Therefore, the mode is 9.
Algebra o In algebra, we use exactly the same kinds of equations we’ve been using all along, with one difference: Now we are replacing some of the numbers with variables, which are represented by a letter (usually x) that means “we don’t know what this number is yet,” or sometimes, “you can use any number” if there is more than one letter in the equation. o
Now is a good time to introduce the basic properties of numbers. There are three of them. Distributive property – When multiplying a number times something added or subtracted in parentheses, you must multiply each item inside the parentheses by that number individually. 2(x+y) = 2x + 2y 5(x–y) = 5x – 5y Associative property – When adding or multiplying multiple things together, it does not matter what order you do it in. (ab)c = a(bc) (a+b)+c = a+(b+c) Commutative property – Switching the order in which addition or multiplication problems are written does not change the outcome. x+y = y+x
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Polynomials are just mathematical expressions. Polynomials is a word meaning “many (poly) terms (nomials)”. The “terms” in a polynomial expression are anything that is separated by a + or a – (even if the term is a constant.) Examples of polynomials include: 9x2 + 3y – 4x (terms are +9x2, +3y, and –4x) x + 5 (terms are +x and +5) 2 + 6 (terms are +2 and +6) If there is only one term, even if that term is a constant (not a variable), it is called a monomial. Examples: 10x3 (term is +10x3) 6x (term is +6x) 4 (term is +4) Quadratic polynomials are polynomials in which there is a term with an exponent of 2, and it is the highest numbered exponent in the polynomial. Quadratic polynomials take the form of ax2 + bx + c, where a, b, and c are constants. Examples: 6x2 + 3x + 10 9x2 + 4 x2 – 9x
7 You may notice the last two don’t look like quadratic polynomials. But they are, because there is an x2 in them. Let’s rewrite them to show how they are actually quadratic polynomials: 9x2 + 0x + 4 x2 – 9x + 0 When you solve a quadratic equation, you will always get two answers. BOTH ANSWERS ARE VALID, and both will make the equation work. You will always begin by setting the quadratic equation equal to zero. Then you can solve these equations in a few different ways. We’ll take these one at a time. Factoring This method involves taking your quadratic equation, ax2 + bx + c, and figuring out which two numbers can be multiplied to equal “c”, and added together to equal “b”. This works in some situations, but is not always the easiest method. However, when it does work, it is very quick. x2 + 7x + 12 = 0 I know 3*4 = 12, and 3+4=7, so my two numbers are 3 and 4. Now I can rewrite the problem like this: x2 + 7x + 12 = (x+3)(x+4) = 0 I set each one of those parentheses equal to zero. x + 3 = 0 ; x = –3 x + 4 = 0 ; x = –4 So the answer is: x = –3, x = –4 Cross-Multiplying (the FOIL method) This is a good time to talk about cross-multiplication, which is how you multiply together two (or more) sets of parentheses that each have multiple terms. I’ll use the solution from the quadratic equation we just solved as an example: (x+3)(x+4) In order to multiply this together, we will use the cross-multiplication method, also called the FOIL method (FOIL stands for First, Outer, Inner, Last). First, we multiply the first terms from each set of parentheses together. Then, multiply together the outer terms. Then, multiply together the inner terms. And finally, multiply together the last terms. First: (x+3)(x+4) x*x = x2 Outer: (x+3)(x+4) x*4 = 4x Inner: (x+3)(x+4) 3*x = 3x Last: (x+3)(x+4) 3*4 = 12 Then, add all the terms together. x2 + 4x + 3x + 12 = x2 + 7x + 12
8 Quadratic formula The quadratic formula is:
Let’s use the same problem from the previous example. x2 + 7x + 12 = 0 Our “a” term is 1 (because x2 = 1x2), our “b” term is 7, and our “c” term is 12. Plug these numbers into the formula. x = –7 ± √(72 – 4(1)(12)) 2(1) I know it looks like a lot of work, but we’ll take it one step at a time. Remember our order of operations, we start with what’s in the parentheses, and then we start with the exponent first. x = –7 ± √(49 – 4(1)(12)) 2(1) Then we multiply the numbers inside the parentheses. x = –7 ± √(49 – 48) 2(1) Now subtract what’s in the parentheses. x = –7 ± √(1) 2(1) And take the square root next. The square root of 1 is 1, because 12=1. You may remember that the square root of 1 is also –1, which is true, but since there is a ± symbol in our formula, we don’t need to worry about it. x = –7 ± 1 2(1) Now, solve the next set of parentheses. 2*1=2, so just write 2. x = –7 ± 1 2 And finally, you can solve for the two answers to the quadratic equation. First, by adding –7 + 1, and second by subtracting –7 – 1. x = –7 + 1 x = –6 x = –3 2 2 x = –7 – 1 2
x = –8 x = –4 2
So we get the same answers as before. x=–3 and x=–4.
9 Completing the square We’ll use a different problem for this one, to make it simpler – x2 + 6x –7 = 0. For this method, we will need to rearrange the equation a bit. x2 + 6x – 7 = 0 x2 + 6x –7 + 7 = 0 + 7 x2 + 6x = 7 Now things will get a little bit confusing, but I’ll explain every step as we go. First, we need to add in a term to make things a little more workable again for what we’re about to do. Take your “b” number, cut it in half, and square that number. x2 + 6x = 7 3 (half of 6) 9 (32) Now add that number to both sides of the equation. You’ll see why in a minute. x2 + 6x + 9 = 7 + 9 x2 + 6x + 9 = 16 We added 9 to both sides to keep them equal to each other. Now we have changed our quadratic equation into an equation where the left side is a perfect squares of something. Let’s find out what that is. We know 9 is the square of 3 – that’s how we got the 9 in the first place, by squaring 3. So while it may not be immediately obvious: x2 + 6x + 9 = (x+3)2 Proof: (x+3)2 = (x+3)(x+3) = x2 + 3x + 3x + 9 = x2 + 6x + 9 This is handy here, because we can replace the left side of our equation with (x+3)2. x2 + 6x + 9 = 16 (x+3)2 = 16 And we know that 16 is equal to ±42, so let’s take the square root of both sides. (x+3)2 = 16 x+3 = ±4 x = –3 ± 4 Now we can solve like before, by adding and subtracting 4 from –3. x = –3 + 4 = 1 x = –3 – 4 = –7 So our answers are x=1, x=–7.
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Graphing algebraic equations o This is called a coordinate plane, or a coordinate graph.
The two lines are called the y-axis and the x-axis. We can use this graph to show what our equations look like, and this can help us to visualize what they mean. For example: The graph of y = 2x looks like this:
11 This is the line that goes through all of the coordinates for the line y = 2x. Coordinates are written as ordered pairs, which take the form of (x, y). The x and y of each ordered pair is generated by the equation of the line. y = 2x If x = 1, then y = 2(1), or y = 2. If x = 2, then y = 2(2), or y = 4. If x = –1, then y = 2(–1), or y = –2. And so on. The first ordered pair would be written as (1, 2), the second as (2, 4), the third as (–1, –2), etc. When we plot these coordinates on the graph, we can begin to see the line emerging.
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There are two main forms that straight-line equations will be written in: Slope–intercept form: y=mx+b Where m is the slope of the line (in other words, the tilt of the line), and b is the y-intercept (in other words, the number where the line crosses the y-axis.)
Point–slope form: y–y1=m(x–x1) Where m is the slope of the line, and (x1, y1) is the coordinate of a point that is on that line.
If you’re told to graph an equation like 3y-2x=4, you first need to solve for y (in other words, rearrange the equation until one side of the equation is nothing but y.) 3y – 2x = 4 (first, add 2x to both sides of the equation) 3y = 2x + 4 (now, divide all terms on both sides of the equation by 3 to get the y by itself.) y = 2/3x + 4/3
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The slope of the line shows how fast the line moves on the y-axis per each step along the x-axis. This is often referred to as rise over run – how much the line rises (or falls) as it runs along the x-axis. It will generally be expressed as an integer or a fraction, and this number will represent, “for each 1 I move to the right on the xaxis, this is how much the line moves up (if the slope is positive) or down (if the slope is negative).” The line we used above, y=2x is in slope–intercept form. In this case, the y-intercept is 0, because the line crosses the y-axis at the point (0, 0). (In other words, when x=0, y=0. To find the y-intercept, always set x equal to 0 and solve the equation.) The slope formula is m=(y – y1)/(x – x1), or slope=rise/run. Let’s use that formula to figure out the slope of a line from two points. Find the slope of the line that passes through the points (5, 2) and (7, 8). One of our ordered pairs has x=7 and y=8. The other ordered pair has x1=5 and y1=2. (We use the subscript so we don’t confuse the points with each other. It doesn’t matter which ordered pair you use for (x, y) and (x1, y1).) Since slope=rise/run, let’s see how much the line rises and how much it runs from one point to the next. y – y1 = 8 – 2 = 6. So our line rose 6. x – x1 = 7 – 5 = 2. So our line ran 2. Let’s plug those numbers into the slope formula. m = rise/run = 6/2 = 3. So our slope is 3.
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Graphing inequalities Let’s say you have the equations y > 1/2x + 2, and y ≤ 1/2x -1. How would you graph those?
The top line represents y > 1/2x + 2, and the bottom line represents y ≤ 1/2x -1. Because the first equation says “greater than, but not equal to”, we use a dashed line. The second equation says, “less than and equal to”, so we use a solid line. Then we shade – above the line for “greater than”, and below the line for “less than.”
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Graphing polynomials Graphing polynomials gives you a line that is curved, instead of straight. Let’s see how, with the equation y=x2–4. First, we will figure out some points along this line. We know the y-intercept is (0, –4) because it is given to us in the equation, which is already in slope-intercept form. Let’s figure out some more points by plugging in random numbers for x. y = x2 – 4 (0, –4) y-intercept y = 12 – 4 = 1 – 4 = –3 (1, –3) 2 y = 2 –4 = 4–4 = 0 (2, 0) 2 y = 3 –4 = 9–4 = 5 (3, 5) 2 y = (–1) – 4 = 1 – 4 = –3 (–1, –3) 2 y = (–2) – 4 = 4 – 4 = 0 (–2, 0) 2 y = (-3) – 4 = 9 – 4 = 5 (–3, 5) Now, we will graph these points.
The actual graph, with the curved line drawn in, looks something like this:
14 A curved line like this is called a parabola, and the point where it turns is called the vertex. To find the vertex of a quadratic equation, use the vertex formula, which is: x=–b/2a. This gives you the x-coordinate of the vertex point. Take the equation y = x2 + 4x + 3. The a in the quadratic equation is 1 (1x2), and the b is 4 (4x). x = –b/2a = –4/2(1) = –4/2 = –2 Then plug the x-coordinate into the quadratic equation. y = x2 + 4x + 3 = (-2)2 + 4(-2) + 3 = 4 – 8 + 3 = –1 So the coordinates of the vertex of this quadratic equation are (–2, –1).
Khan Academy Math http://www.khanacademy.org/math I love Khan Academy. It’s a completely free educational resource on a wide variety of topics, but their math section is wonderful. Thousands of videos explaining in simple terms every aspect of all the math you’ll ever need to know, and practice problems that help to solidify those concepts.
Wolfram Alpha http://www.wolframalpha.com This is one of the best and most well known math resources on the internet. It can solve or graph just about any equation you can throw at it – very, very useful, especially for double-checking your own work.
Desmos Graphing Calculator http://www.desmos.com/calculator If you’re struggling to understand what certain equations look like, or what they represent, this is the best online graphing calculator I’ve found! The interface is simple and intuitive, and you can graph any number of equations on the same graph to compare them, and even customize the colors of each equation to make them stand out. A tip if you’re trying to learn what a formula means: Type the formula in with only variables. Desmos will allow you to add a slider for each variable, so you can change its value and watch how it affects the graph in real time.
Complete List of Online Math Resources http://www.studentguide.org/a-complete-list-of-online-math-resources/ Lastly, a long list of helpful links for studying math, from what this guide covers and beyond, including sections on basic math, educational math games online, algebra, geometry, trigonometry, pre-calculus, and calculus!