# Basic Math & Pre-Algebra For Dummies

**Published: **06-13-2016

**Basic Math & Pre-Algebra For Dummies, 2nd Edition (9781119293637) was previously published as Basic Math & Pre-Algebra For Dummies, 2nd Edition (9781118791981). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.**

Tips for simplifying tricky basic math and pre-algebra operations

Whether you're a student preparing to take algebra or a parent who wants or needs to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you'll build necessary math skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations.

- Explanations and practical examples that mirror today's teaching methods
- Relevant cultural vernacular and references
*Standard For Dummies*materials that match the current standard and design

*Basic Math & Pre-Algebra For Dummies* takes the intimidation out of tricky operations and helps you get ready for algebra!

## Articles From Basic Math & Pre-Algebra For Dummies

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Article / Updated 07-09-2021

The center of a circle is a point that's the same distance from any point on the circle itself. This distance is called the radius of the circle, or r for short. And any line segment from one point on the circle through the center to another point on the circle is called a diameter, or d for short. The diameter As you can see, the diameter of any circle is made up of one radius plus another radius — that is, two radii (pronounced ray-dee-eye). This concept gives you the following handy formula: For example, given a circle with a radius of 5 millimeters, you can figure out the diameter as follows: The circumference Because the circle is an extra-special shape, its perimeter (the length of its "sides") has an extra-special name: the circumference (C for short). Early mathematicians went to a lot of trouble figuring out how to measure the circumference of a circle. Here's the formula they hit upon: Note: Because 2 x r is the same as the diameter, you also can write the formula as C = π x d. The symbol π is called pi (pronounced "pie"). It's just a number whose approximate value is as follows (the decimal part of pi goes on forever, so you can't get an exact value for pi): So given a circle with a radius of 5 mm, you can figure out the approximate circumference: The area of a circle The formula for the area (A) of a circle also uses π: Here's how to use this formula to find the approximate area of a circle with a radius of 5 mm:

View ArticleArticle / Updated 07-07-2021

Math itself is one big concept, and it's chock full of so many smaller mathematical concepts that no one person can possibly understand them all — even with a good dose of studying. Yet certain concepts are so important that they make the Math Hall of Fame: Sets and set theory A set is a collection of objects. The objects, called elements of the set, can be tangible (shoes, bobcats, people, jellybeans, and so forth) or intangible (fictional characters, ideas, numbers, and the like). Sets are such a simple and flexible way of organizing the world that you can define all of math in terms of them. Mathematicians first define sets very carefully to avoid weird problems — for example, a set can include another set, but it can't include itself. After the whole concept of a set is well-defined, sets are used to define numbers and operations, such as addition and subtraction, which is the starting point for the math you already know and love. Prime numbers go forever A prime number is any counting number that has exactly two divisors (numbers that divide into it evenly) — 1 and the number itself. Prime numbers go on forever — that is, the list is infinite — but here are the first ten: 2 3 5 7 11 13 17 19 23 29 . . . It may seem like nothing, but . . . Zero may look like a big nothing, but it's actually one of the greatest inventions of all time. Like all inventions, it didn't exist until someone thought of it. (The Greeks and Romans, who knew so much about math and logic, knew nothing about zero.) The concept of zero as a number arose independently in several different places. In South America, the number system that the Mayans used included a symbol for zero. And the Hindu-Arabic system used throughout most of the world today developed from an earlier Arabic system that used zero as a placeholder. In fact, zero isn't really nothing — it's simply a way to express nothing mathematically. And that's really something. Have a big piece of pi Pi (π): The symbol π (pronounced pie) is a Greek letter that stands for the ratio of the circumference of a circle to its diameter. Here's the approximate value of π: π ≈ 3.1415926535… Although π is just a number — or, in algebraic terms, a constant — it's important for several reasons: Geometry just wouldn't be the same without it. Circles are one of the most basic shapes in geometry, and you need π to measure the area and the circumference of a circle. Pi is an irrational number, which means that no fraction that equals it exactly exists. Beyond this, π is a transcendental number, which means that it's never the value of x in a polynomial equation (the most basic type of algebraic equation). Pi is everywhere in math. It shows up constantly (no pun intended) where you least expect it. One example is trigonometry, the study of triangles. Triangles obviously aren't circles, but trig uses circles to measure the size of angles, and you can't swing a compass without hitting π. Equality in mathematics The humble equals sign (=) is so common in math that it goes virtually unnoticed. But it represents the concept of equality — when one thing is mathematically the same as another — which is one of the most important math concepts ever created. A mathematical statement with an equals sign is an equation. The equals sign links two mathematical expressions that have the same value and provides a powerful way to connect expressions. Bringing algebra and geometry together Before the xy-graph (also called the Cartesian coordinate system) was invented, algebra and geometry were studied for centuries as two separate and unrelated areas of math. Algebra was exclusively the study of equations, and geometry was solely the study of figures on the plane or in space. The graph, invented by French philosopher and mathematician René Descartes, brought algebra and geometry together, enabling you to draw solutions to equations that include the variables x and y as points, lines, circles, and other geometric shapes on a graph. The function: a mathematical machine A function is a mathematical machine that takes in one number (called the input) and gives back exactly one other number (called the output). It's kind of like a blender because what you get out of it depends on what you put into it. Suppose you invent a function called PlusOne that adds 1 to any number. So when you input the number 2, the number that gets outputted is 3: PlusOne(2) = 3 Similarly, when you input the number 100, the number that gets outputted is 101: PlusOne(100) = 101 It goes on, and on, and on . . . The very word infinity commands great power. So does the symbol for infinity (∞). Infinity is the very quality of endlessness. And yet mathematicians have tamed infinity to a great extent. In his invention of calculus, Sir Isaac Newton introduced the concept of a limit, which allows you to calculate what happens to numbers as they get very large and approach infinity. Putting it all on the line Every point on the number line stands for a number. That sounds pretty obvious, but strange to say, this concept wasn't fully understood for thousands of years. The Greek philosopher Zeno of Elea posed this problem, called Zeno's Paradox: To walk across the room, you have to first walk half the distance across the room. Then you have to go half the remaining distance. After that, you have to go half the distance that still remains). This pattern continues forever, with each value being halved, which means you can never get to the other side of the room. Obviously, in the real world, you can and do walk across rooms all the time. But from the standpoint of math, Zeno's Paradox and other similar paradoxes remained unanswered for about 2,000 years. The basic problem was this one: All the fractions listed in the preceding sequence are between 0 and 1 on the number line. And there are an infinite number of them. But how can you have an infinite number of numbers in a finite space? Mathematicians of the 19th century — Augustin Cauchy, Richard Dedekind, Karl Weierstrass, and Georg Cantor foremost among them — solved this paradox. The result was real analysis, the advanced mathematics of the real number line. Numbers for your imagination The imaginary numbers (numbers that include the value i = √ - 1) are a set of numbers not found on the real number line. If that idea sounds unbelievable — where else would they be? — don't worry: For thousands of years, mathematicians didn't believe in them, either. But real-world applications in electronics, particle physics, and many other areas of science have turned skeptics into believers. So, if your summer plans include wiring your secret underground lab or building a flux capacitor for your time machine — or maybe just studying to get a degree in electrical engineering — you'll find that imaginary numbers are too useful to be ignored.

View ArticleArticle / Updated 07-06-2021

A polygon is any shape whose sides are all straight. Every polygon has three or more sides (if it had fewer than three, it wouldn't really be a shape at all). Following are a few of the most common polygons. Triangles The most basic shape with straight sides is the triangle, a three-sided polygon. You find out all about triangles when you study trigonometry. Triangles are classified on the basis of their sides and angles. Take a look at the differences: Equilateral: An equilateral triangle has three sides that are all the same length and three angles that all measure 60 degrees. Isosceles: An isosceles triangle has two sides that are the same length and two equal angles. Scalene: Scalene triangles have three sides that are all different lengths and three angles that are all unequal. Right: A right triangle has one right angle. It may be isosceles or scalene. Quadrilaterals A quadrilateral is any shape that has four straight sides. Quadrilaterals are one of the most common shapes you see in daily life. If you doubt this statement, look around and notice that most rooms, doors, windows, and tabletops are quadrilaterals. Here are a few common quadrilaterals: Square: A square has four right angles and four sides of equal length; also, both pairs of opposite sides (sides directly across from each other) are parallel. Rectangle: Like a square, a rectangle has four right angles and two pairs of opposite sides that are parallel. Unlike the square, however, although opposite sides are equal in length, sides that share a corner — adjacent sides — may have different lengths. Rhombus: Imagine starting with a square and collapsing it as if its corners were hinges. This shape is called a rhombus. All four sides are equal in length, and both pairs of opposite sides are parallel. Parallelogram: Imagine starting with a rectangle and collapsing it as if the corners were hinges. This shape is a parallelogram — both pairs of opposite sides are equal in length, and both pairs of opposite sides are parallel. Trapezoid: The trapezoid's only important feature is that at least two opposite sides are parallel. Kite: A kite is a quadrilateral with two pairs of adjacent sides that are the same length. Isosceles trapezoid: A trapezoid in which the nonparallel sides (the legs) are congruent In the hierarchy of quadrilaterals shown in the above figure, a quadrilateral below another on the family tree is a special case of the one above it. A rectangle, for example, is a special case of a parallelogram. Thus, you can say that a rectangle is a parallelogram but not that a parallelogram is a rectangle (a parallelogram is only sometimes a rectangle). Polygons on steroids — larger polygons A polygon can have any number of sides. Polygons with more than four sides aren't as common as triangles and quadrilaterals, but they're still worth knowing about. Larger polygons come in two basic varieties: regular and irregular. A regular polygon has equal sides and equal angles. The most common are regular pentagons (five sides), regular hexagons (six sides), and regular octagons (eight sides). Every other polygon is an irregular polygon.

View ArticleArticle / Updated 01-15-2020

Numbers starting with a 1 and followed by only 0s (such 10, 100, 1,000, 10,000, and so forth) are called powers of ten, and they're easy to represent as exponents. Powers of ten are the result of multiplying 10 times itself any number of times. To represent a number that's a power of 10 as an exponential number, count the zeros and raise 10 to that exponent. For example, 1,000 has three zeros, so 1,000 = 103 (103 means to take 10 times itself three times, so it equals 10 x 10 x 10). Powers of Ten Expressed as Exponents Number Exponent 1 100 10 101 100 102 1,000 103 10,000 104 100,000 105 1,000,000 106 When you know this trick, representing a lot of large numbers as powers of ten is easy — just count the 0s! For example, the number 1 trillion — 1,000,000,000,000 — is a 1 with twelve 0s after it, so 1,000,000,000,000 = 1012 This trick may not seem like a big deal, but the higher the numbers get, the more space you save by using exponents. For example, a really big number is a googol, which is 1 followed by a hundred 0s. You can write this: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 As you can see, a number of this size is practically unmanageable. You can save yourself some trouble and write 10100. A 10 raised to a negative number is also a power of ten. You can also represent decimals using negative exponents. For example, Although the idea of negative exponents may seem strange, it makes sense when you think about it alongside what you know about positive exponents. For example, to find the value of 107, start with 1 and make it larger by moving the decimal point seven spaces to the right: 107 = 10,000,000 Similarly, to find the value of 10–7, start with 1 and make it smaller by moving the decimal point seven spaces to the left: 10–7 = 0.0000001 Negative powers of 10 always have one fewer 0 between the 1 and the decimal point than the power indicates. In this example, notice that 10–7 has six 0s between them. As with very large numbers, using exponents to represent very small decimals makes practical sense. For example, 10–23 = 0.00000000000000000000001 As you can see, this decimal is easy to work with in its exponential form but almost impossible to read otherwise.

View ArticleArticle / Updated 01-27-2017

You can tell whether a number is divisible by 2, 5, 10, 100, or 1,000 simply by looking at how the number ends — no calculations required. Divisible by 2 Every even number — that is, every number that ends in 2, 4, 6, 8, or 0 — is divisible by 2. For example, the following bold numbers are divisible by 2: Divisible by 5 Every number that ends in either 5 or 0 is divisible by 5. The following bold numbers are divisible by 5: Divisible by 10, 100, or 1,000 Every number that ends in 0 is divisible by 10. The following bold numbers are divisible by 10: Every number that ends in 00 is divisible by 100: And every number that ends in 000 is divisible by 1,000: In general, every number that ends with a string of 0s is divisible by the number you get when you write 1 followed by that many 0s. For example, 900,000 is divisible by 100,000. 235,000,000 is divisible by 1,000,000. 820,000,000,000 is divisible by 10,000,000,000. When numbers start to get this large, mathematicians usually switch over to scientific notation to write them more efficiently.

View ArticleArticle / Updated 04-25-2016

The things contained in a set are called elements (also known as members). Consider these two sets: {Empire State Building, Eiffel Tower, Roman Colosseum} and {Albert Einstein's intelligence, Marilyn Monroe's talent, Joe DiMaggio's athletic ability, Sen. Joseph McCarthy's ruthlessness}. The Eiffel Tower is an element of A, and Marilyn Monroe's talent is an element of B. You can write these statements using a symbol that means "is an element of": However, the Eiffel Tower is not an element of B. You can write this statement using a symbol that means "is not an element of": These two symbols become more common as you move higher in your study of math. Cardinality of sets The cardinality of a set is just a fancy word for the number of elements in that set. When A is {Empire State Building, Eiffel Tower, Roman Colosseum}, it has three elements, so the cardinality of A is three. Set B, which is {Albert Einstein's intelligence, Marilyn Monroe's talent, Joe DiMaggio's athletic ability, Sen. Joseph McCarthy's ruthlessness}, has four elements, so the cardinality of B is four. Equal sets If two sets list or describe the exact same elements, the sets are equal (you can also say they're identical or equivalent). The order of elements in the sets doesn't matter. Similarly, an element may appear twice in one set, but only the distinct elements need to match. Suppose some sets are defined as follows: C = the four seasons of the year D = {spring, summer, fall, winter} E = {fall, spring, summer, winter} F = {summer, summer, summer, spring, fall, winter, winter, summer} Set C gives a clear rule describing a set. Set D explicitly lists the four elements in C. Set E lists the four seasons in a different order. And set F lists the four seasons with some repetition. Thus, all four sets are equal. As with numbers, you can use the equals sign to show that sets are equal: C = D = E = F Subsets When all the elements of one set are completely contained in a second set, the first set is a subset of the second. For example, consider these sets: C = {spring, summer, fall, winter} G = {spring, summer, fall} As you can see, every element of G is also an element of C, so G is a subset of C. The symbol for subset is shown in the following: Every set is a subset of itself. This idea may seem odd until you realize that all the elements of any set are obviously contained in that set. Empty sets The empty set — also called the null set — is a set that has no elements: H = {} As you can see, H is defined by listing its elements, but none are listed, so H is empty. The symbol is used to represent the empty set. You can also define an empty set using a rule. For example, I = types of roosters that lay eggs Clearly, roosters are male and, therefore, can't lay eggs, so this set is empty. You can think of an empty set as nothing. And because nothing is always nothing, there's only one empty set. All empty sets are equal to each other, so in this case, H = I. Furthermore, is a subset of every other set, so the following statements are true: This concept makes sense when you think about it. Remember that 8 has no elements, so technically, every element in 8 is in every other set.

View ArticleArticle / Updated 04-25-2016

Exponents (also called powers) are shorthand for repeated multiplication. For example, 23 means to multiply 2 by itself three times. To do that, use the following notation: In this example, 2 is the base number and 3 is the exponent. You can read 23 as “2 to the third power” or “2 to the power of 3” (or even “2 cubed,” which has to do with the formula for finding the value of a cube). Here’s another example: 105 means to multiply 10 by itself five times That works out like this: This time, 10 is the base number and 5 is the exponent. Read 105 as “10 to the fifth power” or “10 to the power of 5.” When the base number is 10, figuring out any exponent is easy. Just write down a 1 and that many 0s after it: 1 with two 0s 1 with seven 0s 1 with twenty 0s 102 = 100 107 = 10,000,000 1020 = 100,000,000,000,000,000,000 Exponents with a base number of 10 are important in scientific notation. The most common exponent is the number 2. When you take any whole number to the power of 2, the result is a square number. For this reason, taking a number to the power of 2 is called squaring that number. You can read 32 as “three squared,” 42 as “four squared,” and so forth. Here are some squared numbers: Any number (except 0) raised to the 0 power equals 1. So 10, 370, and 999,9990 are equivalent, or equal, because they all equal 1.

View ArticleArticle / Updated 04-25-2016

When people first find out about subtraction, they often hear that you can't take away more than you have. For example, if you have four pencils, you can take away one, two, three, or even all four of them, but you can't take away more than that. It isn't long, though, before you find out what any credit card holder knows only too well: You can, indeed, take away more than you have — the result is a negative number. For example, if you have $4 and you owe your friend $7, you're $3 in debt. That is, 4 – 7 = –3. The minus sign in front of the 3 means that the number of dollars you have is three less than 0. Here's how you place negative whole numbers on the number line. Adding and subtracting on the number line works pretty much the same with negative numbers as with positive numbers. For example, here's how to subtract 4 – 7 on the number line. Placing 0 and the negative counting numbers on the number line expands the set of counting numbers to the set of integers.

View ArticleArticle / Updated 04-25-2016

Plane geometry is the study of figures on a two-dimensional surface — that is, on a plane. You can think of the plane as a piece of paper with no thickness at all. Technically, a plane doesn't end at the edge of the paper — it continues forever. Making some points A point is a location on a plane. It has no size or shape. Although in reality a point is too small to be seen, you can represent it visually in a drawing by using a dot. When two lines intersect, they share a single point. Additionally, each corner of a polygon is a point. Knowing your lines A line — also called a straight line — is pretty much what it sounds like; it marks the shortest distance between two points, but it extends infinitely in both directions. It has length but no width, making it a one-dimensional (1-D) figure. Given any two points, you can draw exactly one line that passes through both of them. In other words, two points determine a line. When two lines intersect, they share a single point. When two lines don't intersect, they are parallel, which means that they remain the same distance from each other everywhere. A good visual aid for parallel lines is a set of railroad tracks. In geometry, you draw a line with arrows at both ends. Arrows on either end of a line mean that the line goes on forever. A line segment is a piece of a line that has endpoints. A ray is a piece of a line that starts at a point and extends infinitely in one direction, kind of like a laser. It has one endpoint and one arrow. Figuring the angles An angle is formed when two rays extend from the same point. Angles are typically used in carpentry to measure the corners of objects. They're also used in navigation to indicate a sudden change in direction. For example, when you're driving, it's common to distinguish when the angle of a turn is "sharp" or "not so sharp." The sharpness of an angle is usually measured in degrees. The most common angle is the right angle — the angle at the corner of a square — which is a 90-degree angle: Angles that have fewer than 90 degrees — that is, angles that are sharper than a right angle — are called acute angles, like this one: Angles that measure greater than 90 degrees — that is, angles that aren't as sharp as a right angle — are called obtuse angles, as seen here: When an angle is exactly 180 degrees, it forms a straight line and is called a straight angle. Shaping things up A shape is any closed geometrical figure that has an inside and an outside. Circles, squares, triangles, and larger polygons are all examples of shapes. Much of plane geometry focuses on different types of shapes.

View ArticleArticle / Updated 04-25-2016

Multiplication and division with negative numbers is virtually the same as with positive numbers. The presence of one or more minus signs (–) doesn't change the numerical part of the answer. The only question is whether the sign is positive or negative. Just remember that when you multiply or divide two numbers, If the numbers have the same sign, the result is always positive. If the numbers have opposite signs, the result is always negative. For example, As you can see, the numerical portion of the answer is always 6. The only question is whether the complete answer is 6 or –6. That's where the rule of same or opposite signs comes in. Another way of thinking about this rule is that the two negatives cancel each other out to make a positive. Similarly, look at these four division equations: In this case, the numerical portion of the answer is always 5. When the signs are the same, the result is positive, and when the signs are different, the result is negative.

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