tag:blogger.com,1999:blog-46305781807111091362017-07-30T01:48:19.711-07:00Math Lesson A DayA Math or Statistics Lesson Each Day...to keep me posted on all those things I shouldn't forget.Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.comBlogger165125tag:blogger.com,1999:blog-4630578180711109136.post-40713842735689735862009-04-20T18:45:00.000-07:002009-04-20T18:47:17.999-07:00Scalene TriangleA scalene triangle has no congruent sides and no congruent angles. That is to say a scalene triangle has no sides or angles which are the same.<br /><br /><a href="http://en.wikipedia.org/wiki/Triangle#Types_of_triangles">For more...</a>Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-70935899735167237632009-04-19T20:07:00.000-07:002009-04-19T20:16:30.262-07:00Multiplying decimalsWhen multiplying decimals:<br /><br />1. Ignore the decimal point<br />2. Add the number of digits to the right of the point of the numbers you are multiplying<br />3. Starting from the right of the answer, place that number of digits before the decimal point (add zeros if you have to)<br /><br />Example:<br /><br />.2 * .4 <br /><br />1. Ignore digits so 2*4 = 8<br />2. Count numbers 1 and 1, so 2 total<br />3. Count over two from right of answer so 0.08<br /><br />Now<br /><br />0.03 * 0.004<br /><br />1. Ignore digits: 3*4 = 12<br />2. Count digits: 2 and 3 so 5 total<br />3. Count over: 0.00012<br /><br />also 1.33 * 3.44<br /><br />1. Ignore digits so 133*344 = 45752<br />2. Count: 2 and 2 so 4<br />3. Count over: 4.5752<br /><br />Nice!Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com1tag:blogger.com,1999:blog-4630578180711109136.post-11085969405637327492009-04-18T19:39:00.000-07:002009-04-18T19:40:38.177-07:00TopologyTopology (Greek Τοπολογία, from τόπος, “place”, and λόγος, “study”) is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others.<br /><br />Ideas that are now classified as topological were expressed as early as 1736, and toward the end of the 19th century a distinct discipline developed, called in Latin the geometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking apart of place”), and later gaining the modern name of topology. In the middle of the 20th century, this was an important growth area within mathematics.<br /><br />The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms, which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line.<br /><br />Topology includes many subfields. The most basic and traditional division within topology is point-set topology, which establishes the foundational aspects of topology and investigates concepts as compactness and connectedness; algebraic topology, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology and graph theory, do not fit neatly in this division.<br /><br />http://en.wikipedia.org/wiki/TopologyPaul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-14090861145856454332009-04-17T18:42:00.000-07:002009-04-17T18:43:14.535-07:00Poisson distributionIn probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.<br /><br />http://en.wikipedia.org/wiki/Poisson_distributionPaul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-21871229463027366312009-04-16T11:04:00.000-07:002009-04-16T11:05:05.032-07:00Radix and BaseIn arithmetic, the radix or base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". The term power strictly refers to the entire expression, but is sometimes used to refer to the exponent.<br /><br />http://en.wikipedia.org/wiki/Base_(mathematics)Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-64174737391329537942009-04-15T20:59:00.000-07:002009-04-15T21:03:08.788-07:00SexagesimalSexagesimal (base-sixty) is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BC, was transmitted to the Babylonians, and is still used—in modified form—for measuring time, angles, and geographic coordinates.<br /><br />The number 60 has twelve factors, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, of which 2, 3, and 5 are prime. With so many factors, many fractions of sexagesimal numbers are simple. For example, an hour can be divided evenly into segments of 30 minutes, 20 minutes, 15 minutes, etc. 60 is the smallest number divisible by every number from 1 to 6.<br /><br />http://en.wikipedia.org/wiki/SexagesimalPaul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-44235904025753035222009-04-14T20:47:00.000-07:002009-04-15T21:03:36.906-07:00DecimalsDecimal notation is the writing of numbers in a base-10 numeral system. Examples are Roman numerals, Brahmi numerals, and Chinese numerals, as well as the Arabic numerals used by speakers of English. Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). Brahmi numerals had symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100 and another for 1000. Chinese has symbols for 1–9, and fourteen additional symbols for higher powers of 10, which in modern usage reach 1044.<br /><br />http://en.wikipedia.org/wiki/DecimalPaul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-80024682288471553342009-04-13T20:34:00.000-07:002009-04-13T20:37:09.265-07:00Proper and Improper FractionsI had almost forgot about this distinction.<br /><br />Proper fractions always have a smaller number in the numerator than in the denominator. (On top than under) <br /><br />Examples of proper fractions<br /><br />1/2 , 2/5 , 6/17 <br /><br />Improper fractions have a larger numerator than denominator like<br /><br />5/2 , 7/4 , 20/17 <br /><br />These fractions can be rewritten with whole numbers included like<br /><br />5/2 = 2(1/2)<br />7/4 = 1(3/4)<br />20/17=1(3/17)Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-75737302848009839092009-04-12T18:33:00.001-07:002009-04-12T18:34:10.515-07:00The Golden RatioIn mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.<br /><br />http://en.wikipedia.org/wiki/Golden_ratioPaul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-87929622656728782102009-04-11T19:57:00.000-07:002009-04-11T20:05:21.412-07:00The Quadratic FormulaThe quadratic formula can help you solve any quadratic equation of the form<br /><br />ax<sup>2</sup> + bx + c<br /><br />To find the solutions to this equation we can use the quadratic formula which is written as follows<br /><br />(-b (+or-) sqrt(b<sup>2</sup>-4ac)) / 2a<br /><br />Let us consider an example of<br /><br />x<sup>2</sup> + 6x + 7<br /><br />a=1<br />b=6<br />c=7<br /><br />so<br /><br />(-6 (+or-) sqrt(6<sup>2</sup>-4*1*7)) / 2*1<br /><br />=<br /><br />-6 (+or-) sqrt(36 - 28) / 2<br /><br />=<br /><br />-6 (+or-) sqrt(8) / 2<br /><br />We can simplify the square root so we get<br /><br />(-6 (+or-) 2sqrt(2)) / 2<br /><br />=<br /><br />-3 +or- sqrt(2)<br /><br />and that is our final answer.Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-83903181814976604472009-04-10T17:41:00.000-07:002009-04-10T17:46:11.827-07:00Solving Equations with a Square RootWhen we solve an equation by taking a square root, we have to consider both a positive and negative outcome.<br /><br />For example consider -3 and 3<br /><br />now 3<sup>2</sup> = 9 and -3<sup>2</sup>=9<br /><br />Thus when we have an equation such that<br /><br />x<sup>2</sup> = 9 to find x we need to take the square root of 9, however, then we have to say x= + or - 3 that is to say positive or negative 3 since it could be either and we don't know.Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-34737332465113479592009-04-09T19:01:00.001-07:002009-04-09T19:01:54.443-07:00Multiplying and dividing square roots (radicals)Square roots act just like other numbers when you multiply and divide them, consider the example<br /><br />3sqrt(5) * 5sqrt(6) = 15sqrt(30)<br /><br />Similarly<br /><br />15sqrt(30) / 5sqrt(6) = 3sqrt(5)Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-37593548674839035572009-04-08T15:08:00.000-07:002009-04-08T15:38:55.218-07:00Adding and Subtracting RadicalsWhen adding and subtracting radicals you treat the radicands as variables.<br /><br />Example<br /><br />3sqrt(5) + 4sqrt(5) = 7sqrt(5)<br /><br />However, we cannot add together a radicand that is different as in<br /><br />3sqrt(5) + 4sqrt(2)Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-37850279392859173552009-04-07T17:58:00.000-07:002009-04-07T18:18:40.222-07:00The square root of a fractionBy definition we cannot derive the square root of a fraction. Thus we must find a way to get a fraction out of the radical.<br /><br />Consider the square root of 1/2<br /><br />sqrt(1/2)<br /><br />how can we get the fraction out? We have to multiply and make the denominator a perfect square. But what we multiply to the bottom we must also multiply to the top<br /><br />sqrt((1/2)*(2/2))<br /><br />=<br /><br />1/2*sqrt(2)<br /><br />Now we have got the fraction out of the radical and created a square root we can rationalize.<br /><br />Another example<br /><br />sqrt(3/5)<br /><br />To get the denominator out we multiply by (5/5) (which is equal to 1)<br /><br />sqrt((3/5)*(5/5))<br /><br />=<br /><br />(1/5)*sqrt(3*5)<br /><br />=<br /><br />(1/5)*sqrt(15)Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com1tag:blogger.com,1999:blog-4630578180711109136.post-48974539981909026452009-04-06T19:55:00.000-07:002009-04-06T19:59:16.027-07:00Simplifying RadicalsLike so much of algebra, it is good to know how to simplify radicals for purposes of canceling out or combining like terms.<br /><br />Consider the square root of 27 or sqrt(27) there is no whole number that equals sqrt(27) but we can write the radical as sqrt(9*3) and that is equal to 3*sqrt(3)<br /><br />Thus we have simplified the radical for purposes of mathematical calculation or canceling.Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-16308758493369636152009-04-05T20:20:00.000-07:002009-04-05T20:21:07.633-07:00Square roots are radicalIt is true, taking the square root of an expression can also be called "The radical"<br /><br />We will write square root as sqrt on this website, thus <br /><br />sqrt(4) = 2<br /><br />and<br /><br />sqrt(9x<sup>2</sup>y<sup>10</sup>) ?<br /><br />To solve this, it is good to factor under the radical ( or factor the square root)<br /><br />So we get<br /><br />sqrt(9x<sup>2</sup>y<sup>10</sup>)<br /><br />=<br /><br />(3xy<sup>5</sup>)Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-83328276483169821392009-04-04T18:44:00.000-07:002009-04-04T18:56:05.476-07:00Solving quadractics by factoringOne way to solve a quadratic equation like:<br /><br />(4x<sup>2</sup> - 4)=0 is by factoring and setting both factors equal to zero. Because a quadratic contains a x<sup>2</sup> they often have two solutions.<br /><br />(4x<sup>2</sup> - 4)=0 can be factored to<br /><br />(2x - 2)(2x + 2)=0<br /><br />Now set both factors equal to zero<br /><br />2x-2=0<br />2x+2=0<br /><br />we get<br />x=1 and x=-1<br />substituting 1 or -1 for x will solve the quadratic equation (4x<sup>2</sup> + 4)=0Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-63516338425585027472009-04-03T18:44:00.000-07:002009-04-03T18:47:18.161-07:00Quadratic EquationA quadratic equation is described as an equation where the highest exponent is 2. <br />The graph of a quadratic is a smooth curve known as a parabola.<br /><br />All of the following are quadratic equations.<br />x<sup>2</sup> + 4 = 0<br /><br />x<sup>2</sup> + 4x + 3 = 0<br /><br />3x<sup>2</sup> + 34x + 7 = 50Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-62047195043428107452009-04-02T20:42:00.000-07:002009-04-02T20:56:16.273-07:00Factoring a trinomialAs I said yesterday, I really think factoring comes down to trial and error till the process is internalized. As an example today we will factor trinomials.<br /><br />Consider<br /><br />5x<sup>2</sup> - 8x - 21<br /><br />This is quite a complicated trinomial to factor, lets start with a guess<br /><br />First off, we know that to get 5x<sup>2</sup> we need to multiply 5x and x, so that gives us our first two terms:<br /><br />(5x + ) (x - )<br /><br />As a further guess I also alternated the signs. <br /><br />Now we can try guess what two numbers can multiply to give us -21. How about 7 and -3?<br /><br />(5x + 7) (x - 3)<br /><br />Checking with the foil method we get<br />5x<sup>2</sup> - 8x - 21Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-66078252335084525782009-04-01T08:41:00.000-07:002009-04-01T08:51:51.378-07:00Factoring with the difference of squaresI think factoring is something which becomes internal, you see a problem, make a guess, and then check. The best method is trial and error till it becomes intuitive.<br /><br />Still the difference of squares method is often taught, and so I will show it here.<br /><br />Basically the difference of squares is always factored in the following form:<br /><br />(x+y)(x-y)<br /><br />Which equals (x<sup>2</sup> - y<sup>2</sup>)<br /><br />Example<br /><br />x<sup>2</sup> - 9<br /><br />9 is a perfect square so we can use the memorized formula<br /><br />(x-3)(x+3)<br /><br />Again, I prefer gaining an intuitive understanding of factoring, but memorizing a rule like this can help till you gain an intuitive understanding.Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-3985704001794651992009-03-31T20:25:00.000-07:002009-03-31T20:28:52.797-07:00Factoring binomials using the greatest common factorOne way to factor binomials is by searching for the greatest common factor.<br /><br />Consider<br /><br />(5x * 25)<br /><br />In this case the greatest common factor is 5 and the phrase can be written as<br /><br />5(x*5)Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-23264100206981067892009-03-30T20:31:00.000-07:002009-03-30T20:39:27.465-07:00Multiplying a trinomial by a binomialMultiplying a trinomial by a binomial is a lot like <a href="http://mathproofaday.blogspot.com/2009/03/multiplying-binomial-by-binomial.html">multiplying a binomial by a binomial</a>. You multiply the first term by all the factors of the second term, then multiply the outer(last) term by all the factors of the second term, then simplify.<br /><br />Consider the example:<br /><br />(x+5) (5x<sup>2</sup> + 3x + 6)<br /><br />First we multiply our first term (x) by every term in the trinomial (5x<sup>2</sup> + 3x + 6)<br /><br />this gives us:<br /><br />(5x<sup>3</sup> + 3x<sup>2</sup> + 6x)<br /><br />next we multiply our outer term (5) by every term in the trinomial (5x<sup>2</sup> + 3x + 6)<br /><br />this gives us:<br />(25x<sup>2</sup> + 15x + 30)<br /><br />so we have<br />(5x<sup>3</sup> + 3x<sup>2</sup> + 6x) + (25x<sup>2</sup> + 15x + 30)<br /><br />we can simplify by adding like terms to get:<br />(5x<sup>3</sup> + 28x<sup>2</sup> + 21x + 30)Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-36194246275159992332009-03-29T21:35:00.000-07:002009-03-29T21:41:32.637-07:00Multiplying a binomial by a binomialThe most common way to multiply binomials is what is called the FOIL method<br /><br /><b>F</b>irst<br /><b>O</b>uter<br /><b>I</b>nner<br /><b>L</b>ast<br /><br />Let us look at an example<br /><br />(x+4) (x+1)<br /><br />These are both binomials, to multiply them we first multiply the first two terms to get x<sup>2</sup><br /><br />Then we still take the first x and multiply it by the outer number: 1, to get x.<br /><br />So far we have<br />x<sup>2</sup> + x<br /><br />Now we do the inner number: 4<br /><br />4 times x is 4x <br /><br />and finally the last number 4 times 1 is 4<br /><br />So in total we have<br />x<sup>2</sup> + x + 4x + 4<br /><br />which can be simplified by combining the like x terms to <br /><br />x<sup>2</sup> + 5x + 4Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-45075172728235035102009-03-28T19:59:00.000-07:002009-03-28T20:08:07.444-07:00PolynomialsPolynomials can be anything from a single number to a variable to a combination of numbers and variables<br /><br />Monomials have one term.<br />Such as... 8x<sup>4</sup> , 6 , or 2xy<br /><br />Binomials have 2 terms which are not like.<br />Such as... 2wz - 4dt , 4x<sup>2</sup> - 3x , 4c - 2d<br /><br />Trinomials have 3 terms which are not like.<br />Such as... 4bt - 5yu + 9o , 3x<sup>2</sup> - 2x + 9 , 5t + 7y - 8uPaul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0tag:blogger.com,1999:blog-4630578180711109136.post-91617932795200761192009-03-27T15:29:00.000-07:002009-03-27T15:35:38.967-07:00Dividing exponents<a href="http://mathproofaday.blogspot.com/2009/03/exponent.html">Yesterday</a> we learned that when you multiply exponents you add the number in the exponent, today we see that when you divide exponent you subtract the number in the exponent.<br /><br />Consider<br /><br />x<sup>7</sup> / <sup>3</sup><br /><br />What is this equal to?<br /><br />x*x*x*x*x*x*x / x*x*x = x*x*x*x or x<sup>4</sup><br /><br />x<sup>7</sup> / <sup>3</sup> = x<sup>7-3</sup> = x<sup>4</sup><br /><br />what about<br /><br />x<sup>3</sup> / <sup>7</sup><br /><br />= 1 / x<sup>7-3</sup> = 1 / x<sup>4</sup><br /><br />We take the reciprocal because the exponent is greater in the divisor, or denominator.Paul Househttp://www.blogger.com/profile/07987167402950129463noreply@blogger.com0