Title Math-Readings-1.pdf ✅

Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math I - 1 - Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! BASIC ARITHMETIC AND ALGEBRA POINTERS Whole (natural) numbers Natural numbers – numbers, which appear as a result of calculus of single subjects: peoples, animals, birds, trees, different wares and so on. Series of natural numbers: 1, 2, 3, 4, 5, ... is continued endlessly and is called natural series. Arithmetical operations Addition – an operation of finding a sum of some numbers: 11 + 6 = 17. Here 11 and 6 – addends, 17 – the sum. If addends are changed by places, a sum is saved the same: 11 + 6 = 17 and 6 + 11 = 17. Subtraction – an operation of finding an addend by a sum and another addend: 17 – 6 = 11. Here 17 is a minuend, 6 – a subtrahend, 11 – the difference. Multiplication. To multiply one number n (a multiplicand ) by another m ( a multiplier ) means to repeat a multiplicand n as an addend m times. The result of multiplying is called a product. The operation of multiplication is written as: n x m or n · m. For example, 12 x 4 = 12 + 12 + 12 + 12 = 48. In our case 12 x 4 = 48 or 12 · 4 = 48. Here 12 is a multiplicand, 4 – a multiplier, 48 – a product. If a multiplicand n and a multiplier m are changed by places, their product is saved the same: 12 · 4 = 12 + 12 + 12 + 12 = = 48 and 4 ·12 = 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 48. Therefore, a multiplicand and a multiplier are called usually factors or multipliers. Division – an operation of finding one of factors by a product and another factor: 48 : 4 = 12. Here 48 is a dividend, 4 – a divisor, 12 – the quotient. At dividing integers a quotient can be not a whole number. Then this quotient can be present as a fraction. If a quotient is a whole number, then it is called that numbers are divisible, i.e. one number is divided without remainder by another. Otherwise, we have a division with remainder. For example, 23 isn’t divided by 4 ; this case can be written as: 23 = 5 · 4 + 3. Here 3 is a remainder. Raising to a power. To raise a number to a whole (second, third, forth, fifth etc.) power means to repeat it as a factor two, three, four, five and so on. The number, repeated as a factor, is called a base of a power; the quantity of factors is called an index or an exponent of a power; the result is called a value of a power. A raising to a power is written as:
Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math I - 2 - Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! 3 5 = 3 · 3 · 3 · 3 · 3 = 243 . Here 3 – a base of the power, 5 – an exponent (an index) of the power, 243 – a value of the power. The second power is called a square, the third one – a cube. The first power of any number is this number. Extraction of a root – an operation of finding a base of a power by the power and its exponent: Here 243 – a radicand, 5 – an index (degree) of the root, 3 – a value of the root. The second root is called a square root, the third root – a cube root.The second degree of square root isn’t written: Addition and subtraction, multiplication and division, raising to a power and extraction of a root are two by two mutually inverse operations. Order of operations. Brackets If brackets are absent, the following order of operations is right: 1) raising to a power and extraction of a root (one after another); 2) multiplication and division (one after another); 3) addition and subtraction (one after another). If brackets are present, at first all operations inside brackets are executed according to the aforesaid order, and then the rest of the operations out of brackets are executed (in the same order). E x a m p l e. Calculate the next expression: ( 10 + 23 · 3 ) + 43 – ( 16 : 2 – 1 ) · 5 – 150 : 52 . S o l u t i o n. At first, powers must be calculated and changed by theirs values: (10 + 8 · 3) + 64 – (16: 2 – 1) · 5 – 150: 25;
Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math I - 3 - Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! after this, multiplication and division in the brackets and out of them are executed: ( 10 + 24 ) + 64 – ( 8 – 1 ) · 5 – 6 ; now, additions and subtractions in the brackets are executed: 34 + 64 – 7 · 5 – 6 ; finally, after the rest of the multiplication 7 · 5 = 35 we receive: 34 + 64 – 35 – 6 = 57 Laws of addition and multiplication Commutative law of addition: m + n = n + m. A sum isn’t changed at rearrangement of its addends. Commutative law of multiplication: m · n = n · m. A product isn’t changed at rearrangement of its factors. Associative law of addition: (m + n) + k = m + (n + k) = m + n + k. A sum doesn’t depend on grouping of its addends. Associative law of multiplication: (m · n) · k = m · (n · k) = m · n · k. A product doesn’t depend on grouping of its factors. Distributive law of multiplication over addition: (m + n) · k = m · k + n · k. This law expands the rules of operations with brackets (see the previous section). Prime and composite numbers Numbers, which aren’t divisible by any numbers except 1 and itself, are called prime numbers. Numbers, which have also other factors, are called composite numbers. There is an infinite set of prime numbers. The set of them till 200 is: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.