Exam Study Notes: Elements in Mathematics, Economics, Basic Engineering Sciences and ME Laws

ALGEBRA

• Function Symmetry:
  • If f(t) = f(-t), the function has even symmetry.
• Significant Figures:
  • 1.4140 has four significant figures.
• Roots of an Equation:
  • If the roots of an equation are zero, they are classified as trivial solutions.
• Convergent Series:
  • A convergent series is a sequence of decreasing numbers where the succeeding term is lesser than the preceding term.
• Axioms in Algebra:
  • If a = b then b = a, this illustrates the symmetric axiom.
• Probability:
  • If A and B are independent events, P(A) = Pa, and P(A cap B) = P{ab}, then P(B) = frac{P{ab}}{Pa}.
• Second Derivative of a Curve:
  • If the second derivative of a curve's equation equals the negative of the equation, the curve is a sinusoid.
• Law of Cosines:
  • To find the angle of a triangle given only the lengths of the sides, use the Law of Cosines.
• Equal Equations:
  • Two or more equations are equal if and only if they have the same solution set.
• Square Matrix Determinant:
  • In any square matrix, when the elements of any two rows are exactly the same, the determinant is zero.
• Signs of Natural Functions:
  • For angles between 90° and 180°, the cosine is negative.
• Inverse Natural Function:
  • The inverse natural function of the cosecant is the sine.
• Cumulative Frequency Distribution:
  • The graphical presentation of a cumulative frequency distribution in a set of statistical data is called an Ogive.
• Statement of Truth:
  • A statement of truth that follows with little or no proof from a theorem is a corollary.
• Arithmetic Progression:
  • A sequence of numbers where successive terms differ by a constant.
• Frequency Curve:
  • A frequency curve composed of a series of rectangles constructed with the steps as the base and the frequency as the height is a histogram.
• Ratio and Proportion:
  • The ratio or product of two expressions in direct or inverse relation with each other is called the constant of variation.
• Harmonic Progression:
  • A sequence of terms whose reciprocals form an arithmetic progression.
• Matrix:
  • An array of mxn quantities representing a single number system composed of elements in rows and columns is a matrix.
• Binary Number System:
  • The binary number system, using base 2, is also known as the Dyadic number system.
• Rational Number:
  • The number 0.123123123… is a rational number.
• Roman Numerals:
  • MCMXCIV is the Roman numeral equivalent to 1994.
• Divergent Series:
  • A sequence of numbers where the succeeding term is greater than the preceding term is called a divergent series.
• Like Terms:
  • Terms that differ only in numeric coefficients are like terms.
• Argand Diagram:
  • In complex algebra, we use an Argand diagram to represent the complex plane.
• Imaginary Number:
  • 7 + 0i is an imaginary number.
• Axioms vs. Postulates:
  • Axioms are propositions of a general logical nature, while postulates are propositions concerning objects and constructions.
• Lemma:
  • A lemma is an ancillary theorem whose result is not the target for the proof.
• Axiom Definition:
  • Axioms are statements accepted without proof, from the Greek "axioma" meaning "worth", "correct," "true," or "perfect."
• Probability:
  • The number of successful outcomes divided by the number of possible outcomes is the probability.
• Axioms in Logical Reasoning:
  • In mathematical and logical reasoning, axioms are used as the basis for the formulation of statements called theorems.
• Two-Digit Number Representation:
  • If x is the unit digit and y is the tens digit, the number is represented as 10y + x.
• Axiom:
  • A statement of truth admitted without proof is an axiom.
• Hypothesis:
  • The part of a theorem assumed to be true is the hypothesis.
• Corollary:
  • A statement of truth which follows with little or no proof from the theorem is a corollary.
• Postulate:
  • The construction or drawing of lines and figures is admitted without proof is called postulates.
• Commutative Law of Multiplication:
  • The product of two or more numbers is the same in whatever order they are multiplied.
• Substitution Law of Identity:
  • If a = b, then b can replace a in any equation.
• Reflexive Law of Identity:
  • If a = a, then it illustrates the Reflexive Law.
• Transitive Law:
  • If a = b and b = c, then a = c. This illustrates the transitive law.
• Distributive Law:
  • The axiom relating addition and multiplication is the distributive law.
• Conjecture:
  • A mathematical statement which has neither been proved nor denied by counterexamples is a conjecture.
• Lemma:
  • A proved proposition useful mainly as a preliminary to proving a theorem is a lemma.
• Algebraic Expression:
  • Any combination of symbols and numbers related by the fundamental operations of algebra is an algebraic expression.
• Multinomial:
  • The algebraic expression consisting a sum of any number of terms is called a multinomial.
• Rational Equation:
  • An equation that is satisfied by all values of the variable for which the members of the equation are defined is a rational equation.
• Literal Numbers:
  • The numbers which are represented with letters; in another words, equations in which some or all of the known quantities are represented by letters.
• Conditional Equations:
  • Equations whose members are equal only for certain values of the unknown are conditional equations.
• Monomial:
  • An algebraic expression consisting of one term is a monomial.
• Redundant Equation:
  • An equation which, because of some mathematical process, has acquired an extra root is a redundant equation.
• Irrational Equation:
  • An equation in which the variables appear under the radical symbol is an irrational equation.
• Defective Equation:
  • Any equation which, because of some mathematical process, has fewer roots than its original is a defective equation.
• Rational Algebraic Expression:
  • An algebraic expression that can be represented as a quotient of two polynomials is a rational algebraic expression.
• Open Sentence:
  • A statement containing one or more variables, becoming either true or false when variables are given specific values.
• Integral Rational Term:
  • Any algebraic term that consists of the product of possible integral powers of numbers and a factor not containing them.
• Implicit Function:
  • An equation in x and y not easily solved for y in terms of x is an implicit function.
• Term:
  • In algebra, consists of products and quotients of ordinary numbers and letters representing numbers.
• Binomial:
  • An expression of two terms is called a binomial.
• Degree of a Polynomial:
  • The degree of a polynomial or equation is the maximum sum of the exponents.
    • Example: What is the degree of the polynomial 3x^2y^4 + 2x^3z^3 - 4yz^2 ? Answer: The degree is 6th (2+4).
• Complex Fraction:
  • Any fraction that contains one or more fractions in either numerator or denominator, or both, is called a complex fraction.
• Unit Fraction:
  • A common fraction with unity for numerator and a positive integer as denominator (i.e. 1/n).
• Proper Fraction:
  • If the absolute value of the numerator of a fraction is smaller than the denominator, it is a proper fraction
• Decimal Fraction:
  • A number consisting of an integer part and a decimal part less than unity that follows the decimal marker, which may be a point or a comma, ex: 0.123.
• Natural Numbers:
  • Considered as the "counting numbers".
• Modulus of a Complex Number:
  • For a complex number a + bi, the real number \sqrt{a^2 + b^2} is the modulus or absolute value of the complex number.
• Irrational Number:
  • A number represented by a non-terminating, non-repeating decimal is an irrational number.
• Completeness Axiom:
  • The completeness axiom proved that the real number system has numbers other than rational numbers.
• Dispersion:
  • The concept of spread of a random variable or a set of observations is called dispersion.
• Product of Complex Numbers:
  • The product of two complex numbers is found by multiplying each term of one by every term of the other.
• Rational Number:
  • A number which can be expressed as a quotient of two integers (division of zero excluded) is called a rational number.
• Prime Number Divisors:
  • A prime number has exactly two divisors.
• Repeating Decimal:
  • A number containing a non-terminating but repeating decimal is a rational number.
• Square-Free Integer:
  • A positive integer which has no perfect-square factor greater than 1 is a square-free integer.
• Numbers:
  • Numbers are used to describe a magnitude (quantity) and a position.
• Numerals:
  • Numerals are symbols or combinations of symbols which describe a number.
• Prime Number Definition:
  • A prime number is an integer greater than 1 which has 1 and itself as its only positive divisors.
• Composite Number Definition:
  • An integer which is the product of two integers, both different from 1 and -1, is called a composite number.
• Divisors of Composite Number:
  • A composite number has at least three divisors.
• Relatively Prime:
  • Two natural numbers a and b are relatively prime if their greatest common divisor is 1.
• Imaginary Numbers:
  • Imaginary numbers are not integers.
• Imaginary Number with Even Exponent:
  • When an imaginary number is raised to an even exponent, it becomes a real number.
• Pure Imaginary Number:
  • The complex number is in the form of a + bi. If a = 0, the resulting number is a pure imaginary number.
• Cardinal Numbers:
  • Numbers used to count the objects or ideas in a given collection are called cardinal numbers.
• Ordinal Numbers:
  • Numbers which are used to state the position of individual objects in a sequence are called ordinal numbers.
• Perfect Number:
  • An integer number that is equal to the sum of all its possible divisors except the number itself is a perfect number.
• Abundant Number:
  • An integer where teh sum of all its possible divisors, except the number itself, is greater than the integer.
• Defective Number:
  • An integer wherein the sum of all its possible divisors, except the number itself, is less than the integer.
• Ratio Meaning:
  • The term "ratio" comes from the Latin verb "ratus", meaning "to estimate."
• Proportion Extremes:
  • In a proportion of four quantities, the first and fourth terms are referred to as the extremes.
• Smallest Perfect Number:
  • 6 is the smallest perfect number possible.
• Even Perfect Numbers:
  • All perfect numbers are even numbers.
• Amicable Numbers:
  • Two integer numbers are said to be amicable numbers if each is the sum of all possible divisors of the other.
• Antecedent:
  • The first term of a ratio is called the antecedent.
• Consequent:
  • The second term of a ratio is called the consequent.
• Friendly Numbers:
  • Amicable numbers are also known as friendly numbers.
• Smallest Pair of Friendly Numbers:
  • The smallest pair of friendly numbers is 220 and 284.
• Twin Primes:
  • Prime numbers that appear in pairs and differ by 2 (i.e. 3 and 5, 11 and 13 etc.) are called twin primes.
• Goldbach Conjecture:
  • The statement that every even integer greater than 2 can be written as the sum of two primes is known as Goldbach's conjecture.
• Fundamental Theorem of Arithmetic:
  • Every positive integer greater than 1 is a prime or can be expressed as a unique product of primes and powers”. This is known as the fundamental theorem of arithmetic.
• Vinogradov's Theorem:
  • “Every sufficiently large odd number can be express as a sum of three prime numbers". This is known as Vinogradov's theorem
• Mean Proportional:
  • The mean proportional is the square root of the product of the extremes.
• Equal Means in Proportion:
  • If the means of a proportion are equal, their common value is called the mean proportional.
• Dirichlet Theorem:
  • The theorem that in every arithmetic progression a, a +d, a = 2d….. where a and d are relatively prime, is known as Dirichlet's Theorem.
• Inequality:
  • A statement that one mathematical expression is greater than or less than another.
• Absolute Inequality:
  • If an equality is truw for all values fo the variable, it is a absolute inequality.
• Preserved Inequality:
  • If the same number is added to both sides if an inequality is a preserved inequality.
• Positive Number and Inequality:
  • An inequality is preserved if both sides are multiplied by a positive number.
• Negative Number and Inequality:
  • An inequality is reversed if both sides are multiplied by a negative number.
• Dichotomy:
  • Division of a population or sample into two groups based either on measurable variables or on attributes.
• Matrix Multiplication:
  • A 3 x 2 matrix can be multiplied to a 2 x 5 matrix, not vice versa since multiplication of matrices isn't commutative.
• Interchanged Determinant:
  • If two rows of a determinant are interchanged, the determinant changes sign.
• Matrices Operation:
  • Division is not an operation of matrices.
• Square Matrix:
  • If there are as many equations as unknowns, the matrix of the coefficient is a square matrix.
• Cramer's Rule:
  • A method of solving linear equations with several unknowns simultaneously using determinants.
• Denominator in Cramer's Rule:
  • Using Cramer's rule, the determinant of the coefficient is always the denominator of the quotient.
• Identical Elements in Square Matrix:
  • In any square matrix, when the elements of any two rows are exactly the same the determinant is zero.
• Surd:
  • An irrational number which is a root of a positive integer or fraction is called a surd.
• Index of Radical:
  • The symbol \sqrt[n]{b} means the principal nth root. "n" is called the index.
• Radicand:
  • In the preceding item, b is called the radicand.
• Radical Symbol:
  • The symbol \sqrt{\ } is called the radical symbol.
• Rules of Combining Radicals:
  • The rules of combining radicals follow the rules for fractional exponents.
• Zero Determinant:
  • When corresponding elements of two rows of a determinant are proportional, then the value of the determinant is zero.
• Positive Principal nth Root:
  • When a number has both a positive and negative nth root, the principal nth root is the positive root.
• Matrix:
  • An array of mxn quantities which represent a single number and are composed of elements in rows and columns is known as a matrix.
• Interchanging Rows in Determinant:
  • When two rows are interchanged in position, the value of the determinant will be multiplied by -1.
• Principal nth Root of Negative Number:
  • The principal nth root of a negative number is the negative root if n is odd.
• Conjugate to Eliminate Surd:
  • To eliminate a surd, multiply it by its conjugate.
• Multiplying Row by Constant:
  • If every element of a row (or column) is multiplied by a constant, k, then the value of the determinant is multiplied by k.
• Surd:
  • A radical which is equivalent to a non-terminating and non-repeating decimal.
• Discriminant of Quadratic Equation:
  • For the quadratic equation Ax^2 + Bx + C = 0, the discriminant is B^2 - 4AC.
• A Surd:
  • A radical expressing an irrational number.
• Mixed Surd:
  • A surd which contains at least one rational term is called a mixed surd.
• Pure Surd:
  • A surd that contains no rational number is a pure surd. Example: \sqrt{3} or \sqrt{\sqrt{3} + \sqrt{2}}.
• Rationalizing the Denominator:
  • The process of removing a surd from a denominator is to rationalize the denominator.
• Pure Quadratic Equation:
  • A quadratic equation of the form ax^2 + c = 0, without the coefficient of the first-degree term, is a pure quadratic equation.
• Product of Roots of Quadratic Equation:
  • In the quadratic equation Ax^2 + Bx + C = 0, when the two roots are multiplied, the result is C/A.
• Discriminant of Quadratic Equation:
  • The discriminant determines the nature of the roots of a quadratic equation.
• Real Roots of Cubic Equation:
  • The real roots of a cubic equation are the points of intersection of the graph of the equation with the x-axis.
• Cubic Equation with Zero Discriminant:
  • For a cubic equation, if the discriminant is equal to zero, we produce three real roots, of which two are equal.
• Cubic Equation with Three Distinct Real Roots:
  • For a cubic equation, we produce three distinct real roots only if the discriminant is less than zero.
• Cubic Equation:
  • If the discriminant is greater than zero, the roots are one real and two conjugate complex roots.
• Sequence:
  • A succession of numbers in which one number is designated as first, another as second, another as third and so on is called a sequence.
• Series:
  • An indicated sum a1 + a2 + a_3 + … is called a series.
• Geometric Series of Repeating Decimal:
  • The repeating decimal 0.333… is a geometric series of a_1 = 0.3 and r = \frac{1}{10}.
• Harmonic Progression:
  • A progression whose reciprocal forms an arithmetic progression is a harmonic progression.
• Geometric Means:
  • The number between two geometric terms.
• Partial Sum:
  • The sum of the first n terms of a series is called the nth partial sum.
• Arithmetic Series:
  • The sum of the terms of an arithmetic progression is the arithmetic series.
• Harmonic Mean:
  • The harmonic mean between a and b is \frac{2ab}{a+b}.
• Triangular Numbers:
  • Pyramid numbers are numbers which can be drawn as dots and arranged in triangular shapes (i.e. 1, 3, 6, 10, 15, 21…). They are also called Triangular Numbers.
• Fibonacci Numbers:
  • A sequence of numbers where the number is equal to the sum of the two preceding numbers is called a Fibonacci sequence.
    • Example: sequence such as 1, 1, 2, 3, 5, 8, 13, 21…
• Multiplicative Inverse:
  • The multiplicative inverse of the integer 5 is \frac{1}{5}.
• Additive Identity Element:
  • 0 is the additive identity element.
• Multiplicative Identity Element:
  • 1 is the multiplicative identity element.
• Arithmetic Mean:
  • The arithmetic mean of a and b is \frac{a+b}{2}.
• Geometric Mean:
  • The geometric mean of a and b is \sqrt{ab}.
• Additive Identity:
  • The number 0 such that 0 + a = a for all a is called the additive identity.
• Additive Inverse:
  • The additive inverse of a complex number a + bi is -a - bi.
• Opppsites:
  • All real numbers have additive inverses, commonly called opposites.
• Reciprocals:
  • All real numbers except zero have multiplicative inverse, commonly called reciprocals.
• No Multiplicative Inverse for Zero:
  • The number zero has no multiplicative inverse.
• Additive Inverse of a + bi:
  • The additive inverse of a + bi is -a - bi.
• Multiplicative Inverse of a + bi:
  • The multiplicative inverse of a + bi is \frac{a}{a^2 + b^2} - i \frac{b}{a^2 + b^2}.
• Cubic Numbers:
  • A sequence 1, 8, 27, 64, 125, 216… is known as Cubic Numbers.
    • (Answer C if the same choices)
• ** Tetrahedral Numbers:**
  • A sequence 1, 4, 10, 20, 35, 56.. is known as Tetrahedral Numbers.
• Pascal's Triangle Binomial Expansion:
  • One property of a binomial expansion (x+y)^n is that the number of terms = n + 1.
• Coefficient of Second Term:
  • The coefficient of the second term of the expansion of (x + y)^n is always equal to n.
• Pascal's Tringle by Sum:
  • Numbers in Pascal's triangle are each obtained "by getting the sum of the two numbers directly above it."
• Binomial Expansion with Negative Sign:
  • If the sign between the terms of the binomial is negative, its expansion will have signs which are alternate starting with positive.
• Term Coefficient in Binomial Expansion:
  • In the absence of Pascal's triangle, the coefficient of any term of the binomial expansion can be obtained by dividing the product of coefficient of the preceding term and exponent of x of the preceding term by the exponent of y + 1 of the preceding term.
• Permutation:
  • Is the arrangement of the objects in specific order.
• Combination:
  • Is the arrangement of objects regardless of the order they are arranged.
• Degree of Permutation:
  • The number of elements in the collection being permuted is the degree of the permutation.
• Probability Definition:
  • The ratio of the successful outcomes over the total possible outcomes is called probability.
• Value of Probability :
  • The value of probability of any outcome will never be equal to nor exceed 1.
• Mutually Exclusive Events Probability:
  • If two events A and B are mutually exclusive events and the probability that A will happen is Pa and the probability that B will happen is Pb, then the probability that A or B happen is Pa + Pb.
• Independent Events Probability:
  • A and B are two independent events. The probability that A can occur is p and that for both A and B to occur is q. The probability that event B can occur is \frac{q}{p}.
• Probability of Non-Occurrence:
  • If the probability of occurance of a is Pa, what is the probability that a will not occur? Answer: 1 - Pa.
• Fundamental Principle of Counting:
  • The fundamental principle of counting states that if one thing can be done in "m" different ways and another thing can be done in "n" different ways, then the two things can be done in m \times n different ways.
• Statistical Depiction of Probability Concepts:
  • The statistical depiction of probability concepts is the Venn diagram.
• Range:
  • The range is the difference between the highest score and the lowest score in the distribution.
• Variance:
  • The second power of the standard deviation is called variance.
• Ogive:
  • A graph of cumulative frequency distribution plotted at class marks and connected by straight lines is called Ogive.
• Median:
  • A point in the distribution of scores at which 50 percent of the scores fall below and 50 percent of the scores fall above is called median.
• Mode:
  • A number that occurs most frequently in a group of numbers is called mode.
• Factors:
  • Each of two or more numbers which is multiplied together to form a product are called factors.
• Power:
  • When the factors of a product are equal, the product is called a power of the repeated factor.
• Function Definition:
  • A relation in which every ordered pair (x, y) has one and only one value of y that corresponds to the value of x is called a function.
• Approximate Value:
  • A value which is not exact but might be accurate enough for some specific considerations is an approximate value.
• Reliable Digit:
  • If the absolute error does not exceed a half unit in the last gigit, this digit is usually referred to as the reliable digit.
• Most Significant Digit:
  • The most significant digit of the number 0.2015 is 2.
• Accuracy:
  • Accuracy is stated in the magnitude of the absolute or relative error of the approximated value.
• Leading Digit:
  • The first non-zero digit from the left of the number is leading digit.
• Significant Figure:
  • A significant figure is any one of the digit from 1 to 9 inclusive, and 0 except when it is used to place a decimal.
• Evolution:
  • In algebra, the operation of the root extraction is called evolution.
• Power:
  • The operation of raising to the integral power known as involution.
• False Statement:
  • The false statement : The objects in a set are called its elements; ever number is either rational or irrational; the additive inverse of number a is \frac{1}{a}; The negative of zero is zero.
    • So, the additive inverse of number a is \frac{1}{a} , it is false.
• Arithmetic Sum:
  • Any one of the individual constants of an expressed sum of constant is called addend.
• Multiplicand:
  • In the equation n \times m = q, n is called the multiplicand.
• Parameter:
  • A symbol holding a place for an unspecified constant is called a parameter.
• False Statement About Significant Figures:
  • The false statement about significant figures is: any zero not needed for placing a decimal point is NOT significant.
• Sum of Number and Reciprocal:
  • The sum of any point number and its reciprocal is always greater than 2.
• Absolute Value With Exponent Infinity:
  • What is the absolute value of a number less than one but greater than negative one raised to exponent infinity? Answer: Zero.
• Even Expression:
  • If a is an odd number and b is an even number, which expression must be even? The answer is a times b.
• Augend:
  • In the equation 5 + 2 = 7, 5 is known as augend.
• Complex Number:
  • A number of the form a + bi, with a and b real constants and i is the square root of -1, is a complex number.
• Positive Result:
  • The absolute value of a non-zero number is always positive.
• Rational Number Relations:
  • For any two rational number a/b and c/d, which the relations is true? The coreect answer is \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}.
• Equal Rational Numbers:
  • Two rational numbers \frac{a}{b} and \frac{c}{d} are said to be equal if ad = bc.
• Division by Infinity:
  • Any number divided by infinity equals 0.
• Numbers Theory:
  • The study of the properties of positive integers is known as Number of Theory.
• Least Common Multiple:
  • A polynomial exactly divisible by two or more polynomials is called the least common multiple.
• Linear Factors:
  • A polynomial with real coefficient can be factored into real linear factors and irreducible quadratic factors.
• False Statement:
  • The false statement: a - b = (a + b)(a - ab + b^2).
  • It should have been written more like: a - b = (a + b)(a^2 - ab + b^2)
• Scientific Notation:
  • A number is said to be in scientific notation when it is written as the product of a number having the decimal point just after the leading digit, and a power of 10.
• Linear Polynomial:
  • If the degree of the numerator is one more than the degree of the denominator, the quotient is a linear polynomial.
• Transcendental Numbers:
  • A number which cannot be a root of an integral rational equation is called transcendental number.
• For Every Value/Law of Addition/Subraction:
  • For every low of addition and substraction, there is a parallel flow for multiplication and division except division by zero.
• Reciprocal and Nots Roots Algebraic Equation:
  • Refers to the numbers which are not roots of any algebraic equation are trasncendental numbers.
• Invere of unity:
  • All number multipled by it's reciprocal equals unity. - All number multiplied unit for its reciprocal is equal to one.
• False Statement 1:
  • False statement: ab ba isthe associative law from multiplicaiton, should be ab x bc= ba
• Euler's Number:
  • The number is denoted as an eaqual to 2.718… is called Eulers number
• Factorial Notiation:
  * A notation that represents product od all posstive integerts from 1 ton which number and including:
• Factorial:
  • The notation n! called the factorial
• Simplify Fraction and Factorials:
  * \frac{n!}{(n-1)!} =n.
• Christian Kramp
  • The factorial symbol (!) was indtroduced in 1808 by Christian Kramp.
    • Divergent Series
      • If an infinite series has no sum,it is reffered to an divergetn.
• Sum of Interms in A:.
• *Goldback Conjecture:
  • Conjecture that every even number (except two) equals the sum of two prime numbers.
• **Febonacci Number:
  *The unending sequence of intehers formed according to the rule taht each itneger is the sum of the preceeding integer:
• Farment numbers is a result to a prime :
  • Fermat Theoreom
    If > the eqaution X+Y+Z can be solved in postive integer *xzy
    #TRIGONOMETTRY:
• Sin(A) cos(B)-cos(A) sin(B)
  The equaivlance is sin (A-B)
• The sum angles in sphere
  The sum of angles in ocant is 540
• The angular distance of pint
  The angular distance is from North to is called co altitude.
• CSc 520 Deequal
  The cos 20 equai co what value
• Sine of 820
  Sine =0.866
• *Median Of Triangle :
  The median of angles intersect as