Mathematical Concepts and Definitions

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Questions and Answers

What is a statement accepted as true without proof referred to in mathematics?

Axiom

Define what a postulate is in the context of geometric construction.

A postulate is a statement accepted without proof, concerning the drawing of lines and figures.

What characterizes a conjecture in mathematical terms?

A conjecture is a mathematical statement that has not been proved or disproved.

What is the name of a proved proposition that is primarily useful for proving a theorem?

Lemma Signup and view all the answers

How does the definition of a theorem differ from that of an axiom?

A theorem is a statement that has been proven, while an axiom is accepted as true without proof. Signup and view all the answers

What does the commutative law of multiplication state?

The commutative law of multiplication states that the product of two numbers is the same regardless of the order. Signup and view all the answers

What law is illustrated by the statement 'if a = b, then b can substitute a in any equation'?

Substitution law Signup and view all the answers

What type of algebraic expression consists of a sum of multiple terms?

Multinomial Signup and view all the answers

What is the term for a sequence of terms whose reciprocals form an arithmetic progression?

Harmonic Progression Signup and view all the answers

What is the determinant of a square matrix if any two of its rows are exactly the same?

Zero Signup and view all the answers

What is another name for the binary number system that uses a base of 2?

Dyadic number system Signup and view all the answers

What type of number is represented by the decimal 0.123123123123…?

Rational number Signup and view all the answers

What do we call a statement that is accepted without proof?

Axiom Signup and view all the answers

In the context of a matrix, what is an array of m x n quantities representing elements organized in rows and columns called?

Matrix Signup and view all the answers

What do we call terms in algebra that differ only in their numeric coefficients?

Like terms Signup and view all the answers

What is defined as the number of successful outcomes divided by the number of possible outcomes?

Probability Signup and view all the answers

What type of symmetry does the function f(t) = f(-t) exhibit?

Even symmetry Signup and view all the answers

Identify a number with four significant figures from the options given.

1.4140 Signup and view all the answers

What is the base of the natural logarithm, also known as the Naperian logarithm?

2.72 Signup and view all the answers

Which law would you use to find an angle in a triangle given the lengths of all three sides?

The law of cosines Signup and view all the answers

What type of progression is defined as a sequence of numbers that differ by a constant amount?

Arithmetic progression Signup and view all the answers

If the roots of an equation are zero, what are they classified as?

Zeros of function Signup and view all the answers

Which axiom does the statement 'If a=b then b=a' illustrate?

Symmetric axiom Signup and view all the answers

In the context of convergent series, what must be true about each succeeding term?

Less than Signup and view all the answers

Study Notes

Mathematics Concepts

Symmetry:
- An even function satisfies the condition f(t) = f(-t).
- This indicates symmetry about the y-axis.
Significant Figures:
- 0.01414 has four significant figures.
Naperian Logarithm:
- Its base is approximately 2.72.
Second Derivative of a Curve:
- If the second derivative is equal to the negative of the original equation, the curve is a sinusoid.
Triangles:
- To find a triangle's angles given only side lengths, use the law of cosines.
Trigonometric Functions (90-180 Degrees):
- The cosine function is negative for angles between 90 and 180 degrees.
Inverse Trigonometric Functions:
- The inverse natural function of cosecant is sine.
Cumulative Frequency Distribution:
- The graphical representation of a cumulative frequency distribution is called an ogive.
Corollaries:
- A corollary is a statement of truth that readily follows from a theorem with little or no proof.
Arithmetic Progression:
- An arithmetic progression is a sequence of numbers where each successive term differs by a constant.
Histogram:
- A frequency curve composed of rectangles, where the steps form the base and the frequency is the height, is called a histogram.
Roots of Equations:
- If the roots of an equation are zero, they're classified as trivial roots.
Convergent Series:
- In a convergent series, successive terms are always less than the preceding term.
Symmetric Axiom:
- If a = b, then b = a. This demonstrates the symmetric axiom.
Independent Events:
- If A and B are independent events with probabilities P(A) and P(B) respectively, the probability that both A and B occur is P(A) x P(B). The probability of B occurring is P(B).
Equal Equations:
- Two or more equations are equal if and only if they have the same solution set.
Determinants (Square Matrices):
- If two rows of a square matrix are identical, its determinant is zero.
Ratio and Proportion:
- The ratio or product of two expressions in a direct or inverse relationship is called a constant of variation.
Harmonic Progression:
- A sequence whose reciprocals are in arithmetic progression.
Matrix:
- A matrix is an array of m x n quantities, arranged in rows and columns, representing a single number system.
Binary Number System:
- Also known as the dyadic number system, it is based on the base 2 placeholder.
Rational Numbers:
- A number represented as 0.123123123... is a rational number (repeating decimal).
Roman Numerals:
- MCMXCIV = 1994
Divergent Series:
- A sequence of numbers where each succeeding term is greater than the preceding term.
Like Terms:
- Terms that differ only in their numerical coefficients.
Argand Diagram:
- In complex algebra, the complex plane is commonly represented using an Argand diagram.
Real Numbers:
- 7 + Oi is a real number.
Probability:
- The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.
Two-Digit Numbers:
- A two-digit number with units digit x and tens digit y is represented as 10y + x.
Axiom:
- A statement of truth accepted without proof.
Hypothesis:
- The assumed true part of a theorem.
Postulate:
- A statement relating to geometric construction, accepted without proof.
Conjecture:
- A mathematical statement that has neither been proved nor disproven.
Lemma:
- A proven proposition used as a step in the proof of a theorem.
Axioms and Postulates:
- Axioms are statements of general logical principles (e.g., about equality or inequality), while postulates deal with properties of objects & constructions.
Lemma (ancillary theorem):
- A lemma is an ancillary theorem whose main result isn't the target of the proof.
Worth of Axioms:
- Axioms come from the Greek word "axioma" meaning worth.
Hypothesis and Theorem:
- Axioms are the foundation for formulating hypotheses and theorems.
Commutative Law of Multiplication:
- The order of multiplication does not affect the product.
Substitution Law:
- If two values are equal, one can replace one with the other in an equation, preserving the equality
Reflexive Law:
- Any quantity is equal to itself (a=a).
Transitive Law:
- If a = b and b = c, then a = c.
Distributive Law:
- The law relating addition and multiplication (not multiplication and addition).
Algebraic Expression:
- Any combination of numbers and symbols related by the fundamental operations of algebra.
Multinomial: A sum of any number of terms (unlike binomials).
Rational Equation:
- An equation that is satisfied by all values of the variable provided the members of the equation are defined. (This means the equation is true for all allowed values, not just certain values).
Literal Equation:
- An equation which contains some or all of the known quantities represented by letters (variables).
Irrational Equation:
- An equation in which a variable is under a radical.
Redundant Equation:
- An equation that acquires an extra root through mathematical processes.
Defective Equation:
- An equation with fewer roots than its original form through mathematical process.
Rational Algebraic Expression:
- An algebraic expression that can be expressed as a quotient of two polynomials.
Open Sentence:
- A statement containing one or more variables that becomes either true or false when given specific values for the variables.
Integral Rational Term:
- A term that results from multiplying possible integral powers of numbers.
Implicit Function:
- An equation in x and y that's not easily solved for y in terms of x.
Variables as Literal Numbers:
- Letters used to represent numbers in equations or expressions.
Conditional Equations:
- Equations whose equality holds only for certain values of unknowns.
Monomial:
- An algebraic expression consisting of a single term.
Term (algebra):
- A product or quotient of numbers or variables in algebraic expressions
Expression (algebra):
- A combination, through multiplication or division, of variables and ordinary numbers.
Binomial:
- An expression consisting of two terms.
Degree of a Polynomial Equation:
- The highest sum of exponents of the variables in a polynomial expression.
Complex Fraction:
- A fraction that has fractions in the numerator or denominator, or both.
Ordinary Fraction / Common Fraction:
- A fraction with an integer numerator and denominator.
Unit Fraction:
- A common fraction where the numerator is 1.
Proper Fraction:
- A fraction where the absolute value of the numerator is smaller than the denominator.
Decimal Fraction:
- Numbers containing a decimal part, less than unity, following a decimal point or comma.
Mixed Number:
- A number with an integer part and a proper fraction part.
Counting Numbers / Natural Numbers:
- Numbers used for counting.
Irrational Number:
- A number that cannot be expressed as a quotient of two integers.
Rational Number:
- A number that can be represented as a quotient of two integers (ex. fractions, repeating decimals).
Integers:
- Whole numbers and their negatives.
Imaginary Number:
- A number of the form bi where b is a real number and i represents the imaginary unit.
Absolute Value / Magnitude / Modulus:
- The distance of a complex number from the origin (0, 0) in the complex plane.
Product of Complex Numbers:
- The product of two complex numbers is found by multiplying each term in the first complex number with each term in the second complex number.
Rational Number (definition):
- Any number expressible as a quotient of two integers.
Prime Number:
- A number greater than 1 that is only divisible by 1 and itself.
Composite Number:
- An integer greater than 1 that is not prime.
Relatively Prime / Coprime Numbers:
- Two natural numbers with a greatest common divisor of 1.
Cardinal Numbers:
- Numbers used to count objects.
Ordinal Numbers:
- Numbers used to indicate position within a sequence.
Perfect Number:
- An integer equal to the sum of all its positive divisors excluding itself
Abundant Number:
- A number whose sum of its positive divisors excluding itself is greater than itself.
Defective Number:
- A number whose sum of its positive divisors excluding itself is smaller than itself.
Amicable Numbers:
- Two numbers where each is equal to the sum of the proper divisors of the other. (Also called friendly numbers).
Twin Primes:
- Pairs of prime numbers that differ by 2 (e.g., 3 and 5, 5 and 7).
Goldbach Conjecture:
- Every even integer greater than 2 is the sum of two prime numbers.
Fundamental Theorem of Arithmetic:
- Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers and their powers.
Vinogradov's Theorem:
- Every sufficiently large odd integer is expressible as the sum of three odd prime numbers
Ratio: - comes from Latin “ratus” meaning to divide

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Test your knowledge on fundamental mathematical concepts such as postulates, conjectures, theorems, and various laws. This quiz also covers topics related to algebra and matrices, helping you solidify your understanding of these essential topics in mathematics.