Table of Contents
Mathematics is like a secret code that helps us understand the world around us. It’s all about numbers, shapes, and patterns. Learning math can help us solve problems, make sense of data, and even have fun with puzzles and games. Have you ever wondered how to count, add, or subtract? Or how shapes fit together? By learning the vocabulary of math, you’ll unlock the power to solve problems, think logically, and discover the beauty of numbers. Are you ready to embark on this exciting math adventure?
Basic Arithmetic
1. Addition (noun) /əˈdɪʃ(ə)n/: The process of combining two or more numbers to get a total.
Example: If you have 2 apples and I give you 3 more, the addition of these numbers means you now have 5 apples.
2. Subtraction (noun) /səbˈtrakʃ(ə)n/: The process of taking away one number from another.
Example: If you start with 10 cookies and eat 3, the subtraction of these numbers leaves you with 7 cookies.
3. Multiplication (noun) /ˌmʌltɪplɪˈkeɪʃ(ə)n/: The process of adding a number to itself a certain number of times.
Example: If you have 5 groups of 4 pencils each, you have 20 pencils in total, which is the result of multiplication.
4. Division (noun) /dɪˈvɪʒ(ə)n/: The process of splitting a number into equal parts.
Example: If you have 12 candies and want to share them equally among 3 friends, division tells you that each friend gets 4 candies.
5. Whole numbers (noun) /hoʊl ˈnʌmbər/: A number without fractions or decimals, including zero and positive numbers.
Example: 5, 10, and 15 are all examples of whole numbers.
6. Integers (noun) /ˈɪntɪdʒər/: A whole number that can be either positive, negative, or zero.
Example: -3, 0, and 4 are all examples of integers.
7. Decimals (noun) /ˈdesɪməl/: A number that represents a fraction or a part of a whole, written with a dot (.)
Example: 0.5 represents half and is an example of a decimal.
8. Fractions (noun) /ˈfrækʃ(ə)n/: A number that represents part of a whole or part of a set.
Example: If you eat 1 of 2 equal parts of a pizza, you’ve eaten 1/2 of it, which is a fraction.
9. Percentages (noun) /pərˈsentɪdʒ/: A way of expressing a number as a fraction of 100.
Example: If 20 people out of 100 surveyed like ice cream, then 20% of the people like ice cream, which is an example of percentage.
10. Order of operations (noun) /ˈɔːrdər ɒv ˈɒpəˈreɪʃənz/: Rules that tell the sequence in which operations in an expression to be done (parentheses, exponents, multiplication and division, and addition and subtraction).
Example: In the expression 5 + 2 x 3, the order of operations tells you to do the multiplication before the addition, so the answer is 11, not 21.
Algebra
1. Variables (noun) /ˈveərɪəbl/: Symbols that represent unknown numbers or values.
Example: In the equation x + 3 = 7, x is a variable.
2. Equations (noun) /ɪˈkweɪʒ(ə)n/: Mathematical statements that assert the equality of two expressions.
Example: In the statement 3x – 2 = 4, 3x – 2 and 4 are expressions and the entire statement is an equation.
3. Expressions (noun) /ɪkˈsprɛʃ(ə)n/: Combinations of numbers, variables and operations without an equals (=) sign.
Example: In the expression 2y + 5, 2y and 5 are combined using the operation of addition.
4. Inequalities (noun) /ˌɪnɪˈkwɔːlɪtiz/: Mathematical statements that compare two quantities using less than, greater than, less than or equal to, or greater than or equal to.
Example: In the inequality x > 5, x is a variable and the inequality states that x is greater than 5.
5. Functions (noun) /ˈfʌŋkʃ(ə)n/: Mathematical relationships between two quantities where one quantity depends on the other.
Example: In the function y = 3x + 2, y is dependent on the value of x.
6. Graphing (noun) /ˈɡræfɪŋ/: The process of representing data or equations visually using lines, bars, points, or other symbols on a coordinate plane.
Example: Graphing the function y = x^2 gives a curve called a parabola.
7. Polynomials (noun) /ˌpɒlɪˈnoʊmiːəl/: Mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.
Example: The polynomial 2x^2 + 3x – 2 is composed of three terms.
8. Quadratic equations (noun) /ˌkwɒdrəˈtɪk ɪˈkweɪʒ(ə)n/: Equations that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
Example: The quadratic equation x^2 – 5x + 6 = 0 has solutions that can be found using the quadratic formula.
9. Systems of equations (noun) /ˈsɪstɪm ɒv ɪˈkweɪʒ(ə)n/: A set of two or more equations that are solved together.
Example: The system of equations x + y = 7 and x – y = 1 can be solved to find the values of x and y.
10. Exponents (noun) /ɪkˈspoʊnənt/: Numbers that denote the power to which the base number is to be raised.
Example: In the expression 2^3, 3 is the exponent, which means 2 is multiplied by itself 3 times.
Geometry
1. Points, lines, and planes (noun) /pɔɪnts, laɪnz ænd pleɪnz/: Basic concepts in geometry where a point is a location, a line is a straight path that extends in two opposite directions, and a plane is a flat surface that extends without end.
Example: In a rectangular room, the corners are points, the edges are lines, and the walls, floor, and ceiling are planes.
2. Angles (noun) /ˈæŋɡəlz/: A measure of rotation or bend between two lines that meet at a common point called the vertex.
Example: When you open a book, the opening creates an angle.
3. Triangles (noun) /ˈtraɪæŋɡəlz/: Polygons with three edges (or sides) and three vertices (or corners).
Example: A slice of pie shaped like a triangle has three sides and three corners.
4. Quadrilaterals (noun) /kwɒdɹəˈlatərəl/: Polygons that have four edges (sides) and four vertices (corners).
Example: A rectangular piece of paper is a quadrilateral because it has four sides and four corners.
5. Circles (noun) /ˈsɝːkəlz/: A shape with all points in a plane at a constant distance from a fixed point called the center.
Example: The sun appears as a circle in the sky.
6. Polygons (noun) /ˈpɒlɪɡɒn/: A shape made of straight lines, and the shape is “closed” (all the lines connect up).
Example: A hexagon is a polygon with six sides.
7. Perimeter (noun) /pəˈrɪmɪtər/: The total distance around the boundary of a shape.
Example: If you walk around the edge of a park, you have walked the perimeter of the park.
8. Area (noun) /ˈeəriːə/: The amount of space inside the boundary of a flat (2-dimensional) object.
Example: The area of a rectangular room can be found by multiplying its length by its width.
9. Volume (noun) /ˈvɒl.juːm/: The amount of space a 3-dimensional object takes up.
Example: The volume of a cube can be found by multiplying its length, width, and height.
10. Coordinate geometry (noun) /koʊˈɔːrdɪnət dʒiːˈɒmɪtri/: A branch of geometry where points are defined within a given space by coordinates, usually as pairs (x, y) or triplets (x, y, z).
Example: In coordinate geometry, we might describe the location of a point on a graph by its coordinates, such as (2,3).
Trigonometry
1. Sine, cosine, and tangent (noun) /saɪn, ‘koʊsaɪn, tæn’dʒənt/: They are mathematical functions commonly used in trigonometry. Sine (abbreviated sin) and cosine (abbreviated cos) represent the ratios of sides of a right triangle to its hypotenuse, while tangent (abbreviated tan) is the ratio of sine to cosine.
Example: In a right triangle, the sine, cosine, and tangent of an angle can help us find unknown side lengths.
2. Right triangles (noun) /raɪt ‘traɪæŋɡəlz/: Triangles that have one angle equal to 90 degrees.
Example: A ladder leaning against a wall forms a right triangle with the ground and the wall.
3. Trigonometric functions (noun) /ˌtrɪɡənoʊˈmɛtrɪk ˈfʌŋkʃənz/: Mathematical functions resulting from the ratios of sides of a right triangle. The primary ones are sine, cosine, and tangent.
Example: Trigonometric functions such as sine, cosine, and tangent are useful for solving problems in physics and engineering.
4. Pythagorean theorem (noun) /ˌpɪθæɡəˈriːən ˈθiːrəm/: A mathematical principle that states in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Example: If you know the lengths of two sides of a right triangle, you can use the Pythagorean theorem to find the length of the third side.
5. Unit circle (noun) /ˈjuːnɪt ˈsɝːkl/: A circle with a radius of 1, typically used in trigonometry for defining trigonometric functions.
Example: The unit circle helps us understand the relationship between angles and the trigonometric functions.
6. Trigonometric identities (noun) /ˌtrɪɡənoʊˈmɛtrɪk aɪˈdɛntɪtiz/: Mathematical equations involving trigonometric functions that are true for every value of the variable.
Example: One of the most common trigonometric identities is sin²x + cos²x = 1.
7. Trigonometric ratios (noun) /ˌtrɪɡənoʊˈmɛtrɪk ‘reɪʃioʊz/: These are the ratios of the sides of a right triangle, given a specific angle, commonly referred to as sine, cosine, and tangent.
Example: The trigonometric ratios of a specific angle in a right triangle are used to relate the angle to the lengths of the sides of the triangle.
8. Law of sines (noun) /lɔː ɒv saɪnz/: In any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides and angles.
Example: The law of sines can be used to find unknown sides or angles in a triangle.
9. Law of cosines (noun) /lɔː ɒv ‘koʊsaɪnz/: In any triangle, the square of a side length is equal to the sum of the squares of the other two lengths, minus twice their product multiplied by the cosine of the included angle.
Example: The law of cosines is a generalization of the Pythagorean theorem that works for any type of triangle, not just right triangles.
10. Trigonometric equations (noun) /ˌtrɪɡənoʊˈmɛtrɪk iː’kweɪʒənz/: Equations involving trigonometric functions and variables.
Example: Trigonometric equations can be solved to find the values of the variables that make the equation true.
Statistics and Probability
1. Data analysis (noun) /ˈdeɪtə əˈnælɪsɪs/: The process of inspecting, cleaning, transforming, and modeling data to discover useful information.
Example: Data analysis can help us understand trends and make informed decisions.
2. Mean, median, and mode (noun) /miːn, ‘miːdiən, moʊd/: These are measures of central tendency in statistics. The mean is the average, the median is the middle value, and the mode is the most frequently occurring value in a dataset.
Example: In a set of test scores, the mean, median, and mode can provide different perspectives on the students’ performance.
3. Range (noun) /reɪndʒ/: The difference between the highest and lowest values in a set of data.
Example: The range of ages at the summer camp was 10 years, with the youngest camper being 7 and the oldest being 17.
4. Variance (noun) /’veəriəns/: A measure of how much values in a data set differ from the mean.
Example: If the test scores have a high variance, it means the students’ performances varied a lot.
5. Standard deviation (noun) /ˌstændərd diːviˈeɪʃən/: A measure that tells us how much individual data points differ from the mean on average.
Example: If the standard deviation of a set of test scores is low, it means most students scored close to the average.
6. Probability (noun) /ˌprɑːbəˈbɪlɪti/: The chance or likelihood that a particular event will occur.
Example: In a coin toss, the probability of getting heads is 0.5 or 50%.
7. Experimental probability (noun) /ɪkˌspɛrɪˈmɛntəl prɑːbəˈbɪlɪti/: The ratio of the number of times an event occurs to the total number of trials or times the activity is performed.
Example: If we roll a die 60 times and get a six 10 times, the experimental probability of getting a six is 10/60 or 1/6.
8. Theoretical probability (noun) /ˌθiːəˈrɛtɪkəl prɑːbəˈbɪlɪti/: The probability that is calculated by considering all possible outcomes.
Example: The theoretical probability of getting a six when rolling a fair six-sided die is 1/6 because there’s one six and six possible outcomes.
9. Sampling methods (noun) /ˈsæmplɪŋ ˈmɛθədz/: The ways in which we select individuals from the population to be in the sample for a study or experiment.
Example: In a survey about student opinions, different sampling methods could include random sampling, stratified sampling, or systematic sampling.
10. Statistical graphs and charts (noun) /stəˈtɪstɪkəl ɡræfs ænd tʃɑːrts/: Visual representations of data that can include bar graphs, pie charts, histograms, and scatter plots.
Example: Statistical graphs and charts are helpful for understanding data patterns and trends.
Calculus
1. Limits (noun) /ˈlɪmɪts/: A concept in calculus where a sequence, function, or series approaches a specific value. E
xample: When we study limits, we’re trying to find out what a function is approaching as it gets closer to a certain point.
2. Derivatives (noun) /dɪˈrɪvətɪvs/: These are the fundamental tool in calculus for determining how a function changes at any given point.
Example: By calculating the derivative of a function, we can find the slope of the function at any point.
3. Integration (noun) /ˌɪntɪˈɡreɪʃən/: This is the reverse process of differentiation, used in calculus to calculate areas under curves, among other things.
Example: Integration is like adding up an infinite number of infinitesimally small parts.
4. Differentiation (noun) /ˌdɪfəˌrɛnʃiˈeɪʃən/: The process in calculus of finding the derivative, which represents the rate at which a quantity is changing at a given point.
Example: Differentiation helps us find the rate of change or the slope at any point on the curve of a function.
5. Differential equations (noun) /ˌdɪfəˈrɛnʃl ɪˈkweɪʒəns/: Equations that involve derivatives and are used to describe rates of change and relationships between variables.
Example: Physicists use differential equations to model the movement of planets and galaxies.
6. Rates of change (noun) /reɪts ɒv tʃeɪndʒ/: The speed at which a variable changes over a specific period of time.
Example: In calculus, rates of change are typically represented by the derivative of a function.
7. Maxima and minima (noun) /ˈmæksɪmə ænd ˈmɪnɪmə/: The highest points (maxima) and the lowest points (minima) on the curve of a function.
Example: By finding the maxima and minima of a function, we can understand where the function reaches its peak or valley.
8. Optimization (noun) /ˌɒptɪmaɪˈzeɪʃən/: The process of making a system or design as effective or functional as possible, often through the use of mathematical techniques.
Example: Optimization in calculus can help find the most efficient way to use resources or minimize costs.
9. Area under a curve (noun) /ˈɛəriə ˈʌndər ə kɜːrv/: The integral of a function from a to b which gives the total area between the x-axis and the graph of the function from x=a to x=b.
Example: The process of integration is often used to calculate the area under a curve.
10. Applications of calculus (noun) /ˌæplɪˈkeɪʃənz ɒv ˈkælkjʊləs/: The practical ways that calculus is used in real life, including physics, engineering, economics, statistics, and medicine.
Example: One of the applications of calculus is in physics, where it is used to describe the motion of celestial bodies and particles.
Number Theory
1. Prime numbers (noun) /praɪm ˈnʌmbərz/: Whole numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder.
Example: 2, 3, 5, 7, and 11 are examples of prime numbers.
2. Factors and multiples (noun) /ˈfæktərz ənd ˈmʌltɪpəlz/: Factors are numbers that can be multiplied together to get another number, while multiples are numbers that can be divided by another number without leaving a remainder.
Example: The factors of 12 are 1, 2, 3, 4, 6, and 12, and the first four multiples of 4 are 4, 8, 12, and 16.
3. Divisibility rules (noun) /dɪˈvɪzəˌbɪlɪti ˈrulz/: Guidelines that help determine if a number can be evenly divided by another number without actually performing the division.
Example: One of the divisibility rules is that if the last digit of a number is even, the number is divisible by 2.
4. Prime factorization (noun) /praɪm ˌfæktərɪˈzeɪʃən/: The process of breaking down a number into its prime factors, which are the prime numbers that multiply together to form that number.
Example: The prime factorization of 24 is 2 × 2 × 2 × 3, or 2³ × 3¹.
5. GCD and LCM (noun) /ˈdʒiːˈsiːˈdiː ənd ˈɛlˈsiːˈɛm/: GCD (Greatest Common Divisor) is the largest number that divides two or more numbers exactly, while LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers.
Example: The GCD of 8 and 12 is 4, and the LCM of 8 and 12 is 24.
6. Rational and irrational numbers (noun) /ˈræʃənəl ənd ɪˈræʃənəl ˈnʌmbərz/: Rational numbers are numbers that can be expressed as a fraction, where both the numerator and denominator are integers, while irrational numbers cannot be expressed as a fraction.
Example: 1/2 and -3 are rational numbers, while the square root of 2 and pi are irrational numbers.
7. Real numbers (noun) /riːl ˈnʌmbərz/: All numbers that can be represented on a number line, including rational and irrational numbers.
Example: -2, 0, 1/3, and pi are all examples of real numbers.
8. Complex numbers (noun) /kəmˈplɛks ˈnʌmbərz/: Numbers that have a real part and an imaginary part, usually written as a + bi, where a and b are real numbers, and i is the imaginary unit.
Example: 3 + 4i and -2 – 5i are examples of complex numbers.
9. Sequences and series (noun) /ˈsikwənsəz ənd ˈsɪriz/: Sequences are ordered lists of numbers, while series are sums of the terms in a sequence.
Example: The arithmetic sequence 2, 4, 6, 8 is a sequence, and the sum of its terms, 2 + 4 + 6 + 8 = 20, is a series.
10. Number patterns (noun) /ˈnʌmbər ˈpætərnz/: Regular arrangements or rules that help predict the next number in a sequence or list of numbers.
Example: In the number pattern 3, 6, 9, 12, the rule is to add 3 to the previous number to get the next number in the sequence.
Mathematical Logic
1. Propositions (noun) /ˌprɒpəˈzɪʃənz/: Statements that can be either true or false, but not both, often used as basic building blocks in logic and mathematics.
Example: In logic, “A is a multiple of 3″ and “B is an even number” are examples of propositions.
2. Truth tables (noun) /truθ ˈteɪbəlz/: A chart used to show all possible truth values of logical expressions or statements, often used to study the relationships between propositions.
Example: When learning about logic, students often create truth tables to understand how different logical connectives work.
3. Logical connectives (noun) /ˈlɒdʒɪkəl kəˈnɛktɪvz/: Symbols or words used to combine propositions and create more complex logical expressions, such as “and,” “or,” “not,” and “if…then.”
Example: In the logical expression “A and B,” the word “and” is a logical connective.
4. Implication (noun) /ˌɪmplɪˈkeɪʃən/: A logical connective that represents the “if…then” relationship between two propositions, where the truth of the first proposition implies the truth of the second proposition.
Example: In the statement “If it rains, then I will stay indoors,” the implication is that rain causes me to stay indoors.
5. Conjunction (noun) /kənˈdʒʌŋkʃən/: A logical connective that combines two propositions, creating a new proposition that is true only when both original propositions are true.
Example: In the statement “It is cold and it is snowing,” the conjunction “and” connects the two propositions.
6. Disjunction (noun) /dɪsˈdʒʌŋkʃən/: A logical connective that creates a new proposition that is true when at least one of the original propositions is true.
Example: In the statement “I will go to the park or I will go to the movies,” the disjunction “or” connects the two propositions.
7. Negation (noun) /nɪˈɡeɪʃən/: A logical operation that reverses the truth value of a proposition, turning true statements into false ones and vice versa.
Example: The negation of the statement “It is raining” is “It is not raining.”
8. Equivalence (noun) /ɪˈkwɪvələns/: A relationship between two propositions that have the same truth value, meaning they are either both true or both false.
Example: The statements “If it is raining, then the ground is wet” and “If the ground is not wet, then it is not raining” have an equivalence relationship.
9. Logical reasoning (noun) /ˈlɒdʒɪkəl ˈriːzənɪŋ/: The process of using logic, rules, and principles to draw conclusions and solve problems.
Example: To solve a difficult math problem, the student relied on her logical reasoning skills to find the correct solution.
10. Proof techniques (noun) /pruf ˈtɛkniːks/: Methods used in mathematics and logic to demonstrate the truth or validity of a statement or proposition, such as direct proofs, indirect proofs, and proof by contradiction.
Example: The teacher taught her students various proof techniques to help them understand and prove mathematical theorems.
Congratulations on exploring the world of math vocabulary! Now you have the tools to understand and communicate about numbers, shapes, and patterns. Whether you’re counting objects, solving equations, or measuring shapes, math will be your trusted companion. Remember, math is all around us, from calculating the cost of items at the store to planning your schedule. Embrace the power of math and keep exploring its wonders!
