Name: number types on the real line
Uploaded: 2025-11-26T05:44:44.000Z
Duration: 1 min 27 s
Channel: JoVE

실수 시스템에는 유리수와 무리수를 포함하여 수직선에 배치할 수 있는 모든 값이 포함됩니다.유리수에는 분수로 쓸 수 있는 값이 포함됩니다. 예를 들어, 과일을 두 개의 동일한 부분으로 자르면 유리수인 절반이 됩니다. 여기에는 음수 4분의 3과 같은 음수 분수도 포함됩니다.유리수에는 0.5와 같이 끝나는 소수점 또는 0.33과 같이 숫자 위에 막대가 있는 반복되는 소수가 포함됩니다.유리수 안에는 정수의 하위 집합이 있습니다. 정수에는 음수, 0 및 양수가 포함됩니다. 쟁반에 담긴 과일을 세면 1, 2, 3 등의 자연수가 나옵니다. 자연수에 0을 더하면 정수가 됩니다. 음수를 포함하도록 확장하면 정수의 전체 집합이 제공됩니다.실수 체계의 또 다른 부분인 무리수는 단위 제곱의 대각선을 제공하는 2의 제곱근과 같은 값을 나타냅니다.이 숫자에는 영원히 계속되는 반복되지 않는 소수가 있으며 분수로 쓸 수 없습니다.실수 시스템에는 계산, 측정 또는 비교에 사용되는 모든 값이 포함됩니다.

실수 체계

Real Number System

&#49892;&#49688; &#52404;&#44228;&#45716; &#49688;&#47484; &#49464;&#44144;&#45208;, &#52769;&#51221;&#54616;&#44144;&#45208;, &#48708;&#44368;&#54616;&#45716; &#45936; &#49324;&#50857;&#46104;&#45716; &#47784;&#46304; &#49688;&#47484; &#54252;&#54632;&#54633;&#45768;&#45796;. &#51088;&#50672;&#49688;&#45716; 1, 2, 3 &#46321;&#44284; &#44057;&#51008; &#44032;&#51109; &#44592;&#52488;&#51201;&#51064; &#49480;&#49688;&#51077;&#45768;&#45796;. &#51221;&#49688;&#45716; 0&#44284; &#51020;&#51032; &#51221;&#49688;&#47484; &#54252;&#54632;&#54616;&#50668; &#51060; &#51665;&#54633;&#51012; &#54869;&#51109;&#54633;&#45768;&#45796;. (&#8230;, &#8722;3, &#8722;2, &#8722;1, 0, 1, 2, &#8230;). &#50976;&#47532;&#49688;&#45716; &#46160; &#51221;&#49688;&#51032; &#48708; m/n (&#45800;, n&#8800;0)&#47196; &#45208;&#53440;&#45244; &#49688; &#51080;&#45716; &#49688;&#47484; &#47568;&#54616;&#47728;, 1/2&#50752; &#44057;&#51008; &#48516;&#49688;, 46&#44284; &#44057;&#51008; &#51221;&#49688;, 0.17&#44284; &#44057;&#51008; &#50976;&#54620;&#49548;&#49688;, 0.317171717&#52376;&#47100; &#48152;&#48373;&#46104;&#45716; &#49692;&#54872;&#49548;&#49688;&#47484; &#54252;&#54632;&#54633;&#45768;&#45796;. &#49692;&#54872;&#49548;&#49688;&#45716; &#45824;&#49688;&#51201; &#48169;&#48277;&#51012; &#49324;&#50857;&#54616;&#50668; &#48516;&#49688;&#47196; &#45796;&#49884; &#45208;&#53440;&#45244; &#49688; &#51080;&#49845;&#45768;&#45796;.
&#47924;&#47532;&#49688;&#45716; &#51221;&#49688;&#51032; &#48708;&#47196; &#45208;&#53440;&#45244; &#49688; &#50630;&#45716; &#49688;&#51077;&#45768;&#45796;. &#49548;&#49688; &#51204;&#44060;&#45716; &#47924;&#54620;&#55176; &#51060;&#50612;&#51648;&#47732;&#49436;&#46020; &#48152;&#48373;&#46104;&#51648; &#50506;&#49845;&#45768;&#45796;. &#50696;&#47196; &#8730;2, &#8730;5, &#960;, e &#48143; 3/&#960;^2 &#46321;&#51060; &#51080;&#49845;&#45768;&#45796;. &#51060;&#47084;&#54620; &#49688;&#45716; &#51221;&#49324;&#44033;&#54805;&#51032; &#45824;&#44033;&#49440; &#44600;&#51060;&#47484; &#44396;&#54616;&#45716; &#47928;&#51228; &#46321; &#44592;&#54616;&#54617;&#51201; &#47589;&#46973;&#50640;&#49436; &#51088;&#51452; &#45208;&#53440;&#45225;&#45768;&#45796;. &#47784;&#46304; &#49892;&#49688;&#45716; &#49688;&#51649;&#49440; &#50948;&#51032; &#54620; &#51216;&#50640; &#45824;&#51025;&#54633;&#45768;&#45796;. &#51060;&#47084;&#54620; &#50672;&#49549;&#51201;&#51064; &#54364;&#54788; &#45909;&#48516;&#50640; &#50976;&#47532;&#49688;&#50752; &#47924;&#47532;&#49688;&#44032; &#54616;&#45208;&#51032; &#49688;&#54617;&#51201; &#53952; &#50504;&#50640;&#49436; &#44277;&#51316;&#54616;&#50668;, &#51060;&#47200;&#44284; &#51025;&#50857; &#49688;&#54617;&#50640;&#49436; &#51221;&#48128;&#54620; &#44228;&#49328;&#44284; &#52769;&#51221;&#51012; &#44032;&#45733;&#54616;&#44172; &#54633;&#45768;&#45796;.

&#49892;&#49688; &#52404;&#44228;&#45716; &#49688;&#47484; &#49464;&#44144;&#45208;, &#52769;&#51221;&#54616;&#44144;&#45208;, ...

Education

교육

JoVE Core

Math Fundamentals

수학 기초

Unit

단위

Foundations of Mathematics

수학의 기초

KEY TERMS AND CONCEPTS

주요 용어 및 개념

실수 시스템

number types on the real line

The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system.

Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two integers, m/n, where n is not 0. This group includes fractions like 1/2, whole numbers such as 46, and decimals that terminate, like 0.17, or repeat, like 0.317171717.

Repeating decimals can be changed into fractions using algebraic methods. Irrational numbers cannot be written as a ratio of integers. Their decimal forms continue forever without repeating. Examples include √2, √5, π, e, and 3/π^2.

These numbers often appear in geometry, such as when finding the diagonal of a square. Every real number can be placed on the number line as a point. This continuous view lets rational and irrational numbers fit into one framework for exact measurement and calculation.

Number Types on the Real Line

real number system

The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two...

real number operations

Real Numbers on the Number Line

Real numbers are the values that can be placed on a continuous number line. The system grew from counting numbers used for enumeration. It later expanded to include zero, which shows the absence of quantity, and negative numbers, which are opposites of counting numbers. The real number system also includes fractions and decimals that represent parts of a whole or equal divisions. When written in decimal form, these numbers may end or repeat in a regular pattern. Together, these values describe...

exponents

Exponent Rules and Scientific Notation

Exponents show repeated multiplication in a short form. They are important in algebra and in many other math topics, including scientific computation, scaling laws, and dimensional analysis. For a nonzero real number a and an integer n, a^n means a multiplied by itself n times. Exponent rules help simplify expressions with powers. These properties make it easier to work with exponential expressions in both symbolic and numerical form. They also support common calculations that use exponents in...

radicals

Simplifying Radicals

Radicals show the root of a number and the exponent needed to get that value. A radical expression has two main parts. The radicand is the number inside the root symbol, and the index tells which root to take. The notation n√a means the principal nth root of a. The index changes the type of root. When n equals 2, the expression is a square root. When n equals 3, it is a cube root. The sign of the radicand also matters. If the index is even, a negative radicand does not give a real result. If...

algebraic expressions

Simplifying Algebraic Expressions

Algebraic expressions use variables, constants, and operations to show mathematical relationships. They help describe patterns and solve problems across many areas of mathematics. Knowing the parts of an expression makes it easier to simplify and work with it correctly. Each algebraic expression has separate pieces that work together. A coefficient is the number attached to a variable, and an exponent shows repeated multiplication. Some expressions have a single part, while others include...

rational expressions

Working with Rational Expression Domains

Rational expressions are algebraic fractions with polynomials in the numerator and denominator. They follow the same basic rules as numerical fractions, but variables make them more delicate to work with. A key step is finding values that make the expression undefined, usually because of division by zero or an undefined radical. To determine the domain, start by identifying the values excluded from the denominator. The domain includes all real numbers except those that make the denominator...

linear equations

Solving Linear Equations

Linear equations are a basic algebra tool for modeling constant rates and everyday relationships. They use constants and a single variable, and each term is either a constant or a product of a constant and that variable. When graphed on a Cartesian coordinate plane, they form a straight line that shows a constant rate of change between two quantities. A linear equation in one variable often has the form ax + b = c, where a, b, and c are constants and x is the variable. Solving the equation...

quadratic equations

Solving Quadratics by Factoring and Formula

Quadratic equations are algebraic equations with a variable squared, a first-power term, and a constant term. They are set equal to zero. In high school math, they often help model area, motion, and optimization problems. The general form of a quadratic equation uses real numbers a, b, and c, with a not equal to zero. The nonzero a value keeps the squared term in the equation. One common way to solve a quadratic is to rewrite it as a product of two linear expressions. In factored form, the...

complex numbers

Complex Numbers in Solving Equations

Complex numbers extend the real number system so equations with negative square roots can still be solved. The real numbers cannot represent the square root of a negative number. That limits some quadratic equations, especially those with negative discriminants. To solve this problem, mathematicians introduced the imaginary unit i, where i = √(-1). A complex number is written as x + yi, where x and y are real numbers. In this form, x is the real part and y is the imaginary part. The key rule...

quadratic equations in the complex number system

quadratic-equations-in-the-complex-number-system

Complex Roots of Quadratic Equations

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect. When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a...

radical equations

Solving Radical Equations with Checks

Radical equations are equations with a variable inside a radical, most often a square root or cube root. These equations appear in science, engineering, and real-world measurement problems that involve nonlinear relationships. Solving them starts by isolating the radical expression. Then the radical is removed by raising both sides to the power that matches the radical’s index. This process can create extraneous solutions, which are values that work in the transformed equation but not in the...

inequalities

Solving Inequalities and Interval Notation

Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value. Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint. Intervals can also extend forever in one...

absolute value inequalities

Absolute Value Inequalities on a Number Line

Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable. An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at...

mathematical modeling: problem solving

Mathematical Modeling for Loan Payments

Mathematical modeling turns a real-life situation into an equation that can be solved. It starts with defining the problem, choosing measurable quantities, and assigning variables to them. Then a matching model is selected and the equation is solved for the unknown value. This process is especially useful in finance, where models help evaluate investments, loans, and repayment plans. One common example is finding the fixed monthly payment on a loan. That calculation uses the standard annuity...

key terms and concepts

Explore natural numbers, integers, rational numbers, and irrational numbers, plus repeating decimals, geometry examples, and how every real number maps to the number line.

Number Types on the Real Line | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Number Types on the Real Line

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