Let A = {-6, -12⁄4, -5⁄8, -√3, 0, ¼, 1, 2π, 3, √12}. List all the elements of A that belong to each set. Real numbers

Guida verificata passo dopo passo

Recall that the set of real numbers includes all rational and irrational numbers, including integers, fractions, and irrational roots, but excludes complex numbers with imaginary parts.

Examine each element of the set \(A = \{-6, -\frac{12}{4}, -\frac{5}{8}, -\sqrt{3}, 0, \frac{1}{4}, 1, 2\pi, 3, \sqrt{12}\}\) to determine if it is a real number.

Note that \(-6\) is an integer, so it is a real number.

Simplify \(-\frac{12}{4}\) to \(-3\), which is an integer and thus a real number.

Recognize that \(-\frac{5}{8}\) is a rational number (a fraction), so it is real.

Understand that \(-\sqrt{3}\) is an irrational number (since \(\sqrt{3}\) is irrational), but still a real number.

Note that \(0\) is a real number.

Recognize that \(\frac{1}{4}\) is a rational number, so it is real.

Note that \(1\) is an integer and thus real.

Understand that \(2\pi\) is a real number because \(\pi\) is irrational but real, and multiplying by 2 keeps it real.

Note that \(3\) is an integer and real.

Simplify \(\sqrt{12}\) to \(2\sqrt{3}\), which is irrational but real.

Conclude that all elements in set \(A\) are real numbers.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Real Numbers

Real numbers include all rational and irrational numbers, encompassing integers, fractions, decimals, and roots. They can be represented on the number line and do not include imaginary or complex numbers. Understanding this set helps identify which elements from a given list are real.

Video consigliato:

Percorso guidato

3:31

Introduction to Complex Numbers

Rational and Irrational Numbers

Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be written as simple fractions and have non-repeating, non-terminating decimals. Recognizing these helps classify elements like -5/8 (rational) and -√3 (irrational) within the real numbers.

Video consigliato:

Percorso guidato

2:58

Rationalizing Denominators

Simplification of Expressions

Simplifying expressions such as fractions and roots is essential to accurately identify and compare numbers. For example, -12/4 simplifies to -3, and √12 simplifies to 2√3, aiding in clearer classification within sets like real numbers.

Video consigliato:

Percorso guidato