Real numbers
Explication
Real numbers (ℝ) include all rational and irrational numbers, making them the most comprehensive set among the options. Natural numbers are only non-negative integers, integers include negatives, and decimal numbers are a subset of rational numbers with finite decimal expansion.
ℕ ⊂ ℤ ⊂ ⅅ ⊂ ℚ ⊂ ℝ
Explication
The number sets are nested as ℕ ⊂ ℤ ⊂ ⅅ ⊂ ℚ ⊂ ℝ, meaning natural numbers are included in integers, finite decimals, rationals, and reals respectively. Understanding this hierarchy helps in grasping the scope of each set.
Open interval excluding endpoints a and b
Explication
The notation ]a; b[ represents an open interval, which includes all numbers between a and b but excludes the endpoints a and b themselves.
ℝ
Explication
The real numbers ℝ include both rationals and irrationals, such as π and √2. These are numbers that cannot be expressed as fractions.
x = c ± d
Explication
The equation |x - c| = d has two solutions: x = c + d and x = c - d, representing the points at a distance d from c on the number line.
| -5 | = 5
Explication
The absolute value of a number is its magnitude, so |-5| = 5, meaning the value disregards the sign.
x is in the interval [c - d; c + d]
Explication
The inequality |x - c| ≤ d means x lies within d units of c, i.e., in the interval [c - d; c + d]. It describes all points within distance d of c.
]a; +∞[
Explication
The notation ]a; +∞[ indicates an open interval from a to infinity, meaning all numbers greater than a.
To make numbers easier to read and compare
Explication
Scientific notation condenses large or tiny numbers into a manageable form, e.g., 3.0 × 10^8, making them easier to read, compare, and use in calculations.
ℤ
Explication
The set ℤ comprises all integers, which include positive, negative, and zero, covering all whole numbers.
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Number sets — inclusion?
ℕ ⊂ ℤ ⊂ ⅅ ⊂ ℚ ⊂ ℝ
Number sets — hierarchy?
ℕ ⊂ ℤ ⊂ ⅅ ⊂ ℚ ⊂ ℝ
Interval notation — types?
Open, closed, infinite intervals
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