Real Numbers - Revision history

Robert.adlington: /* Intervals */

2025-09-24T16:47:38Z

Intervals

← Older revision		Revision as of 16:47, 24 September 2025
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	\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>		\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>
	\|}		\|}
	The interval <math>[a,a]</math>is equal to <math>[a]</math>.		The interval <math>[a,a]</math> is equal to <math>[a]</math>.

	The interval <math>[b,a]</math> where <math>b > a</math> is empty.		The interval <math>[b,a]</math> where <math>b > a</math> is empty.

2025-09-24T16:47:08Z

Number Sets

← Older revision		Revision as of 16:47, 24 September 2025
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	** For example <math>\frac{11}{4} = 2.75 </math>		** For example <math>\frac{11}{4} = 2.75 </math>
	* Q: Rationals - ie all D, plus those that do recur (but may never terminate.		* Q: Rationals - ie all D, plus those that do recur (but may never terminate.
	** For example <math>1/3</math> ( = 0.333333) or <math>143/999</math> ( = 0.143143143)		** For example <math>\frac{1}{3}</math> ( = 0.333333) or <math>\frac{143}{999}</math> ( = 0.143143143)
	* R: Reals - ie all Q plus the irrationals, which are decimals that neither terminate nor recur.		* R: Reals - ie all Q plus the irrationals, which are decimals that neither terminate nor recur.
	** For example <math display="inline">\pi, \sqrt{2}, 3\sqrt{3}, -\sqrt{2}, -\sqrt{5}/2</math>, etc		** For example <math display="inline">\pi, \sqrt{2}, 3\sqrt{3}, -\sqrt{2}, -\frac{\sqrt{5}}{2}</math>, etc

	N is a subset of Z is a subset of D is a subset of Q is a subset of R		N is a subset of Z is a subset of D is a subset of Q is a subset of R

2025-09-24T16:45:34Z

Number Sets

← Older revision		Revision as of 16:45, 24 September 2025
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	** For example -2,-1,0,1,2...		** For example -2,-1,0,1,2...
	* D: Decimals - ie all Z plus fractions that can be written with a finite number of decimals.		* D: Decimals - ie all Z plus fractions that can be written with a finite number of decimals.
	** For example <math>11/4 = 2.75 </math>		** For example <math>\frac{11}{4} = 2.75 </math>
	* Q: Rationals - ie all D, plus those that do recur (but may never terminate.		* Q: Rationals - ie all D, plus those that do recur (but may never terminate.
	** For example <math>1/3</math> ( = 0.333333) or <math>143/999</math> ( = 0.143143143)		** For example <math>1/3</math> ( = 0.333333) or <math>143/999</math> ( = 0.143143143)

2025-09-24T16:43:39Z

Intervals

← Older revision		Revision as of 16:43, 24 September 2025
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	\|}		\|}
	The interval <math>[a,a]</math>is equal to <math>[a]</math>.		The interval <math>[a,a]</math>is equal to <math>[a]</math>.

	The interval <math>[b,a]</math> where <math>b > a</math> is empty.		The interval <math>[b,a]</math> where <math>b > a</math> is empty.

	The intersection of two intervals is always an interval or the empty set.		The intersection of two intervals is always an interval or the empty set.

	If the intersection of two intervals is not empty, then the union of them is an interval.		If the intersection of two intervals is not empty, then the union of them is an interval.

	== Solving Inequalities ==		== Solving Inequalities ==
			If the real number <math>a</math> is :

			* Positive, then all solutions to <math>ax + b > 0</math> is <math>]-b/a, +\inf[</math>.
			* Negative, then all solutions to <math>ax + b > 0</math> is <math>]-\inf, -b/a[</math>.

	== Sign Tables ==		== Sign Tables ==

	== Absolute Values ==		== Absolute Values ==

2025-09-24T16:38:50Z

Intervals

← Older revision		Revision as of 16:38, 24 September 2025
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	\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>		\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>
	\|}		\|}
	The interval <math>[a,a]</math>is equal to <math>[a]</math>		The interval <math>[a,a]</math>is equal to <math>[a]</math>.
			The interval <math>[b,a]</math> where <math>b > a</math> is empty.
	The interval <math>[b,a] where b > a ~~is empty~~</math>		The intersection of two intervals is always an interval or the empty set.
			If the intersection of two intervals is not empty, then the union of them is an interval.

	== Solving Inequalities ==		== Solving Inequalities ==

2025-09-24T16:37:01Z

Intervals

← Older revision		Revision as of 16:37, 24 September 2025
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	\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>		\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>
	\|}		\|}
			The interval <math>[a,a]</math>is equal to <math>[a]</math>

			The interval <math>[b,a] where b > a is empty</math>

	== Solving Inequalities ==		== Solving Inequalities ==

2025-09-24T16:34:52Z

Intervals

← Older revision		Revision as of 16:34, 24 September 2025
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	\|<math>x \in[a,+\infty]</math><br><math>x \in]a,+\infty[</math><br><math>x \in]-\infty,a]</math><br><math>x \in]-\infty,a[</math>		\|<math>x \in[a,+\infty]</math><br><math>x \in]a,+\infty[</math><br><math>x \in]-\infty,a]</math><br><math>x \in]-\infty,a[</math>
	\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>		\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>
	\|-
	\|
	\|
	\|
	\|}		\|}

2025-09-24T16:34:07Z

Intervals

← Older revision		Revision as of 16:34, 24 September 2025
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	\|-		\|-
	\|Open		\|Open
	\|<math>x \in[a,+\infty]</math>		\|<math>x \in[a,+\infty]</math><br><math>x \in]a,+\infty[</math><br><math>x \in]-\infty,a]</math><br><math>x \in]-\infty,a[</math>
	<br><math>x \in]a,+\infty[</math>		\|<math> x \geq a</math><br><math>x > a</math><br><math>x \leq a</math><br><math>x < a</math>
	<br><math>x \in]-\infty,a]</math>
	<br><math>x \in]-\infty,a[</math>
	\|<math>~~a \leq~~ x \~~leq b~~</math><br><math>a ~~\leq x < b~~</math><br><math>~~a <~~ x \leq b</math><br><math>~~a <~~ x < b</math>
	\|-		\|-
	\|		\|

2025-09-24T16:32:30Z

Intervals

← Older revision		Revision as of 16:32, 24 September 2025
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	\|-		\|-
	\|Open		\|Open
	\|<math>x \in[a,+\infty]</math><br><math>x \in[a,b[</math><br><math>x \in]a,b]</math><br><math>x \in]a,b[</math>		\|<math>x \in[a,+\infty]</math>
			<br><math>x \in]a,+\infty[</math>
			<br><math>x \in]-\infty,a]</math>
			<br><math>x \in]-\infty,a[</math>
	\|<math>a \leq x \leq b</math><br><math>a \leq x < b</math><br><math>a < x \leq b</math><br><math>a < x < b</math>		\|<math>a \leq x \leq b</math><br><math>a \leq x < b</math><br><math>a < x \leq b</math><br><math>a < x < b</math>
	\|-		\|-

2025-09-24T16:30:28Z

Intervals

← Older revision		Revision as of 16:30, 24 September 2025
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	\|-		\|-
	\|Open		\|Open
	\|		\|<math>x \in[a,+\infty]</math><br><math>x \in[a,b[</math><br><math>x \in]a,b]</math><br><math>x \in]a,b[</math>
	\|		\|<math>a \leq x \leq b</math><br><math>a \leq x < b</math><br><math>a < x \leq b</math><br><math>a < x < b</math>
	\|-		\|-
	\|		\|