[br]So far, you've been studying sets in a general context. In this lesson, you'll learn about the [color=#0000ff]set of real numbers[/color] and its subsets.[br][br]Below is a [color=#0000ff]tree diagram[/color] of the [color=#0000ff]set of real numbers[/color]. Let's study the real number tree from the roots.[br][br]At the root of the real number tree, we find the set of [color=#0000ff]positive integers[/color], also known as [color=#0000ff]natural numbers[/color] or [color=#0000ff]counting numbers[/color]. Beside it is another root containing a single element—[color=#0000ff]zero[/color]. These two sets together form the set of [color=#0000ff]whole numbers[/color] or [color=#0000ff]nonnegative integers[/color]. If we add the set of [color=#0000ff]negative integers[/color] to the set of [color=#0000ff]whole numbers[/color], we get the set of [color=#0000ff]integers[/color].[br][br]The set of [color=#0000ff]integers[/color] is [color=#0000ff]discrete[/color], meaning each number in the set is individually separate and distinct. However, between any two integers is an infinite number of numbers, some of which may be [color=#0000ff]fractions.[/color] [color=#0000ff]Fractions[/color] may be written as a [color=#0000ff]ratio[/color] of two [color=#0000ff]integers[/color] (say [b][sub][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAiCAYAAACEEVOfAAAA40lEQVRIDe2UsQ1CMQxEjwoaREVFwwJ0lJT0dNTU9MxAwwIswAJMwAZswCigJ3wS+voJ/hUgYclK4uTi5OKL9JM2lHSTtMycvifpIOkePn0HAsBiMhyzIDadxc7jLiCfZvIHPanoxB5v1Zd0CvbOMSZetEssdkW4vRYR3z2xl1TzVdvxR5JqPmgDEaPKTbdlsigtJo7gTDH9eWPcirUckLptE8CdA5mWP4LsnUBUAiBqMWXOwkeTMt8vXXMQYbpTGfyNAarK4XU3Pyyasq1r7PkeZEF8aMvsFSnfxl0ANZ25D9gDNdZD8kGoAzgAAAAASUVORK5CYII=[/img] [/sub][/b]or [sub][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAiCAYAAACEEVOfAAAA/UlEQVRIDe2UsQ0CMQxFHzQgJERFRUNFR0dJSU/HEPT0dDQswAIswARswAaMwAigL/KlCCW5XAcSlqK78/0fO9924CdtCNyBVU32HeAIPMOaNpFEEFgRTrUkbToPO4/bkJzN5E96S9FKPdWqB5yDepfwLX/WrgHsjvDzlmV8948DUFrrVPojoLT6KZJ86nLJHKu2yIHl18AJ/ADU5QKbrGInzeOg4tp2gbi1o+Zp0rIGLIzTVYrFNhJYu+tcPk+3Jsrmg6iLpjFSvLG7vZUQnivVLmmzMEfxT5dB50zaPhzegAGgOZIg2ZtWkQyyclLRF2gyUuyUzFVSx6Ti+wsCP0NzM8e3DAAAAABJRU5ErkJggg==[/img][/sub]) or as [color=#0000ff]terminating[/color] or [color=#0000ff]repeating[/color] [color=#0000ff]decimals [/color](0.5 for [sub][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAiCAYAAACEEVOfAAAA40lEQVRIDe2UsQ1CMQxEjwoaREVFwwJ0lJT0dNTU9MxAwwIswAJMwAZswCigJ3wS+voJ/hUgYclK4uTi5OKL9JM2lHSTtMycvifpIOkePn0HAsBiMhyzIDadxc7jLiCfZvIHPanoxB5v1Zd0CvbOMSZetEssdkW4vRYR3z2xl1TzVdvxR5JqPmgDEaPKTbdlsigtJo7gTDH9eWPcirUckLptE8CdA5mWP4LsnUBUAiBqMWXOwkeTMt8vXXMQYbpTGfyNAarK4XU3Pyyasq1r7PkeZEF8aMvsFSnfxl0ANZ25D9gDNdZD8kGoAzgAAAAASUVORK5CYII=[/img] [/sub]or [sub][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAATCAYAAAB7u5a2AAABL0lEQVQ4EeWTMWoCQBBFvyEgGFKksJA0VrbxACltbVOnE9J7kJS5gPfwBt7AI+QIypP5YczOaiBVyIdhYefPn52ZHem/YS5p1bHJb5sxlvTUsbue+IukQ9he0n2HOJL0Jukz8T8k3XT4JzLCr5IeU9CwCLAo3AdJm+DvCq4GhXMRd+siABFeajiexzWvn4UQpRq8GDLtuYYs3nARRYgkhgNowTVQBfHMrMEl8bLUGPY2ROFU7TslqsRx0JKeOD8GQQ8THnNqYPFp8jAYAn7Sc/49XIx2nmEZDpIY7nn5vUxKJzzESXQGlgUHPTSogruc0D5akYfPvcWbr5idz1GaF+XWikmApBgLBLwT3SppQ55+Dg6Nr3V//7b6cPNSmd+clFWWFhXlSnhQj9sI/52LI0e8WQggqbF+AAAAAElFTkSuQmCC[/img] [/sub]for [sub][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAiCAYAAACEEVOfAAAA/UlEQVRIDe2UsQ0CMQxFHzQgJERFRUNFR0dJSU/HEPT0dDQswAIswARswAaMwAigL/KlCCW5XAcSlqK78/0fO9924CdtCNyBVU32HeAIPMOaNpFEEFgRTrUkbToPO4/bkJzN5E96S9FKPdWqB5yDepfwLX/WrgHsjvDzlmV8948DUFrrVPojoLT6KZJ86nLJHKu2yIHl18AJ/ADU5QKbrGInzeOg4tp2gbi1o+Zp0rIGLIzTVYrFNhJYu+tcPk+3Jsrmg6iLpjFSvLG7vZUQnivVLmmzMEfxT5dB50zaPhzegAGgOZIg2ZtWkQyyclLRF2gyUuyUzFVSx6Ti+wsCP0NzM8e3DAAAAABJRU5ErkJggg==[/img][/sub]; the bar over 3 means the digit repeats indefinitely). Together, [color=#0000ff]integers[/color] and [color=#0000ff]fractions[/color] make up the set of [color=#0000ff]rational numbers[/color], whose main characteristic is that they can be written as a [color=#0000ff]ratio[/color] of two [color=#0000ff]integers[/color]. It's easy to see that [sub][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAiCAYAAACEEVOfAAAA/UlEQVRIDe2UsQ0CMQxFHzQgJERFRUNFR0dJSU/HEPT0dDQswAIswARswAaMwAigL/KlCCW5XAcSlqK78/0fO9924CdtCNyBVU32HeAIPMOaNpFEEFgRTrUkbToPO4/bkJzN5E96S9FKPdWqB5yDepfwLX/WrgHsjvDzlmV8948DUFrrVPojoLT6KZJ86nLJHKu2yIHl18AJ/ADU5QKbrGInzeOg4tp2gbi1o+Zp0rIGLIzTVYrFNhJYu+tcPk+3Jsrmg6iLpjFSvLG7vZUQnivVLmmzMEfxT5dB50zaPhzegAGgOZIg2ZtWkQyyclLRF2gyUuyUzFVSx6Ti+wsCP0NzM8e3DAAAAABJRU5ErkJggg==[/img][/sub] is a [color=#0000ff]fraction[/color], but how about 0 or 1? Well, 0 and 1 can be rewritten as [sub][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAiCAYAAACEEVOfAAABC0lEQVRIDd2UMQ5BQRCGfyJRIQqVRqVzA6VWq1brtVqNC7iACziBG7iBIzgC+fJmkvHs20dCwSSbtzs7//yz/9tZ6edsIelm4yKpU3eClQUvJQ0DuF0FbFjQOQTMzLcOvofp2AJgc4OBUikzaV4aYDdnv7qj/M2BYGuWAaxTIPyUVgsahYxkz55pnhDCzxQVDTmLn0jWU/DCii8qGraLKRkJmkqCBdVYt54ig4NAmAj00Q/72SkCJCXOor66uZWUG5sUe09SbnRToOij8bg+tEatIfkuyB2vVBLsVwaGvQFrQWSaWLrBOyAvwd+Hl5j+GfSWevwr3rqDSX60Nf5KKzegN2LlG1GZ6SMbd6eXSROfxNWfAAAAAElFTkSuQmCC[/img] [/sub]and [sub][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAiCAYAAACEEVOfAAAAgElEQVRIDWNgGJKAl4GB4T4DA4MLMa5nZGBg6GFgYPgPxQqENIE0gBSDbJhJrCaQobpQk0VJ0QRzjfSoJkhQkBR6oLhiZ2BgWAYNvc1QPkgcJzgIVQxLETD6Ak4dg1uinYGBAR/2x+Z8fgYGBnyYA5smZLHRTAgNjeGYc0dcJgQAfCA3QmZNhgMAAAAASUVORK5CYII=[/img][/sub], respectively, so they also satisfy the definition of [color=#0000ff]rational numbers[/color].[br][br]Note, however, that not all numbers between two [color=#0000ff]integers[/color] are [color=#0000ff]rational[/color]; some are [color=#0000ff]irrational numbers[/color]. The exact value of [color=#0000ff]irrational numbers[/color] may be shown as [color=#0000ff]radicals[/color] if they are [color=#0000ff]real algebraic[/color] [sub][img]data:image/png;base64,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[/img] [/sub]or as [color=#0000ff]symbols[/color] if they are [color=#0000ff]transcendental[/color], such as [color=#0000ff]π[/color] and [color=#0000ff]e[/color]. Oftentimes, however, an estimate of these numbers up to the desired number of decimal places is used instead of the exact value since these numbers are nonterminating, nonrepeating decimals.[br][br]At the crown of the tree is the set of [color=#0000ff]real numbers[/color], which overarches the set of [color=#0000ff]rational numbers[/color] and the set of [color=#0000ff]irrational numbers[/color].

Another way of visualizing the set of real numbers is by means of a [color=#0000ff]Venn diagram[/color]. [br][br]Below is a [color=#0000ff]Venn diagram[/color] of the set of real numbers. Using set notation, [color=#0000ff]N [size=150]⊂[/size] (W) [size=150]⊂[/size] Z [size=150]⊂[/size] Q [size=150]⊂[/size] A[sub]R[/sub] [size=150]⊂[/size] R[/color].[br][br][color=#0000ff]NOTE:[/color] There is some disagreement among authors and texts in the treatment of the number zero. In the tree diagram above, zero is part of the set of whole numbers, but not of the set of natural numbers. In the Venn diagram below, zero is included in the set of natural numbers, which explains why there is no circle for the set of whole numbers. In this lesson series, we treat zero as a whole number, but not as a natural number. Hence, in the notation above, we have introduced the set of whole numbers, W, to include the set of natural numbers, N, and the unit set containing zero.

Below is a set of problems involving the set of real numbers and its subsets.

Check the Answer Box below for the correct answers.

In succeeding lessons, you will explore the set of real numbers some more. Did you have FUN in today's lesson?