The Rel Number Line nd Types of Rel Numbers Most people s first understnding of numbers reltes to wht we mth-type people cll counting numbers or more formlly nturl numbers. These re the numbers we use to count 1 or more whole occurrences of similr objects. We lso use these numbers to keep our young-uns busy when they ve sung one too mny renditions of the lphbet song. Pretty soon fter this, we generlly form n understnding of the number zero. When we dd the number zero to the set of nturl numbers we crete set of numbers we cll the whole numbers. Number with hole in it. Nturl Numbers 0, 1, 2,,, 5, 6, 7, 8, 9, 10,, Whole Numbers The next type of numbers we generlly come to understnd re frctions, but we ll put these on hold for now. One dy in your pst, someone rocked your world with the ide of negtive numbers. At first these seem very strnge how cn you hve negtive number of chickens? The nswer, of course, is tht you cn t. So negtive numbers don t mke sense when counting chickens tht s why they re not prt of the set we cll counting numbers. You cn, however, be bove or below se level. If we cll se level n elevtion of zero, one esy wy to distinguish points bove se level from points below se level is to ssign positive vlues to points bove se level nd negtive vlues to points below se level. If we re mesuring in feet, for exmple, 7 would correspond to point tht is 7 feet bove se level while 7 would correspond to point tht is 7 feet below se level. The numbers 7 nd 7 re clled opposite numbers becuse they re equidistnt from zero nd hve opposite signs. One esy wy to illustrte this is with number line. 7 units 7 units negtive numbers positive numbers zero point Exmple 1 Complete ech sentence correctly. The opposite of 12 is. The opposite of 0 is. The Rel Numbers 1
The set of numbers consisting of the nturl numbers, 0, nd the opposites of the nturl numbers is clled the set of integers. OK bck to frctions. Generlly, when we think of frctions we think of one positive integer divided by nother positive integer. We represent hlf of pie with the frction 1 2. We represent 2 out of voters with the frction 2. Well, just s nturl numbers hve opposites, so do positive frctions. For exmple, 17 nd 17 re opposite numbers. 17 17 1 units 1 units Any number tht cn be written s quotient of integers is clled rtionl number. Remember tht frction cnnot hve denomintor of zero! Exmple Show tht the numbers 0, 8., 25, nd 0. re ll rtionl numbers. Every point on the rel number line corresponds to unique rel number nd every rel number corresponds to unique point on the rel number line. Every rel number is either rtionl number or n irrtionl number. 2 T he Rel Numbers
Exmple Plce ech of the following numbers into the sets (boxes) to which they belong. Note tht most of the numbers fit into more thn one set. 15 28 π, 16.812,, 1, 5, 25,, 0,, 9.221257 2 7 Rel numbers Rtionl numbers Irrtionl numbers Integers Whole numbers Nturl numbers (Counting numbers) The Rel Numbers
Exmple 5 Decide whether ech sttement is true or flse. If the sttement is flse, give n exmple tht shows why the sttement is flse. Every integer is nturl number. Every whole number is rtionl number. Some nturl numbers re irrtionl numbers. Equlity nd Inequlities As previous mentioned, every rel number corresponds to unique point on the rel number line. When plotting two different numbers on trditionl number line, one of the two numbers lies to the left of the other number. The number on the left is clled the lesser of the two numbers nd the number on the right is clled the greter of the two number. < b mens tht lies to the left of b long the rel number line. The expression < b is red loud s is less thn b. > b mens tht lies to the right of b long the rel number line. The expression > b is red loud s is greter thn b. = b mens tht nd b re in fct the sme point long the rel number line. The expression = b is red loud s equls b. b mens tht < b or = b. The expression b to b. b mens tht > b or = b. The expression b equl to b. is red loud s is less thn or equl is red loud s is greter thn or < b, > b, b, nd b re clled inequlities (or inequlity sttements). = b is clled n eqution. T he Rel Numbers
Exmple 6 Indicte, roughly, the position of ech of the following numbers on the provided number line fter first deciding on n pproprite scle nd then nswer the subsequent questions using proper mthemticl symbols nd nottion. 8,, π,,,,.1 (note tht through 5 digits π.116 ) 2 pproximtely equls Which is the true sttement: is less thn or is less thn? Which is the true sttement: is less thn or is less thn? Which is the true sttement: π =.1, π >.1, or π <.1? Which is the true sttement: π =.1, π >.1, or π <.1? Which is true: = π, π, or π? Which is true: 8 2 =, 8 2, or 8 2? The Rel Numbers 5
Absolute Vlue The symbols n re red loud s the bsolute vlue of n. The bsolute vlue of the rel number n is the distnce between tht number nd the number zero long the rel number line. Fct 1: The bsolute vlue of number is never negtive. Fct 2: Opposite numbers hve equl bsolute vlues. Exmple 8 For ech given pir, decide which is the true sttement. 9 = 9 or 9 = 9 9 = 9 or 9 = 9 9 = 9 or 9 = 9 9 = 9 or 9 = 9 Exmple 9 Without using your clcultor, determine which of the following is true. 5 < 2. or 5 = 2. or 5 > 2. 6 T he Rel Numbers