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Lesson4 integers


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A lesson about integer numbers

A lesson about integer numbers

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  • 1. Lesson 4 INTEGERSLESSON 4 INTEGERS1. POSITIVE AND NEGATIVE NUMBERSWe can use negative numbers in different situations:·Positions:A submarine which is sailing 700 m below sea level: – 700 mThe second floor of a subterranean garage: – 2·Money:He is €400 overdrawn: – 400·Temperatures:The temperature in Siberia has risen 25ºC below zero: -25ºCNegative numbers are less than zero.Use a number line to order negative numbers. For instance, it is easy to see that -2 is ahigher number than -5 because it is further to the right on the number line.Negative numbers are always written with a ‘-’ sign in front of them and they arecounted from zero to the left. We read them as “negative” or “minus”, i.e., -5 is read as“negative five” or “minus five”.2. THE SET OF INTEGERSIf we include the negative numbers with the whole numbers, we have a new set ofnumbers that are called integers.Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}The Integers include zero, the counting numbers, and the negative of the countingnumbers, to make a list of numbers that stretch in either direction infinitely.1/6 IES UNIVERSIDAD LABORAL Mercedes López
  • 2. Lesson 4 INTEGERSAbsolute Value means only how far a number is from zero:“6″ is 6 away from zero, and “-6″ is also 6 away from zero. So the absolute value of 6 is6, and the absolute value of -6 is also 6.So in practice “absolute value” means to remove any negative sign in front of a number,and to think of all numbers as positive (or zero).To show that you want the absolute value of something, you put “|” marks either side(they are called “bars”), like these examples:|-5| = 5 |7| = 7Opposite. For every positive integer, there’s a negative integer an equal distance fromthe origin. Two integers that lie the same distance from the origin in opposite directionsare called opposites. For example, “negative 5″ is the opposite of “positive 5.”Comparing numbers. Writing numbers down on a Number Line makes it easy to tellwhich numbers are bigger or smaller. • Between a positive number and a negative number, the positive number is always larger:5 > -2 1 > -3 2 > -100 • Between two positive numbers, the larger number is the one with the largest absolute value:7>2 10 > 4 300 > 200 • Between two negative numbers, the larger number is the one with the smallest absolute value:-7 < -2 -10 < -4 -300 < -200Example: John owes $3, Virginia owes $5 but Alex doesn’t owe anything, in fact he has$3 in his pocket. Place these people on the number line to find who is poorest and whois richest.Having money in your pocket is positive. But owing money is negative. So John has “-3″,Virginia “-5″ and Alex “+3″.2/6 IES UNIVERSIDAD LABORAL Mercedes López
  • 3. Lesson 4 INTEGERSNow it is easy to see that Virginia is poorer than John (-5 is less than -3) and John ispoorer than Alex (-3 is smaller than 3), and Alex is, of course, the richest!3. ADDING AND SUBTRACTING INTEGERSUsing the Number LineWhen we add, we move right on the number line.When we subtract, we move left on the number lineExample:To evaluate -3+2 , we start at -3 and move 2 places to the rightThe answer is -1.To evaluate -3-2 , we start at -3 and move 2 places to the leftThe answer is -5.More examples.1Using RulesRule 1 :The sum of two or more positive integers is a positive integer.The sum of two or more negative integers is a negative integer.Example:(+7) + (+ 4) = + 11(-8) + (– 9) = – 17Rule 2 :To find the sum of a positive and a negative integer:Subtract the two numbers (ignore the signs) and then keep the sign of the larger integer.(– 9) + (+ 7) = – 2(– 3) + (+ 5) = + 21 IES UNIVERSIDAD LABORAL Mercedes López
  • 4. Lesson 4 INTEGERS4. ADDITIONS AND SUBTRACTIONS WITH BRACKETSRemember: If a number has no sign it usually means that it is a positive number.Example: 5 is really +5Adding positive numbers is just simple addition. Example: 2 + 3 = 5 is really saying“Positive 2 plus Positive 3 equals Positive 5″You could write it as (+2) + (+3) = (+5)Subtracting positive numbers is just simple subtraction.Example: 6 – 3 = 3 is reallysaying“Positive 6 minus Positive 3 equals Positive 3″You could write it as (+6) – (+3) = (+3). Subtracting a negative is the same as adding. Example: 6-(-3) = 6+3 = 9. Subtracting a positive or Adding a negative is subtraction 15 - (+3) = 12 15 + (-3) = 12Try playing Casey Runner2, you need to know the rules of positive and negative tosucceed!5. MULTIPLICATION AND DIVISIONTo multiply two or more integers we have to multiply the signs and then the absolutevalue of the numbers. The rule for multiplying and dividing the signs is very similar tothe rule for adding and subtracting. When the signs are different the answer is negativeand when the signs are the same the answer is positive:2 IES UNIVERSIDAD LABORAL Mercedes López
  • 5. Lesson 4 INTEGERSNow try a Test Bite36. POWERS AND ROOTSThe sign of a power depends on the base and the exponent.If the base is positive, the power will be positive, as you learnt in lesson 2.If the base is negative, and the exponent is even, the power will be positive: (–3)4 = (–3)(–3)(–3)(–3) = (9)(9) = 81 and the exponent is odd, the power will be negative: (–3)5 = (–3)(–3)(–3)(–3)(–3) = (9)(9)(–3) = –243Be careful about grouping. To avoid confusion, use brackets () in cases like this:With () : (-2)2 = (-2) × (-2) = 4Without () : -22 = -(22) = – (2 × 2) = -4The rules for square roots are as follows:a positive integer has two square roots, one is positive and the other one is negative (itsopposite):a negative integer has no square roots.Combined operations.When expressions have more than one operation, we have to follow rules for the orderof operations. These are the same rules as for natural numbers:Rule 1: First perform any calculations inside the brackets.Rule 2: Do powers and roots.Rule 3: Next perform all multiplications and divisions, working from left to right.Rule 4: Lastly, perform all additions and subtractions.To finish, try this exercises.43 IES UNIVERSIDAD LABORAL Mercedes López
  • 6. Lesson 4 INTEGERSGLOSSARYLook for the following words in this dictionary5 and copy them in your notebook.SignAbsolute valueOppositeEvenOdd5 IES UNIVERSIDAD LABORAL Mercedes López