Rational NumbersNumbers that can be expressed as a quotient of2 integers a/b (where a and b are integers and b is NOT 0) Integers Whole numbers, their opposites and zero. Whole Numbers Zero and natural numbers Natural Numbers The set of counting numbers 1, 2, 3, 4, 5, ...
REMEMBER… INTEGERS…The counting numbers… 1,2,3,4…Their opposites… -1,-2,-3,-4…And the Zero…0Numbers that are at the same distance from 0 areOPPOSITES (page 408)So we have NEGATIVE INTEGERS, POSITIVEINTEGERS AND ZERO (neither negative or positive)
8-2 COMPARING AND ORDERING INTEGERS• Numbers to the right of 0 on a number line are POSITIVE• Numbers to the left of 0 on a number line are NEGATIVE Always the numbers to the right have greater value…• When COMPARING Integers on a number line, the Integer that is farther to the RIGHT is allways greater (examples A and B page 410)
8-3 UNDERSTANDING RATIONAL NUMBERS• Numbers expressed as fractions and decimals.• Just as there are positive and negative integers, there are also positive and negative fractions and decimals.• The greater the magnitude of a negative number, the less its value (because is farther to the left from 0).
How can we order and compare Rational Numbers?• Sometimes it helps to write fractions as decimals.• SOME TIPS…1. If you have Rational Numbers as fractionswith different whole numbers… you cancompare the whole numbers and depending onwhich one is farther to the left you candetermine which rational number is less
more TIPS…2. If you have Rational Numbers as fractions withthe same whole numbers… you can transformthem into decimals. This way is going to be moreeasy to locate the decimals on the number line orjust to compare the decimals.3. If you have Rational Numbers as fractions andothers as decimals… you better transform all thefractions into decimals and then you can compare.
RULES FOR ADDING INTEGERS When Adding Two Integers with the Same sign1. ADD the two numbers.2. Give the answer the same sign. SIGNS WHAT TO DO 4 18 = 22 ADD THE NUMBERS THE SIGN OF THE ANSWER IS -4 ( - 18 ) = - 22 ADD THE NUMBERS THE SIGN OF THE ANSWER IS See example E page 419
RULES FOR ADDING INTEGERS When Adding Two Integers with Different Sign1. SUBTRACT the two numbers.2. Give the answer the sign of the greater number. SIGNS WHAT TO DO SUBTRACT THE NUMBERS THE SIGN OF THE ANSWER IS 17 ( - 29 ) = - 12 OR DEPENDING OF THE SIGN OF THE ADDEND WITH THE GREATER ABSOLUTE VALUE. See example F page 419
IMPORTANTIt is important to understand the difference between a minus sign and a negative sign. They look the same, but one is anoperation between two numbers indicating subtraction and the other tells you that a number is negative.
EXAMPLESNUMBER OR EXPRESSION HOW TO READ IT-7 NEGATIVE SEVEN- ( - 6) THE OPPOSITE OF NEGATIVE SIX3-4 3 MINUS 43–(-4) 3 MINUS NEGATIVE 4-6- 7 NEGATIVE 6 MINUS 7-6–(-7) NEGATIVE 6 MINUS NEGATIVE 7
RULE FOR SUBTRACTING INTEGERSGuess What?Subtracting two Integers, is the same as Adding the opposite of the second number, to the first number!
SO FOLLOW THESE STEPS! Example: – 10 – ( – 5 ) =1. Transform the subtraction into an addition. (Change the subtraction sign to an addition sign). – 10 + ( – 5 ) =2. Change the sign of the second number. – 10 + ( 5 ) =3. Solve the operation following the rules of Adding Integers. (This is why it is said that subtracting … is the same as adding!) – 10 + ( 5 ) = – 5