Rational Numbers & Equations Pages 227 – 246 Chapter 8 Rational Numbers & Equations Pages 227 – 246

The Number Line The numbers can be assigned to points on a line. The numbers are spaced according to their distance from 0. The set of integers = the counting numbers, their opposites, and 0. An integer and its opposite are the same distance from 0.

Absolute Value & Adding Integers The absolute value of a number is its distance from 0 on the number line. The absolute value of 5 is 5. The absolute value of -5 is 5.

Adding More Than Two Integers To add more than two integers, first rearrange the numbers so that the positive numbers come first. Then work from the left. Combine the positive integers two at a time. Then combine the negative integers two at a time. Then add the resulting positive and negative integers to find the final answer.

Adding Rational Numbers A rational number is a number that can be written as a ratio, or a division, of two integers. Rational number include: The integers such as 15 & -26 Fractions & mixed numbers such as ¾ & -7/8; 6 ½ & -7 3/5 Decimals such as 2.25 & -5.5 Rational numbers can be added using the same rules that are used for adding integers. If you have trouble, review the adding of decimals and fractions in Chapter 2 & 3.

Subtracting Integers An addition sign (+) or a subtraction sign (-) can be the sign of an operation or the sign of a number. As the sign of an operation, + means add and – means subtract. As the sign of a number, + means a positive number and – means a negative number. Any subtraction problem can be changed to addition by changing the sight of the operation and the sign of the number being subtracted. The following pattern shows why it makes sense to change signs and add when subtracting a negative number: 10 – (-2) = 12 Therefore 10 + 2 = 12

Subtracting Rational Numbers Rational numbers follow the same rules as integers. You can change the operation and the sign of the number in order to use addition.

Adding & Subtracting Rational Numbers: Mixed Applications The homework can be done by adding & subtracting rational numbers. A negative answer in money means someone owes money or too much money has been taken out of a bank account. A negative temperature means below 0. A negative time means before something is to happen, such as the launching of a spacecraft.

Multiplication of Integers A positive (+) sign can be the sign of a number or the sign of an operation. Similarly a negative (-) sign can be the sign of a number or an operation. When you are doing multiplication, the sign always is the sign of a number. When no sign is written, the number is positive. The product of two positive numbers is positive: 8 x 7 = 56 The product of a positive and a negative number is negative: 8 x (-7) = -56 The product of two negative numbers is positive: (-8) x (-7) = 56

Division of Integers Division follows the same rules as multiplication positive ÷ positive = positive negative ÷ negative = positive positive ÷ negative = negative negative ÷ positive = negative

Multiplication & Division of Rational Numbers Positive and negative fractions, decimals and other rational numbers can be multiplied and divided. Do the operations as you would for positive numbers. The sign follows the rules for integers +/+ = + -/- = + +/- = - -/+ = - The same rules are true of division