Chapter 2.1 Rational Numbers and Chapter 2.2 Adding and Subtracting Rational Numbers

Concept Summary

Real numbers can be classified according to their characteristics. We can use a number line to show these sets of numbers. Natural numbers are the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero: 0, 1, 2, 3, … Integers are the whole numbers and their opposites: – 3, – 2, – 1, 0, 1, 2, 3, …

Rational numbers are numbers that can be expressed in the form, where a and b are both integers and b ≠ 0. When expressed as a decimal, a rational number is either a terminating decimal or a repeating decimal. A terminating decimal has a finite number of digits after the decimal point (for example, 1.25, 2.75, and 4.0). A repeating decimal has a block of one or more digits after the decimal point that repeat continuously (where all digits are not zeros). Examples of rational numbers -

Identify the coordinates of the points graphed on the number line. {-4, 0, 2.5}

Graph the set of numbers on the number line. {integers less than -4 or greater than or equal to 5}

The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|. 5 units |5| = 5|–5| = 5

The set of all numbers that can be represented on a number line are called real numbers. You can use a number line to model addition and subtraction of real numbers. Addition To model addition of a positive number, move right. To model addition of a negative number, move left. Subtraction To model subtraction of a positive number, move left. To model subtraction of a negative number, move right.

Add or subtract using a number line. –3 + 7 Start at 0. Move left to –3. To add 7, move right 7 units –3 +7 –3 + 7 = 4

Add or subtract using a number line. Start at 0. Move left to – (–7) –4 + (–7) = –11 To add –7, move left 7 units. –4 –4 + (–7)

Add or subtract using a number line. Start at 0. Move right to 3. To subtract –6, move right 6 units – (–6) = 9 3 – (–6) – ( – 6)

Add or subtract using a number line. –3 – 7 Start at 0. Move left to –3. To subtract 7, move left 7 units. –3–3 –7– –3 – 7 = –10

Add or subtract using a number line. –5 – (–6.5) Start at 0. Move left to –5. To subtract –6.5, move right 6.5 units –5 –5 – (–6.5) = – ( – 6.5)

Addition of Rational Numbers  To add rational numbers with the same sign, add their absolute values. The sum has the same sign as the addends. For example (3 + 5) and ( )  To add rational numbers with different signs, subtract the lesser absolute value from the greater absolute value. The sum has the same sign as the number with the greater absolute value. For example (-2 + 5) and (-6 + 1)

Two numbers are opposites if their sum is 0. A number and its opposite are additive inverses and are the same distance from zero. They have the same absolute value.

To subtract signed numbers, you can use additive inverses. Subtracting a number is the same as adding the opposite of the number.

Subtract. 5 – ( – 4) 5 − (–4) = To subtract –4, add 4. Same signs: add absolute values. (5 + 4 = 9) Both numbers are positive, so the sum is positive.