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### Number Systems

Lesson 2 Contents

7N1 PLUS

The Real Number System (rational and irrational)

7N1 Distinguish between the various subsets of real numbers (counting/natural numbers, whole numbers, integers, rational numbers, and irrational numbers)

7N2 Recognize the difference between rational and irrational numbers (e.g., explore different approximations of π)

7N3 Place rational and irrational numbers (approximations) on a number line and justify the placement of the numbers

7N17 Classify irrational numbers as non-repeating/non-terminating decimals

Rational (fraction)

Integer

Irrational

Whole

Natural

Natural – counting numbers beginning with 1, 2, 3, …

Whole – counting numbers beginning with 0, 1, 2, 3, …

Integer – All the whole numbers and their opposites {…, –3, –2, –1, 0, 1, 2, 3, …}

Rational – all numbers that can be written as a ratio (fraction)

Irrational – a decimal that never ends and never repeats

Real– all numbers that can be put on a number line

The Real Number System (rational and irrational)

Example 2-1aName all of the sets of numbers to which the real number 17 belongs.

Answer: This number is a natural number, a whole number, an integer, and a rational number.

Name all of the sets of numbers to which the real number belongs.

Answer: Since , this number is an integer and a rational number.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-1aName all of the sets of numbers to which the real number belongs.

Answer: Since , this number is a natural number, a whole number, an integer, and a rational number.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-1aName all of the sets of numbers to which the real number belongs.

Answer: This repeating decimal is a rational number because it is equivalent to .

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-1aName all of the sets of numbers to which the real number belongs.

Answer: It is not the square root of a perfect square so it is irrational.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-1aName all of the sets of numbers to which each real number belongs.

a. 31

b.

c. 4.375

d.

e.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-1bAnswer: natural number, whole number, integer, rational number

Answer: integer, rational number

Answer: rational number

Answer: natural number, whole number, integer, rational number

Answer: irrational number

Replace with <, >, or = to make a true statement.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-2aExpress each number as a decimal. Then graph the numbers.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-2aExpress each number as a decimal. Then compare the decimals.

From least to greatest, the orderis

Answer:

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-2aa. Replace with <, >, or = to make a true statement.

b. Order from least to greatest.

Answer:

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-2bAnswer: >

Solve . Round to the nearest tenth, if necessary.

Write the equation.

Take the square root of each side.

Find the positive and negative square root.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-3aAnswer: The solutions are 13 and –13.

Solve . Round to the nearest tenth, if necessary.

Write the equation.

Take the square root of each side.

Find the positive and negative square root.

Use a calculator.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-3aAnswer: The solutions are 7.1 and –7.1.

Solve each equation. Round to the nearest tenth, if necessary.

a.

b.

7N1 PLUS

The Real Number System (rational and irrational)

Example 2-3bAnswer: 9 and –9

Answer: 4.9 and –4.9

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