NUMBER SYSTEMS REAL NUMBERS as opposed to fake

NUMBER SYSTEMS

REAL NUMBERS (as opposed to fake numbers? )

Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers

Branching Real Numbers Rational Irrational Now let’s break these subsets down further!

Rational Numbers A rational number… • is a real number that can be written as a ratio (fraction) of two integers. • written in decimal form it is either terminating (ends) or repeating (same pattern of numbers).

Examples of Rational Numbers • 16 • 1/2 • 3. 56 • -8 • 1. 3333… • - 3/4 Can you write 16 as a ratio? Now try writing -8 as a ratio!

Irrational Numbers • An irrational number • is a number that cannot be written as a ratio (fraction) of two integers. • written as decimals are nonterminating (never end!) and nonrepeating (no pattern).

Examples of Irrational Numbers • Square roots of non-perfect “squares” • Pi What is the square root of 17? What is another non-perfect square?

Integers One of the subsets of rational numbers So what do integers branch off of?

Integers • Integers are rational numbers because they can be written as fraction with 1 as the denominator. *Remember* Rational numbers can be written as a ratio (or in other words, a fraction)

What are integers? • Integers are the whole numbers and their opposites. • Examples of integers are 6 -12 0 186 -934 What do you notice about these integers?

Whole Numbers One of the subsets of rational numbers and integers So what do whole numbers branch off of?

Whole Numbers • All positive numbers that are counting numbers plus zero. • Examples: 0, 1, 2, 3, 4, 5 …

Natural Numbers One of the subsets of rational numbers, integers and whole numbers So what do natural numbers branch off of?

Natural Numbers • All of your counting numbers • What numbers do you count with?

Classify the following Numbers • What subset(s) do the following numbers belong to? Remember they could belong to more than one! • 0 • Whole, Integers, Rational, and Real

Classify the following Numbers (Con’t) • 4 • Natural, Whole, Integer, Rational, Real • -9 • Integer, Rational, Real • Π • Irrational, Real

Classify the following Numbers (Con’t) • 3/4 or 0. 75 • Rational and Real • • Natural, Whole, Integer, Rational, Real • • Irrational, Real

Imaginary Number • DEFINITION: • Imaginary numbers are square roots of negative numbers.

IMPORTANT: • They are all real numbers multiplied by i= (the square root of -1) EXAMPLE: • 4 i is an imaginary number and its square is -16.

Complex Numbers • DEFINITION: • A complex number is a number that can be written in the form • a+bi, a and b are real numbers and i is the imaginary unit. A complex number is a combination of a real number and an imaginary number.

EXAMPLE: (a+bi) + (c+di) = (a+c) + (b+d)I (3+2 i)+(1+7 i)=(4+9 i)