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1-A8 Warm Up 1. -15 – 6 – [-4 – (-6)] 3. a2 + b – c when a = -6, b = 5, c = -3 Take your test home tonight and get a parent signature. Return the test by tomorrow to get 2 extra credit points!

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**Algebra 1 Glencoe McGraw-Hill JoAnn Evans**

Math 8H 1-8 Number Systems Algebra Glencoe McGraw-Hill JoAnn Evans

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What are real numbers? Real Numbers

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Natural Numbers Natural Numbers: “Counting Numbers” 1, 2, 3, 4, 5, …

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**0, 1, 2, 3, 4, … Whole Numbers: Natural Numbers and 0 Whole Numbers**

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**…-3, -2, -1, 0, 1, 2,… Integers: Positive and Negative Whole Numbers**

Natural Numbers Integers: Positive and Negative Whole Numbers …-3, -2, -1, 0, 1, 2,…

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**Rational Numbers can be written as:**

When written as decimals they either repeat or terminate. Rational Numbers Integers Whole Numbers Natural Numbers

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**Rational Numbers Irrational Numbers**

Ex: These must be represented by a symbol (ex: ), or as a rounded number, or in radical form because the decimal doesn’t repeat or terminate (stop). Integers Whole Numbers Natural Numbers

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**Real Numbers Rational and Irrational Numbers are…**

Rational Numbers Irrational Numbers Real Numbers Integers Whole Numbers Natural Numbers

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**So what isn’t a real number?**

When you divide by zero and get no solution = i (imaginary numbers)

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**Do you remember the closure property?**

A set of numbers is CLOSED under an operation if the result of the operation (the answer) is in the same number set as the two numbers used in the operation.

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**Is the set of whole numbers closed under **

Determine whether each set of numbers is closed under the indicated operation: Is the set of whole numbers closed under the operation of multiplication? (When you multiply a whole number times another whole number, is the answer always a whole number?) Remember, the whole numbers are 0, 1, 2, 3, … Closed

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**Is the set of integers closed under the operation of division? **

(When you divide an integer by another integer, is the answer always an integer?) Remember, integers are all positive and negative whole numbers, including 0. Not closed. A counterexample:

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**Is the set of irrational numbers closed under **

the operation of multiplication? (When you multiply an irrational number times another irrational number, is the answer always an irrational number?) Remember, irrational numbers are numbers like pi or non-repeating and non-terminating decimals. Not closed. A counterexample to try on your calculator:

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**Name the set of numbers to which each real number belongs.**

Example 1 Example 2 Example 3 Example 4 rational irrational natural rational whole number integer integer rational

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**Square roots are written with a radical symbol**

Square roots are written with a radical symbol The number or expression inside a radical symbol is called the radicand. radical symbol radicand

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**A square root is one of two equal factors of a number.**

3 3 = 9 The positive square root of 9 is 3. -3 -3 = 9 The negative square root of 9 is -3.

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**The positive square root of 4 is 2.**

The negative square root of 4 is -2. The positive square root of 225 is 15. The negative square root of 225 is -15. Zero has only one square root! The positive square root of is . The negative square root of is

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**Some square roots aren’t whole numbers**

Some square roots aren’t whole numbers. The square roots of numbers that aren’t perfect squares are IRRATIONAL numbers. In the case of irrational numbers, approximate the square root by rounding the result to two decimal places and replacing the equal sign with a sign. Approximate the square roots of:

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**The square root of a negative radicand is undefined!**

Negative real numbers do not have square roots because two negative numbers multiplied produce a positive number. = undefined The square root of a negative radicand is undefined! Evaluate each expression: 6 -6 undefined

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To graph a set of numbers means to draw, or plot, the points named by those numbers on a number line. The number that corresponds to a point on a number line is called the coordinate of that point. • • •

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**Graph – 4, , – 6 and 0 on a number line**

Graph – 4, , – 6 and 0 on a number line. Order the numbers from least to greatest. • • • • is an irrational number. Find its approximation on your calculator, then place it on the number line. – 6, – 4, 0,

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Real Numbers Real Numbers are all numbers that can be located on Real Number line. This includes all whole numbers, all fractions, all decimals, all roots,

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