Download Real Numbers Unit Test Study Guide The real number system * A

Real Numbers Unit Test Study Guide The real number system * A number that is an irrational number is NEVER rational, an integer, whole number, or natural. * A number that is a natural number, whole number, integer, or rational is NEVER irrational *Terminating decimals are decimals that stop. They can be changed to a fraction. *Repeating decimals are decimals that go forever ​
in a repeating pattern *Irrational decimals go forever but do not have a repetitive pattern. *0.12345678910111213141516171819… is not a repeating pattern. This is a random set of digits! * ᵭ is irrational because it goes forever and does not repeat in a pattern. Anything added to, subtracted from, multiplied by, or divided with ᵭ is irrational * Perfect squares and cubes are rational because they are equivalent in value to whole numbers, integers, or counting numbers. *If the number is expressed as a square root and you cannot make a square with that area, then it is not a perfect square and is therefore irrational. *If a number is natural, it is also an integer, whole number, rational and real. *If a number is a whole number, it is also an integer, rational, and real. *If a number is an integer, it is also rational and real. *If a number is rational, it is also real. *If a number is an integer, it is also real. *If a number is irrational, it is also real. *The only numbers we will see that are not real numbers are square roots that have a negative sign inside of the radical such as √− 4 or √− 38 because it does not fit the definition of a square root. It is impossible to get these numbers using the real number system, so they are imaginary. ****REMEMBER!! Always simplify fractions before you decide what category it falls into! Numbers like 8/4 simplify to 2, so it is not only a rational number, but a natural, whole, and integer number as well! ****REMEMBER!!! Not all negative numbers are integers! ­0.34598345312048… is irrational, NOT an integer! Practice! Write every set of numbers that each number belongs to. Justify if the number is rational by writing it as an equivalent fraction. 1. 0.35 2. 12.34 3. 18/3 4. ­4/2 5. √82 6. √196 3
7. √216 8. 0 9. 82 10. 0.837202... Estimating Irrational Numbers 1. Determine which perfect squares the number falls between on the number line and draw the number line. Label approximately where you think that number should go. 2. Subtract the larger perfect square from the smaller perfect square to see how many units are between the numbers. That is the denominator (base unit) for your estimate. 3. Subtract the number you are estimating from the smaller perfect square to find how many units away that number is. This is your numerator. 4. If necessary, convert to a decimal by dividing the numerator by the denominator. Practice: Estimate the following irrational numbers to the nearest hundredth 1. √19 2. √125 3. √132 4. √48 5. √72 Comparing and Ordering Real Numbers 1. Draw a number line and label it with counting numbers and square roots. 2. Locate where the fractions and decimals go on the number line and label them 3. Locate where the square roots go on the number line and label them. If necessary, estimate the irrational numbers. Practice! 1. Place the following numbers in order from least to greatest: 3.82, √12 , 3 25 2. Place the following numbers in order from greatest to least: ­ 9.5 , ­9 12 , − √82 3. What is the largest integer value less than − √32 ? 4. What is the smallest integer value greater than √152 ? Changing Repeating Decimals to Fractions 1. Look at your number to see what digits are in the repeating pattern and what numbers, if any, are not. 2. Find your numerator a. If all of the digits repeat, then the digits in the pattern are the numerator b. If only some of the digits repeat, then subtract the non­repeating digits from the whole decimal. This is your numerator. 3. Find your denominator a. For each digit that repeats, place a 9 in the denominator. b. For each digit that does not repeat, place a 0 in the denominator 4. Simplify if necessary Practice! Change the following repeating decimals to fractions of equivalent value 1. 0.5 2. 0.76 3. 0.52 4. 0.45 5. 0.04 Square Roots *A square root is the side length of a square that when multiplied by itself results in the given area of the square. *A perfect square is a number that can be the area of a square with a rational or whole number side length. ****MEMORIZE YOUR PERFECT SQUARES UP THROUGH 15!!!*** *To find the side length of a square given the area, find the square root of that number. * s2 = A side squared is equivalent to the area of a square *No number times itself is equivalent to a negative integer, so these numbers are imaginary Practice! Solve each problem. 1. − √81 2. √36 3. s2 = √64 4. A square table has an area of 121 square inches. What is the length of each side? Cube Roots *A cube root is the side length of a cube that when multiplied by itself three times to find the given volume of a cube *A perfect cube is a number that can be the volume of a cube with a rational or integer side length. *To find the side length of a cube given the volume, find the cube root of that number. * s3 = V side cubed is equivalent to the volume of a cube *To find the surface area of a cube given the volume… 1. Find the cube root of the number 2. Find the area of each side of the cube by squaring the length of the side (each side is a square) 3. Multiply that answer by 6 (6 faces on a cube) *To find the volume of a cube given the surface area… 1. Divide the surface area by 6 (6 faces on a cube) 2. Find the square root of this number to find the length of the side of the cube (each side is a square) 3. Cube that number to find the volume of the cube *To find the side length of a cube given the surface area… 1. Divide the surface area by 6 (6 faces on a cube) 2. Find the square root of that number (each side is a square) ***Memorize your perfect cubes through 10!*** Practice! Solve each problem 1. A cube has a volume of 512. What is the length of the side? 2. A cube has a surface area of 294. What is the volume? 3. A cube has a surface area of 96. What is the length of the side? 4. s3 = √8 Now that you have solved my problems, come up with problems of your own or try out other study tips that I have on my website. You need to do more than just this study guide to make sure that you can ​
apply​
your knowledge, not just recall it! I believe in you! YOU CAN DO IT!