Comparing & Ordering Numbers: A Math Breakdown

Hey guys! Let's dive into some math problems. We're going to compare numbers, deal with square roots, and put numbers in order. Sound good? Let's go!

Comparing Numbers: Squaring Up the Challenge

Alright, first up, we're going to compare some numbers. This involves understanding square roots and how they relate to other numbers. It's like a number face-off, and we need to see who's stronger! We'll look at the first set of problems, where we need to figure out which number is bigger or smaller. Remember, the goal is to determine the relationship between two values. We're going to use a mix of estimation and calculation to make sure we're on the right track. This is all about getting comfortable with numbers and their different forms, like decimals and fractions. It's like learning the rules of the game so you can play it well.

a. Comparing √2 and 1.5: Decoding the Square Root

So, let's start with a. We've got the square root of 2 (√2) and 1.5. The key here is understanding what √2 actually is. It's the number that, when multiplied by itself, equals 2. We know that 1 squared (1 * 1) is 1, and 2 squared (2 * 2) is 4. Therefore, √2 has to be somewhere between 1 and 2. Actually, √2 is roughly 1.414. So, if we compare 1.414 and 1.5, we can easily see that 1.5 is bigger. Easy peasy, right?

b. Comparing -√3 and -2: Negative Numbers and Square Roots

Next, we have -√3 and -2. This one involves negative numbers, which can sometimes trip us up. Remember, with negative numbers, the further you are from zero, the smaller the number is. First, let's figure out roughly what -√3 is. √3 is between 1 and 2 (because √1 = 1 and √4 = 2). More precisely, √3 is around 1.732. So, -√3 is approximately -1.732. Now, let's compare -1.732 and -2. On the number line, -1.732 is to the right of -2, which means -1.732 is bigger. So, -√3 is greater than -2. It's like knowing which side of the street you want to be on; it changes your perspective.

c. Comparing 2/3 and √0.36: Fractions and Square Roots

Alright, let's look at 2/3 and √0.36. First, let's simplify √0.36. The square root of 0.36 is 0.6 because 0.6 * 0.6 = 0.36. Now, let's think about 2/3. You can convert this to a decimal by dividing 2 by 3, which gives you approximately 0.666... Now we compare 0.666... and 0.6. Clearly, 2/3 (or 0.666...) is greater than 0.6. This is where your basic math skills and understanding of number forms come in handy.

d. Comparing -√34 and -√35: Bigger Square Roots, Bigger Negative Values

Finally, we have -√34 and -√35. Since both are negative, let's think about the positive versions first. √35 is greater than √34. However, because they are negative, the situation reverses. -√34 is actually greater than -√35. Think of it like owing money: owing less money is always better. The larger the number under the square root, the further from zero, and therefore smaller, the negative value becomes. It's all about keeping track of those minus signs!

Ordering Numbers: Lining 'Em Up

Now, let's put some numbers in order, from smallest to largest. This is like arranging soldiers in a line based on height. We need to be able to identify which number is the lowest value and which is the highest. We'll use a number line in our head (or on paper) to make sure we've got everything right. This is where all those basic rules we talked about earlier come together.

a. Ordering -4, -3, 9, -2.(64), -3.89, -2.6: Decimals and Negatives, Oh My!

First, we've got -4, -3, 9, -2.(64), -3.89, and -2.6. Start by identifying the smallest number. The most negative number is the smallest. That's -4. Next comes -3.89, then -3. Next is -2.6, then -2.(64). Finally, the largest number is 9. So, the ordered list from smallest to largest is: -4, -3.89, -3, -2.6, -2.(64), 9. Remember, the more negative the number, the further it is to the left on the number line and the smaller the value.

b. Ordering -√11, √10, √8, -√10, √7: Mixing Square Roots

Okay, let's order these: -√11, √10, √8, -√10, √7. First, note that the negative square roots are smaller than the positive ones. -√11 is smaller than -√10. For the positive ones, √7 is the smallest, then √8, then √10. Therefore, the order from smallest to largest is -√11, -√10, √7, √8, √10. Always look for the negatives, then order the rest.

c. Ordering 1, √3: Basic Numbers

Finally, we have 1 and √3. We know that √3 is approximately 1.732. Therefore, the ordered list is 1, √3. This one is pretty straightforward since we understand the approximate value of √3. Simple, but important to understand.

Conclusion: Mastering the Number Game

And there you have it! We've compared numbers, dealt with square roots, and put them in order. Remember, the key is to understand the basics of what each number represents, especially when it comes to square roots and negative numbers. Practice makes perfect, so keep working through these types of problems, and you'll become a number-ordering pro in no time! Keep practicing, guys, and you'll be acing these problems in no time. Good luck, and have fun with math!