Signed Numbers: Positive and Negative Values

0:00 Welcome to this explainer. Okay, today we're tackling something absolutely crucial for the ASVAB mathematics knowledge subtest. And that is signed numbers. Look, if you've ever felt a little, you know, uneasy when a bunch of negative signs suddenly show up in an equation, trust me, you are in the exact right place. We all know the ASVAB doesn't allow calculators. That means every single operation has to be done by hand or right in your head. But seriously, don't worry, by the time we wrap up here, we're going to build your confidence so you can handle these questions smoothly, completely on your own. All right, let's take a really quick look at our roadmap. We've got six bite-size steps. The basics of signed numbers, adding and subtracting rules, multiplying, dividing, and zero, advanced operations and context, worked examples and traps, and then we'll finish up with some check yourself practice questions. So let's jump right into section one, the basics of signed numbers. To really master signed numbers, it helps to just picture a horizontal number line. Zero sits right there in the middle. Positive numbers go off to the right and negative numbers go off to the left.

1:00 The farther right a number is, the larger it is. Pretty straightforward, right? Now let's talk about absolute value. That's the one written with those two vertical bars. Absolute value is honestly just a measurement of distance from zero. And think about it, because it's a distance, it can literally never be a negative number. You can't walk negative miles. So the absolute value of negative 15 is 15. And the absolute value of positive 15 is also 15, because both of those numbers are exactly 15 units away from zero. Now this right here brilliantly illustrates a common trap that trips up a lot of test takers. When you're looking at negative numbers, a bigger absolute value actually means a smaller overall number. I know, that sounds weird, but why is that? It's because it sits further to the left on the number line. So negative 10 is actually less than negative one. Even though 10 feels bigger than one, negative 10 is way farther left, which makes its true value smaller. Adding and subtracting rules, let's see how we actually use these numbers.

2:03 Okay, here is your golden rule for addition. When you're adding numbers, always check their signs first. For same signs, when both numbers share a sign, like negative six plus negative eight, you literally just add their sizes together. Six plus eight is 14, and keep that common negative sign. Two losses just make a bigger loss, right? So the answer is negative 14. But check out different signs. For negative 14 plus nine, the signs differ. Here, you subtract the smaller size from the larger one. 14 minus nine is five. Then you just keep the sign of the number with the larger absolute value. Since 14 is larger than nine and it was negative, our final answer is negative five. Now, for subtraction, the crucial point is this. Subtracting a signed number is basically just adding the opposite. Let's take seven minus negative four. You just turn that subtraction into addition and flip that negative four to a positive four. So it becomes seven plus four, which equals 11. Subtracting a negative always, always increases your result. It's just a simple mental trick that instantly turns a kind of confusing subtraction problem into an easy addition one.

3:11 All right, section three, multiplying, dividing, and zero. Let's add these to our toolkit. The absolutely great news here is that multiplication and division follow the exact same sign rules. If the signs are the same, your result is positive. For example, negative six times negative four equals a positive 24. If the signs are different, your result is negative. But hey, what if you're multiplying a really long string of numbers? Don't panic. Just count the negative factors. An even count of negative numbers gives you a positive answer. And an odd count gives you a negative answer. Figure out the sign first. Then just multiply the actual numbers. Now, let's take a second to look at the unique behavior of zero, because zero is a special case on the ASVAB. It's neither positive nor negative. If you multiply anything by zero, you get zero. But division, that's where they're going to try to trick you. Zero divided by any non-zero number, like negative five, is just zero. But if zero is in the denominator, meaning you're trying to divide a number by zero, like negative five divided by zero, the answer is undefined.

4:12 You simply cannot divide by zero. There's no answer. Let's step it up a notch with section four, advanced operations, and context. Okay, pay close attention, because this is one of the most classic traps on the entire math subtest. Parentheses are what decide what gets raised to the power. If we have negative two enclosed in parentheses raised to the fourth power, the base is that whole negative two. Since it's an even power, the negative cancels itself out, giving us a positive 16. But look at negative two to the fourth power without parentheses. Here, the base is only the two. You calculate two to the fourth power to get 16, and then that negative sign just gets attached to the very end, leaving you with negative 16. Always, always check for parentheses. When you run into word problems, the signs are often hiding in plain English. Words like deposits, gains, or a temperature rise, those mean positive numbers. Withdrawals, losses, or temperature drops, those are your negative numbers. And if a question asks for a change, remember it's always the final value minus the starting value. But most importantly, I really want you to adopt this strategy.

5:18 Sign first, arithmetic second. Figure out if your answer should be positive or negative before doing the heavy mental math. A lot of the wrong multiple choice options have the right number, but the wrong sign. This one defensive habit will literally help you cross out wrong answers instantly. Worked examples and traps. Let's see this in action. Let's take this one careful mental step at a time, just like you would on the test. Step one, we need to evaluate negative eight plus negative six minus negative three. For step two, let's tackle that tricky subtraction at the end. We know subtracting a negative is the exact same as adding a positive, so let's rewrite the equation as negative eight plus negative six plus three. Much better. Step three, let's combine the negative numbers first. Negative eight plus negative six gives us a bigger negative, negative 14. Finally, negative 14 plus three. The signs are different, so we subtract three from 14 to get 11, and we keep the sign of the larger absolute value, which was the negative 14. So the final answer is negative 11.

6:22 All right, let's look at another one. And this one features that exponent trap we talked about. Step one, evaluate eight minus three times negative two squared. Order of operations absolutely protects us here. We have to do the exponent before we multiply or subtract. So in step two, negative two inside parentheses squared is a positive four. Mentally rewriting, we have eight minus three times four. Step three, multiply first, three times four is 12. And finally, in step four, we do the subtraction. Eight minus 12 gives us our answer, which is negative four. So let's quickly summarize these common pitfalls. Think of this as your mental checklist. First, don't confuse negative three squared without parentheses with negative three squared inside parentheses. Huge difference. Second, don't forget to change minus a negative into a plus. Third, remember that on a number line, negative 10 is further left. So it's actually smaller than negative two. And finally, absolute value is distance. It can never ever be a negative answer. Keep these tucked in your brain and you're going to spot the test's tricks way before you fall for them.

7:24 And now, section six, check yourself practice questions. It's time to prove to yourself just how much you've mastered. What is negative nine plus four? Are the options a negative five, b negative 13, c 13, or d five? Go ahead and pause this explainer right now. Solve it in your head using the rules we just went over and then hit play when you're ready for the answer. Okay, if you picked a negative five, excellent job. Let's break down exactly why. We are adding numbers with different signs. The rule says we subtract their absolute values. Nine minus four is five. Then we look at the original numbers to see which had the larger absolute value. Nine is bigger than four and that nine was negative. So we keep the negative sign. The answer is indeed negative five. Let's try one more. Question two, which of the following has the least value? A negative six, b negative one, c the absolute value of negative 14, or d negative 11. Take a moment, pause the explainer, find the smallest value out of the group, and lock in your answer.

8:33 And the correct answer is d negative 11. Did you happen to spot the traps in that one? Option c is an absolute value, which means it actually evaluates to a positive 14. So that is definitely not the smallest. Out of the remaining negative numbers, negative 11 is the farthest to the left on the number line. Remember, among negatives, the one with the bigger absolute value is actually the smaller number. See? With just these few foundational rules, you really can conquer any signed number question without a calculator. But you know, signed numbers are really just the beginning. So what other hidden math tricks are waiting to be unlocked in your ASVAB prep? Keep practicing, keep building that confidence, and thanks so much for joining me for this explainer.