Question

Complete the sentence. The absolute value of 19 is__________________

Short answer

Answer: The absolute value of 19 is 19.

Step-by-step solution

Understand the Absolute Value Function

Absolute value is a function that calculates the distance of a number from zero. It is denoted by a pair of vertical bars surrounding a number, like |x|. The absolute value of a positive number or zero is equal to the number itself, while the absolute value of a negative number is equal to the opposite of the number. In mathematical terms: |x| = x, if x >= 0 |x| = -x, if x < 0

Apply the Absolute Value Function to the Given Number

The given number is 19, which is a positive number. As per the definition of absolute value, |19| = 19. Now, complete the sentence: The absolute value of 19 is 19.

Key concepts

Mathematical Functions

In mathematics, a function is like a special machine that takes an input, does something to it, and then gives back an output. Imagine it as a lemon squeezer; you put in a lemon (input), the squeezer does its job, and out comes lemon juice (output). Mathematical functions work similarly but with numbers and symbols instead of lemons.

Functions are usually written as an equation, like f(x) = x + 2, where f(x) represents the function and x is the input. Whatever you put in place of x, the function will add 2 to it and give you the result. The absolute value function is a bit unique because it tells us about the 'size' of a number without regard to whether it's positive or negative. This is very useful in times when only the magnitude of a number is important. Take away point: functions are fundamental building blocks in mathematics that transform an input to an output based on certain rules.

Distance from Zero

When we talk about distance from zero on a number line, we're discussing how far a number is from the starting point, which we call zero. This concept is vital since it helps us measure both position and length regardless of the direction.

Imagine you're standing on a long straight road. The place where you're standing is marked with a big zero. If you walk forward 10 steps or backwards 10 steps, you'll still be 10 steps away from where you started. That's the idea behind 'distance from zero'. It doesn't matter which direction you move; the distance remains the same. In numbers, this is exactly what the absolute value function does — it tells us the distance of any number from zero, ignoring whether it's to the left (negative) or the right (positive) of zero on a number line.

Positive Numbers

Positive numbers are all the numbers to the right of zero on the number line. They are numbers like 1, 2, 3, and so on, and they represent things that can be counted, like apples or days. They are the 'happy' numbers that indicate you have something rather than owing it.

If you think of temperature, positive numbers represent the warm degrees. On a bank account, positive numbers mean you have money to spend, not debt. When you see a positive number, it simply means there's more of something rather than less. Importantly, the absolute value of a positive number is the number itself because it's already at a 'positive distance' from zero — no changes needed to express its magnitude.