Question

If human height were quantized in 1 -foot increments, what would happen to the height of a child as she grows up: (i) The child's height would never change, (ii) the child's height would continuously get greater, (iii) the child's height would increase in "jumps" of 1 foot at a time, or (iv) the child's height would increase in jumps of 6 inches?

Short answer

The correct answer is (iii) the child's height would increase in "jumps" of 1 foot at a time, as height is quantized in 1-foot increments, and the child's height would only change when she attains each whole feet value.

Step-by-step solution

Understanding Quantization

Quantization is the process of converting continuous values or signals into discrete values. In this exercise, human height is quantized in 1-foot increments, meaning that height can only be represented as whole feet (e.g., 4 feet, 5 feet, 6 feet, etc.) and not any values in between (e.g. 4.5 feet, 5.75 feet, etc).

Applying Quantization to Child's Growth

Since height is quantized in 1-foot increments, as the child grows, her height would only increase when she attains each whole feet value. It would not change at any values between those whole feet increments.

Identifying the Correct Option

Based on our understanding of quantization and how it affects the child's growth, let's evaluate each option: (i) The child's height would never change: This is incorrect, as the child's height would change when she reaches the next whole feet increment. (ii) The child's height would continuously get greater: This is also incorrect, as the child's height would not change continuously but would increase only when the next 1-foot increment is obtained. (iii) The child's height would increase in "jumps" of 1 foot at a time: This is the correct option, as the height would increase only when the child achieves each 1-foot increment. (iv) The child's height would increase in jumps of 6 inches: This is incorrect, as the height is quantized in 1-foot increments, not 6-inch increments. So the correct answer is (iii) the child's height would increase in "jumps" of 1 foot at a time.

Key concepts

discrete values

When we talk about quantization, we're essentially discussing the conversion of something that can vary smoothly — like height — into a set of distinct, separate values called discrete values. Imagine turning the continuous range of human heights into specific steps on a ladder. Instead of smoothly transitioning from one height to another, heights after quantization are restricted to certain fixed points.
In our exercise, human height is transformed into these discrete values by allowing only increments of whole feet. So, if someone stands at 4 feet and grows a bit, this will not reflect in their quantized height until they reach 5 feet. This process of creating discrete values simplifies complexity and is common in digital systems where continuous data needs representation in a finite form.

growth increments

Growth increments refer to the way in which height changes over time. With quantization, these increments become more like clear steps on a staircase rather than a smooth climb up a hill. In this exercise, the increments are defined by the measurement unit, which is feet.
Each growth increment corresponds to a clear, distinct change — a jump from one whole number of feet to the next. For the child in our example, their height increments occur at these whole foot levels. Essentially, as they grow, their quantized height remains at the lower foot mark until they entirely "step up" to the next foot. This discrete step or "jump" from one foot to another embodies the concept of growth increments within quantization.

height representation

Height representation in the realm of quantization simplifies how we perceive and record height. Instead of acknowledging every fractional inch of growth, quantization forces a rounding principle to the nearest foot for simplicity. It's the same as looking at height with bigger brush strokes rather than detailed shading.
This type of representation would show a child's height as static until they grow an entire foot. A child could be 4 feet tall, but any growth wouldn’t reflect until they hit the 5-foot mark. This type of representation helps streamline data but can also hide intermediate changes that occur before the next quantization level. Using such representation is common in digital conversions and computing, where memory and processing prefer simpler, fixed values.