Question

If human height were quantized in 1 -cm 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 increase, (iii) the child's height would increase in jumps of \(6 \mathrm{~cm}\), or (iv) the child's height would increase in "jumps" of \(1 \mathrm{~cm}\) at a time?

Short answer

The correct answer is (iv): the child's height would increase in "jumps" of 1 cm at a time, as this scenario is consistent with the concept of human height being quantized in 1-cm increments.

Step-by-step solution

Understanding quantized height in 1-cm increments

If human height is quantized in 1-cm increments, this means that the height can only increase in discrete steps of 1 cm. In other words, an individual's height can only be an integer multiple of 1 centimeter (e.g. not \({245.5 cm}\), but only \({245 cm}\) or \({246 cm}\)).

Analyzing growth pattern options

Now let's examine the given options one by one and determine which one describes the growth pattern of a child according to the quantized height in 1-cm increments: (i) If "the child's height would never change," this would mean the child would not grow at all, which is unrealistic. (ii) "The child's height would continuously increase" implies that growth is constantly happening, as in a smooth, continuous manner. However, this doesn't match with the concept of quantized height in 1-cm increments, which involves discrete, step-by-step growth. (iii) While "the child's height would increase in jumps of 6 cm" involves distinct increases in height, it doesn't match the given scenario of human height being quantized in 1-cm increments. (iv) "The child's height would increase in 'jumps' of 1 cm at a time" is consistent with the concept of quantized height in 1-cm increments, as the child's height would increase in discrete steps of 1 cm.

Selecting the correct option

Based on our analysis, the best possible option that describes the growth pattern of a child when human height is quantized in 1-cm increments is: (iv) the child's height would increase in "jumps" of 1 cm at a time.

Key concepts

Growth pattern

A growth pattern refers to the way in which development or increase occurs over time. In the context of height, this means how a child's stature changes as they age. Typically, children grow in a more fluid, gradual manner, continuously extending as time progresses. However, if height were quantized or limited to specific increments, this natural flow would be altered. When considering quantized growth, imagining an even, predictable growth pattern becomes challenging because height adjustments would occur in defined units. In our exercise, we explore such scenarios by looking at the idea of height growing in chunks or jumps, rather than a smooth curve. This creates a more step-wise development pattern, where gradual increase is replaced by sudden advancements each time a certain threshold is reached.

Discrete steps

Quantization involves breaking a continuous phenomenon into discrete (or distinct) steps or levels. In the case of height measurement, a person's growth would be represented not as a smooth curve but as a series of incremental "jumps." These jumps occur because, with quantization, any progress made between these steps is not recorded. Thus, the height only visibly increases when the next quantized level is achieved. Our scenario deals with 1-cm increments, meaning height grows by 1 cm chunks.

• It simplifies measurements as each subsequent leap must be exactly 1 cm more than the last recognizable point.
• This model means that the growth in height is precise and consistently paced, always adjusting by the same measurable amount.

Height measurement

Height measurement in a quantized form contrasts with regular measurement by focusing on preset units. Standard measurement might yield a continuous spectrum of possible values, such as 145.1 cm, 145.2 cm, and so on. However, with quantized height, the allowable measurements are restricted, resulting in a simplified scale. For instance, the height can only be measured in whole centimeters—145 cm, 146 cm, etc. This concept is closely linked to quantization because it provides a clear illustration of how discrete steps are implemented. Key points about quantized height measurements include:

• Measuring tools or methods in this system would only recognize full increments.
• The resolution of the measurement is dictated by the size of the quantized unit, which simplifies but also limits precision.
• By using these measurements, we are embracing a model that benefits from simplification for ease of tracking distinct, evenly spaced changes.

This approach ensures clarity in how height differences are recorded, providing standard increments that can ease comparison across different individuals or time periods.