Units and scale

A number needs a meaning

A delay of 5 is incomplete: five seconds and five milliseconds describe different durations. A quantity value combines a numerical value with a unit. Write 5 ms with a space between the number and unit symbol.

The quantity is the duration; the unit is how we express it. A distance can be described in meters or centimeters without changing the distance. Likewise, changing a duration from seconds to milliseconds changes its numerical representation, not the time that passes.

For this chapter, use the variables and operation order from mathematical reasoning. The new habit is to carry units through every step instead of attaching them after the calculation.

Move between scales

A prefix multiplies a unit. The symbol s means second; the prefix m means one thousandth, so one millisecond is one thousandth of a second.

The SI prefixes below use powers of ten. A positive power counts repeated factors of ten. A negative exponent denotes the reciprocal: 10^-3 = 1/1000. The minus sign in an exponent does not make the duration negative.

Unit symbols are case sensitive. The table's m and M have different meanings. Always read a prefix together with the unit that follows it.

For the units we use here, 1 s = 1000 ms and 1 ms = 1000 μs. A smaller unit requires more of those units to express the same duration. That gives a useful check before doing arithmetic: converting seconds to milliseconds increases the numerical value; converting back decreases it.

Convert by multiplying by one

A conversion factor is a ratio of equal quantities. Because 1 s and 1000 ms are the same duration, the ratio 1 s / 1000 ms equals one. Multiplying by it preserves the quantity while changing its written unit.

Convert 3 ms to seconds:

3 ms × (1 s / 1000 ms)
= (3 / 1000) s
= 0.003 s

The ms unit appears in both numerator and denominator and cancels. If instead you multiply by 1000 ms / 1 s, the ms units do not cancel; that is a signal that the ratio is facing the wrong way for this conversion.

Now convert 2500 μs to milliseconds:

2500 μs × (1 ms / 1000 μs) = 2.5 ms

Here are equivalent representations to compare:

The units belong to the column headings. Every row describes one duration in three forms.

Add quantities in a common unit

You cannot add the raw numbers 3 and 2500 and then choose a unit. First make the units agree:

3 ms + 2.5 ms = 5.5 ms
3 ms + 2500 μs = 5.5 ms

The result is also 0.0055 s. The wrong shortcut, 2503 ms, treated microseconds as if they were milliseconds.

Quantities must also describe compatible things. Two durations can be added; a duration and a distance cannot be added into a meaningful total. An equation with incompatible units needs repairing before numerical calculation.

Derive a rate

Consider a machine that finishes one item every 4 ms, with no gaps and no overlapping work. One second contains 1000 ms. Dividing 1000 ms by 4 ms per item gives 250 items in a second:

rate = 1 item / 0.004 s = 250 items/s

The duration per item and the number of items per second are reciprocals under these assumptions. They answer different questions. A larger duration per item gives a smaller rate: doubling the duration to 8 ms reduces the rate to 125 items/s.

The reciprocal shortcut depends on the machine's schedule. If there is a 1 ms pause after every 4 ms item, one cycle takes 5 ms and the sustained rate becomes 200 items/s. If work overlaps, completion rate need not equal the reciprocal of one item's elapsed time. Keep the model visible when interpreting a rate.

Estimate before calculating

One task lasts 2 ms and another 700 μs. Seven hundred microseconds is less than one millisecond, so their sum should fall between 2 ms and 3 ms. Converting gives 2 ms + 0.7 ms = 2.7 ms. The estimate catches a misplaced factor of a thousand.

You can check a result in three ways: inspect the units, ask whether the magnitude is plausible, and reverse the operation. For 2.7 ms, subtracting the first task's 2 ms leaves 0.7 ms, which converts back to 700 μs.

Try it yourself

1. Convert 6 ms to seconds and microseconds.
2. Add 500 μs and 1.5 ms.
3. Under the no-gap, no-overlap model, find the rate for one item every 2 ms.
4. Find the sustained rate if each 2 ms item is followed by a 3 ms pause.
5. Someone reports 3 ms + 2500 μs = 2503 ms. Explain the error before recalculating.

Answers and limits

For the first, divide by 1000 to reach seconds and multiply by 1000 to reach microseconds. The conversions are 0.006 s and 6000 μs.

For the second, 500 μs = 0.5 ms. Add 0.5 ms + 1.5 ms; the sum is 2 ms.

For the third, 2 ms = 0.002 s and 1 / 0.002 = 500. The modeled rate is 500 items/s.

For the fourth, count the entire cycle: 2 ms + 3 ms = 5 ms. One thousand milliseconds divided by five gives 200 items/s.

For the fifth, the two input numbers were expressed in different units. Convert 2500 μs to 2.5 ms, then add to obtain 5.5 ms. The sum should be a little larger than each input duration, which the erroneous answer fails by a large margin.

These calculations use stipulated durations. Measurement uncertainty and binary storage prefixes need separate lessons.

Next: signals and states.