Units and rounding(Mechanics)

Units

The following table gives some of the SI base units that will be used throughout the Mechanics section.

Unit Name

Unit Symbol

Quantity

Formal Definition

Dimension Symbol

metre

$\text{m}$

length

The distance travelled by light in a vacuum during a time interval of $^1/_{299,792,458}$ of a second. ($299,792,458\text{ms}^{-1}$ is the speed of light.)

$\text{L}$

kilogram

$\text{kg}$

mass

The mass of the international prototype of the kilogram.

$\text{M}$

second

$\text{s}$

time

The duration of $9,192,631,770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

$\text{T}$

Rounding

Rounding means reducing the number of decimal places (d.p.) a number has but keeping its value close to what it was. The purpose is to make the number simpler. Suppose we want to round a number to $2$ decimal places (or $2$d.p.). To do this, we look only at the next digit to the right (the third decimal digit). If this is:

  • $\geq 5$ we increase the second decimal digit by $1$.
  • $\lt 5$ we leave the second decimal digit as it is.
Worked

Round $251.6731$ to:

a) $3$ decimal places

b) $2$ decimal places

c) $1$ decimal place.

Solution

a) The fourth decimal digit is $1$. Since $1\lt 5$, we leave the third decimal digit alone: $251.6731$ to $3$d.p. is $2.673$.

b) The third decimal digit is $3$. Since $3\lt 5$, we leave the second decimal digit alone: $251.6731$ to $2$d.p. is $2.67$.

b) The second decimal digit is $7$. Since $7\gt 5$, we increase the first decimal digit by one: $251.6731$ to $1$d.p. is $2.7$.

Significant Figures

Sometimes we are asked to round a number to a given number of “significant figures” (s.f.) or “significant digits”. Here are the basic rules for significant digits:

1) All nonzero digits are significant. 2) All zeroes between significant digits are significant. 3) All zeroes which are both to the right of the decimal point and to the right of all non-zero significant digits are significant.

The rules for rounding are the same. If the next number is:

  • Equal to or greater than $5$, we round up.
  • $4$ or less, we don't round up.
Worked 1

Round $702.019$ to

a) Five significant digits

b) Four significant digits

c) One significant digit

Solution

a) $702.02$

b) $702.0$

c) $700$

Worked 2

Round $0.097065$ to four, three, and two significant digits:

a) Four significant digits

b) Three significant digits

c) Two significant digits

Solution

a) $0.09707$

b) $0.0971$

c) $0.097$

Worked

a) Round $4.0032$ to 3 d.p.

b) Round $1.1674$ to 2 s.f.

c) Round $0.00123$ to 3 d.p.

d) Round $0.00123$ to 3 s.f.

Solution

a) \[4.003\]

b) \[1.2\]

c) \[0.001\]

d) \[0.00123\]

External Resources