1.1 Numbers – Rounding

1. Notation for Sets of Numbers

Remember the following known sets of numbers:

$\mathbb{N} = \{0, 1, 2, 3, 4, \dots\}$ natural
$\mathbb{Z} = \{0, \pm1, \pm2, \pm3, \dots\}$ integers
$\mathbb{Q} = \left\{\frac{a}{b} : a, b \in \mathbb{Z}, b \neq 0\right\}$ rational (fractions of integers)
$\mathbb{R} = \text{rational} + \text{irrational}$ real

Known irrational numbers:

$\sqrt{2}, \sqrt{3}, \sqrt{5}$ and all $\sqrt{a}$ where $a$ is not a perfect square.
$\pi = 3.14159\dots$
$e = 2.7182818\dots$

To indicate particular subsets we use the indices $+$, $-$, $*$ as follows:

  • $\mathbb{Z}^+ = \{1, 2, 3, \dots\}$ positive integers
  • $\mathbb{Z}^- = \{-1, -2, -3, \dots\}$ negative integers
  • $\mathbb{Z}^* = \{\pm1, \pm2, \pm3, \dots\}$ non-zero integers i.e. $\mathbb{Z}^* = \mathbb{Z} \setminus \{0\}$

Similar notations apply for the other sets above.

ℝ (Real Numbers) Irrationals (ℚ') π, e, √2, √3 ℚ (Rational Numbers) 1/2, -0.75, 4.333... ℤ (Integers) ..., -2, -1 ℕ (Naturals) 0, 1, 2, 3...

2. Intervals of Real Numbers

For intervals of real numbers, the following notations are used:

  • $x \in [a,b]$ for $a \le x \le b$
  • $x \in ]a,b[$ or $x \in (a,b)$ for $a < x < b$
  • $x \in [a,b[$ or $x \in [a,b)$ for $a \le x < b$
  • $x \in [a,+\infty[$ or $x \in [a,+\infty)$ for $x \ge a$
  • $x \in ]-\infty,a]$ or $x \in (-\infty,a]$ for $x \le a$
  • $x \in ]-\infty,a] \cup [b,+\infty[$ for $x \le a$ or $x \ge b$

3. Decimal Places vs Significant Figures

The next topic covers a – not so pleasant – discussion about rounding of numbers. The numerical answer to a problem is not always exact, requiring the use of some rounding.

Consider the number 123.4567

There are two ways to round up the number by using fewer digits:

  • To a specific number of decimal places (d.p.)
  • To a specific number of significant figures (s.f.): for the position of rounding, we start counting from the first non-zero digit.

Numbers can also be rounded up before the decimal point (e.g. to the nearest integer, 10, or 100).

Notice that the number at the critical position:

  • remains as it is if the following digit is $0, 1, 2, 3, 4$
  • increases by $1$ if the following digit is $5, 6, 7, 8, 9$
To Decimal Places To Significant Figures Before the Decimal
to 1 d.p. $\rightarrow$ 123.5 to 6 s.f. $\rightarrow$ 123.457 nearest integer $\rightarrow$ 123
to 2 d.p. $\rightarrow$ 123.46 to 5 s.f. $\rightarrow$ 123.46 nearest 10 $\rightarrow$ 120
to 3 d.p. $\rightarrow$ 123.457 to 4 s.f. $\rightarrow$ 123.5 nearest 100 $\rightarrow$ 100
to 3 s.f. $\rightarrow$ 123 to 2 s.f. $\rightarrow$ 120 to 1 s.f. $\rightarrow$ 100

EXAMPLE 1

Consider the number 0.04362018

To decimal places:

  • to 2 d.p. $\rightarrow$ $0.04$
  • to 3 d.p. $\rightarrow$ $0.044$
  • to 4 d.p. $\rightarrow$ $0.0436$
  • to 6 d.p. $\rightarrow$ $0.043620$

To significant figures:

  • to 2 s.f. $\rightarrow$ $0.044$
  • to 3 s.f. $\rightarrow$ $0.0436$
  • to 4 s.f. $\rightarrow$ $0.04362$
  • to 5 s.f. $\rightarrow$ $0.043620$

⚠️ Important Remark

In the final IB exams the requirement is to give the answers either in exact form or to 3 s.f.

  • Exact form: $\sqrt{2}$ $\rightarrow$ to 3 s.f: $1.41$
  • Exact form: $2\pi$ $\rightarrow$ to 3 s.f: $6.28$
  • Exact form: $12348$ $\rightarrow$ to 3 s.f: $12300$

4. The Scientific Form $a \times 10^k$

Any number can be written in the form:

$a \times 10^k \quad \text{where} \quad 1 \le a < 10$

We simply move the decimal point after the first non-zero digit.

For example, the number $123.4567$ can be written as $1.234567 \times 10^2$
Indeed, $1.234567 \times 10^2 = 1.234567 \times 100 = 123.4567$
Notice that the decimal point moved 2 positions to the left $\Rightarrow k = 2$
Even for a "small" number, say $0.000012345$, an expression can be found:
$1.2345 \times 10^{-5}$
Notice that the decimal point moved 5 positions to the right $\Rightarrow k = -5$

NOTICE:

  • They may ask to give the number in scientific form but also to 3 s.f. Then:
    $1.2345 \times 10^2 \approx 1.23 \times 10^2$ $1.2345 \times 10^{-5} \approx 1.23 \times 10^{-5}$
  • Many calculators use the symbol E for the scientific notation:
    The notation 1.2345E+02 means $1.2345 \times 10^2$ The notation 1.2345E-05 means $1.2345 \times 10^{-5}$

EXAMPLE 2

(a) Give the scientific form of the numbers:
$x = 100000 \quad y = 0.00001 \quad z = 4057.52 \quad w = 0.00107$

Solution:

$x = 1 \times 10^5$ $y = 1 \times 10^{-5}$ $z = 4.05752 \times 10^3$ $w = 1.07 \times 10^{-3}$

(b) Give the standard form of the numbers:
$s = 4.501 \times 10^7 \quad t = 4.501 \times 10^{-7}$

Solution:

$s = 45010000$ $t = 0.0000004501$

EXAMPLE 3

Consider the numbers $x = 3 \times 10^7$ and $y = 4 \times 10^7$. Give $x+y$ and $xy$ in scientific form.

Solution:

$x+y = 7 \times 10^7$ [add 3+4, keep the same exponent]
$xy = 12 \times 10^{14}$ [multiply 3×4, add exponents]
$= 1.2 \times 10^{15}$ [modify $a$ so that $1 \le a < 10$]

EXAMPLE 4

Consider the numbers $x = 3 \times 10^7$ and $y = 4 \times 10^9$. Give $x+y$ and $xy$ in scientific form.

Solution:

For addition, we must modify $y$ (or $x$) in order to achieve similar forms:

$x = 3 \times 10^7$ $y = 4 \times 10^9 = 400 \times 10^7$ $x+y = 403 \times 10^7$ [add 3+400, keep the same exponent] $= 4.03 \times 10^9$ [modify $a$ so that $1 \le a < 10$]

For multiplication, there is no need to modify $y$:

$xy = 12 \times 10^{16}$ [multiply 3×4, add exponents] $= 1.2 \times 10^{17}$ [modify $a$ so that $1 \le a < 10$]