1.1 Numbers – Rounding

1. Notation for Sets of Numbers

Remember the following known sets of numbers used throughout the IB Mathematics AA course:

Natural numbers: $$\mathbb{N}=\{0,1,2,3,4,\dots\}$$
Integers: $$\mathbb{Z}=\{0,\pm1,\pm2,\pm3,\dots\}$$
Rational numbers (fractions of integers): $$\mathbb{Q}=\left\{\dfrac{a}{b}:a,b\in\mathbb{Z},\ b\neq0\right\}$$
Real numbers: $$\mathbb{R}=\text{rational}+\text{irrational}$$

Known irrational numbers:

$$\sqrt{2},\quad \sqrt{3},\quad \sqrt{5}$$ 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 $+$, $-$, and $*$ as follows:

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

Similar notations apply for the other sets above. For example, $\mathbb{R}^+$ means positive real numbers, and $\mathbb{Q}^*$ means non-zero rational numbers.

Real Numbers ($\mathbb{R}$)
Irrational Numbers
$\pi, e, \sqrt{2}, \sqrt{3}$
non-terminating
non-repeating decimals
$\mathbb{Q}$ Rational Numbers
fractions of integers
$\dfrac{1}{2}, -0.75, 4.333\dots$
$\mathbb{Z}$ Integers
$\dots, -2, -1, 0, 1, 2, \dots$
$\mathbb{N}$ Naturals
$0, 1, 2, 3, \dots$

2. Intervals of Real Numbers

For intervals of real numbers we use the following notations:

Interval notation Inequality form
$x\in[a,b]$ $a\leq x\leq b$
$x\in]a,b[$ or $x\in(a,b)$ $a< x < b$
$x\in[a,b[$ or $x\in[a,b)$ $a\leq x < b$
$x\in[a,+\infty[$ or $x\in[a,+\infty)$ $x\geq a$
$x\in]-\infty,a]$ or $x\in(-\infty,a]$ $x\leq a$
$x\in]-\infty,a]\cup[b,+\infty[$ $x\leq a$ or $x\geq b$
$-\infty$
$+\infty$
$-6$
$-4$
$-2$
$0$
$2$
$4$
$6$
$[a,b]$: closed endpoints
$(a,b)$: open endpoints
$[a,b)$: left closed, right open
$[a,+\infty)$: $x \geq a$
$(-\infty,a] \cup [b,+\infty)$: outside interval

3. Decimal Places vs Significant Figures

I have to continue my notes with a not so pleasant discussion about rounding of numbers. The numerical answer to a problem is not always exact and we have to use 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.): count digits after the decimal point.
  • To a specific number of significant figures (s.f.): for the position of rounding, we start counting from the first non-zero digit.
  • We can also round up before the decimal point, for example to the nearest integer, nearest $10$, or nearest $100$.

Notice that the number at the critical position:

  • remains as it is if the following digit is $0,1,2,3,$ or $4$.
  • increases by 1 if the following digit is $5,6,7,8,$ or $9$.
To decimal places (d.p.) To significant figures (s.f.) Before the decimal point
to $1$ d.p. $\rightarrow$ $123.5$ to $6$ s.f. $\rightarrow$ $123.457$ to the nearest integer $\rightarrow$ $123$
to $2$ d.p. $\rightarrow$ $123.46$ to $5$ s.f. $\rightarrow$ $123.46$ to the nearest $10$ $\rightarrow$ $120$
to $3$ d.p. $\rightarrow$ $123.457$ to $4$ s.f. $\rightarrow$ $123.5$ to the 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 significant figures
to $2$ d.p. $\rightarrow$ $0.04$ to $2$ s.f. $\rightarrow$ $0.044$
to $3$ d.p. $\rightarrow$ $0.044$ to $3$ s.f. $\rightarrow$ $0.0436$
to $4$ d.p. $\rightarrow$ $0.0436$ to $4$ s.f. $\rightarrow$ $0.04362$
to $6$ d.p. $\rightarrow$ $0.043620$ 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. For example:

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

4. The Scientific Form $a\times10^k$

Any number can be written in the form:

$$a\times10^k$$ where $1\leq a<10$

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

For example, the number $$123.4567$$ can be written as $$1.234567\times10^2.$$ Indeed, $$1.234567\times10^2=1.234567\times100=123.4567.$$ Notice that we moved the decimal point $2$ positions to the left $\Rightarrow k=2$.
Even for a "small" number, say $$0.000012345$$ we can find such an expression: $$1.2345\times10^{-5}.$$ Notice that we moved the decimal point $5$ positions to the right $\Rightarrow k=-5$.

NOTICE

  • They may ask us to give the number in scientific form but also to $3$ s.f. Then: $$1.2345\times10^2\cong1.23\times10^2$$ $$1.2345\times10^{-5}\cong1.23\times10^{-5}$$
  • Many calculators use the symbol E for the scientific notation: The notation $1.2345\text{E}+02$ means $1.2345\times10^2$
    The notation $1.2345\text{E}-05$ means $1.2345\times10^{-5}$

EXAMPLE 2

(a) Give the scientific form of the numbers $$x=100000,\qquad y=0.00001,\qquad z=4057.52,\qquad w=0.00107$$ (b) Give the standard form of the numbers $$s=4.501\times10^7,\qquad t=4.501\times10^{-7}$$

Solution:

(a) $$x=1\times10^5$$ $$y=1\times10^{-5}$$ $$z=4.05752\times10^3$$ $$w=1.07\times10^{-3}$$
(b) $$s=45010000$$ $$t=0.0000004501$$

EXAMPLE 3

Consider the numbers $$x=3\times10^7 \quad \text{and} \quad y=4\times10^7$$ Give $x+y$ and $xy$ in scientific form.

Solution:

$$\begin{aligned} x+y&=7\times10^7 \quad \text{[add $3+4$, keep the same exponent]} \\ xy&=12\times10^{14} \quad \text{[multiply $3\times4$, add exponents]} \\ &=1.2\times10^{15} \quad \text{[modify $a$ so that $1\leq a<10$]} \end{aligned}$$

EXAMPLE 4

Consider the numbers $$x=3\times10^7 \quad \text{and} \quad y=4\times10^9$$ Give $x+y$ and $xy$ in scientific form.

Solution:

For addition we must modify $y$ (or $x$) in order to achieve similar forms: $$\begin{aligned} x&=3\times10^7 \\ y&=4\times10^9=400\times10^7 \end{aligned}$$ Hence, $$\begin{aligned} x+y&=403\times10^7 \quad \text{[add } 3+400 \text{, keep the same exponent]} \\ &=4.03\times10^9 \quad \text{[modify } a \text{ so that } 1\leq a<10 \text{]} \end{aligned}$$
For multiplication there is no need to modify $y$: $$\begin{aligned} xy&=12\times10^{16} \quad \text{[multiply } 3\times4 \text{, add exponents]} \\ &=1.2\times10^{17} \quad \text{[modify } a \text{ so that } 1\leq a<10 \text{]} \end{aligned}$$