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.
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 $x\geq b$ |
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:
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 $y$:
$$\begin{aligned} xy&=12\times10^{16} \quad \text{[multiply } 3\times4 \text{, add exponents]} \\ &=1.2\times10^{17} \quad \text{[modify } a \text{ so that } 1\leq a<10 \text{]} \end{aligned}$$