Rounding & Scientific Notation

Section 1 of 5

Rounding to decimal places

To round to a given number of decimal places: look at the next digit after the last one you want to keep.

Rounding rule:
Look at the digit immediately after the place you're rounding to.
• If it's $5$ or more → round up.
• If it's $4$ or less → leave the last kept digit as is.

(i) Round $\text{€}25.169$ to $2$ decimal places.

Keep two digits after the point: $25.16\,$ then look at the next digit ($9$).

$9 \geq 5$ → round up the $6$ to a $7$.

$\text{€}25.169 \;\approx\; \text{€}25.17$

(ii) Round $\text{€}78.996$ to $2$ decimal places.

Keep $78.99\,$ then look at the next digit ($6$).

$6 \geq 5$ → round up the $9$ to a $10$ — this carries.

$78.99 + 0.01 = 79.00$

$\text{€}78.996 \;\approx\; \text{€}79.00$

Keep the trailing zeros — $\text{€}79.00$ shows the answer is to the nearest cent. Writing just $\text{€}79$ loses that information.

Try this: round $\text{€}15.347$ to $2$ decimal places.

Look at the third decimal: $7$. $7 \geq 5$ → round the $4$ up to $5$.

$\text{€}15.35$

Try this: round $\text{€}12.495$ to $2$ decimal places.

Third decimal is $5$ → round up the $9$ to $10$ → carries to make $12.50$.

$\text{€}12.50$

Try this: round $4.6987$ to $2$ decimal places.

Third decimal is $8$ → round up the $9$ to $10$ → carries: $4.69 + 0.01 = 4.70$.

$4.70$

Try this: round $0.0463$ to $2$ decimal places.

Third decimal is $6$ → round up the $4$ to $5$.

$0.05$

Section 2 of 5

Significant figures

Significant figures count from the first non-zero digit. Leading zeros (before the first non-zero digit) are not significant — they're placeholders.

To round to $n$ significant figures:
1. Find the first non-zero digit. Count $n$ digits from there.
2. Look at the next digit and round.
3. For whole-number place values, replace remaining digits with zeros.
4. For decimals, drop the digits after the rounding point.

(i) Write $83{,}541$ to $2$ significant figures.

First two s.f.: $\mathbf{8}\mathbf{3}\,541$ — keep "$83$".

Next digit is $5$ → round up: $83 \to 84$.

Replace remaining digits with zeros (to preserve place value).

$83{,}541 \;\approx\; 84{,}000$

(ii) Write $0.0000054321$ to $2$ significant figures.

Leading zeros aren't significant. First non-zero digit is $5$.

First two s.f.: $\mathbf{5}\mathbf{4}\,321$ — keep "$54$".

Next digit is $3$ → round down (keep as is).

$0.0000054321 \;\approx\; 0.0000054$

Two-stage trick: convert mentally to scientific notation first, round the front part, then convert back. $0.0000054321 = 5.4321 \times 10^{-6}$ → $5.4 \times 10^{-6} = 0.0000054$.

Try this: write $2{,}478$ to $2$ significant figures.

First two s.f.: $24$. Next digit is $7$ → round up: $24 \to 25$.
Pad with zeros: $2{,}500$.

$2{,}500$

Try this: write $0.04562$ to $2$ significant figures.

First non-zero is $4$. First two s.f.: $45$. Next digit $6$ → round up: $45 \to 46$.

$0.046$

Try this: write $12{,}894$ to $3$ significant figures.

First three s.f.: $128$. Next digit is $9$ → round up: $128 \to 129$.
Pad with zeros: $12{,}900$.

$12{,}900$

Try this: write $0.000789$ to $1$ significant figure.

First non-zero is $7$. Next digit is $8$ → round up: $7 \to 8$.

$0.0008$

Section 3 of 5

Scientific notation — writing it

Scientific (or standard) notation writes any number as $a \times 10^{n}$ where $1 \leq a < 10$ and $n$ is a whole number.

Must learn: the form $a \times 10^{n}$ with $1 \leq a < 10$.
$a = 4.5$ ✓ $a = 9.99$ ✓ $a = 0.45$ ✗ (too small) $a = 10.2$ ✗ (too big)

How to do it: move the decimal point so just one non-zero digit sits in front of it. Count the moves — that's $|n|$. Moves left → $n$ positive. Moves right → $n$ negative.

(i) Write $45{,}000$ in scientific notation.

$45{,}000 = 45000.0$

Move the decimal $4$ places left: $4.5$

$45{,}000 = 4.5 \times 10^{4}$

(ii) Write $0.00037$ in scientific notation.

Move the decimal $4$ places right (to land after the $3$): $3.7$

Right → negative exponent.

$0.00037 = 3.7 \times 10^{-4}$

(iii) Write $6{,}800{,}000$ in scientific notation.

Move the decimal $6$ places left: $6.8$

$6{,}800{,}000 = 6.8 \times 10^{6}$

Try this: write $6{,}500$ in scientific notation.

Move decimal $3$ places left: $6.5$.

$6.5 \times 10^{3}$

Try this: write $0.000042$ in scientific notation.

Move decimal $5$ places right: $4.2$. Negative exponent.

$4.2 \times 10^{-5}$

Try this: write $7{,}800{,}000$ in scientific notation.

Move decimal $6$ places left: $7.8$.

$7.8 \times 10^{6}$

Try this: convert $1.23 \times 10^{5}$ back to an ordinary number.

Move decimal $5$ places right (positive exponent): $1.23 \to 123000$.

$123{,}000$

Section 4 of 5

Multiplying in scientific notation

Numbers in scientific notation multiply easily — handle the front parts separately from the powers of ten.

Multiplication rule:
$(a \times 10^{p}) \times (b \times 10^{q}) = (a \times b) \times 10^{p+q}$
Multiply the front numbers; add the exponents.

(i) Simplify $(2.1 \times 10^{3})(1.2 \times 10^{2})$, giving the answer in the form $a \times 10^{n}$ with $1 \leq a < 10$.

Front: $2.1 \times 1.2 = 2.52$

Powers: $10^{3} \times 10^{2} = 10^{3+2} = 10^{5}$

Combine: $2.52 \times 10^{5}$

Check: $1 \leq 2.52 < 10$ ✓

$(2.1 \times 10^{3})(1.2 \times 10^{2}) = 2.52 \times 10^{5}$

(ii) Simplify $(4 \times 10^{3})(5 \times 10^{2})$. Adjust if needed.

Front: $4 \times 5 = 20$

Powers: $10^{3} \times 10^{2} = 10^{5}$

Combine: $20 \times 10^{5}$ — but $20 \geq 10$, so the form is wrong.

Rewrite: $20 = 2 \times 10^{1}$

So $20 \times 10^{5} = 2 \times 10^{1} \times 10^{5} = 2 \times 10^{6}$

$(4 \times 10^{3})(5 \times 10^{2}) = 2 \times 10^{6}$

When the front product is $\geq 10$, write it in scientific notation itself and combine the powers.

Try this: simplify $(3 \times 10^{2})(2 \times 10^{5})$.

$3 \times 2 = 6$; $10^{2} \times 10^{5} = 10^{7}$.
$6 \times 10^{7}$ — front is in range, done.

$6 \times 10^{7}$

Try this: simplify $(1.5 \times 10^{-2})(2 \times 10^{4})$.

$1.5 \times 2 = 3$; exponents: $-2 + 4 = 2$.
$3 \times 10^{2}$

$3 \times 10^{2}$

Try this: simplify $(6 \times 10^{4})(7 \times 10^{3})$. Give the answer in correct standard form.

$6 \times 7 = 42$; exponents: $4 + 3 = 7$.
$42 \times 10^{7} = 4.2 \times 10^{1} \times 10^{7} = 4.2 \times 10^{8}$

$4.2 \times 10^{8}$

Section 5 of 5

Dividing in scientific notation

Division works in the same style as multiplication, but the exponents subtract.

Division rule:
$\dfrac{a \times 10^{p}}{b \times 10^{q}} = \dfrac{a}{b} \times 10^{p-q}$
Divide the front numbers; subtract the exponents (top minus bottom).

(i) Simplify $\dfrac{6 \times 10^{5}}{2 \times 10^{-3}}$.

Front: $\dfrac{6}{2} = 3$

Powers: $10^{5} \div 10^{-3} = 10^{5 - (-3)} = 10^{8}$

Mind the sign: subtracting a negative adds.

$\dfrac{6 \times 10^{5}}{2 \times 10^{-3}} = 3 \times 10^{8}$

(ii) Simplify $\dfrac{1.5 \times 10^{4}}{5 \times 10^{6}}$. Adjust to standard form.

Front: $\dfrac{1.5}{5} = 0.3$

Powers: $10^{4} \div 10^{6} = 10^{4-6} = 10^{-2}$

Combine: $0.3 \times 10^{-2}$ — but $0.3 < 1$, not in correct form.

Rewrite: $0.3 = 3 \times 10^{-1}$

So $0.3 \times 10^{-2} = 3 \times 10^{-1} \times 10^{-2} = 3 \times 10^{-3}$

$\dfrac{1.5 \times 10^{4}}{5 \times 10^{6}} = 3 \times 10^{-3}$

When the front quotient is $< 1$, write it in scientific notation itself (it'll have a negative exponent) and combine.

Try this: simplify $\dfrac{8 \times 10^{6}}{4 \times 10^{2}}$.

$\dfrac{8}{4} = 2$; exponents: $6 - 2 = 4$.
$2 \times 10^{4}$

$2 \times 10^{4}$

Try this: simplify $\dfrac{9 \times 10^{3}}{3 \times 10^{-2}}$.

$\dfrac{9}{3} = 3$; exponents: $3 - (-2) = 5$.
$3 \times 10^{5}$

$3 \times 10^{5}$

Try this: simplify $\dfrac{2 \times 10^{3}}{8 \times 10^{5}}$. Give the answer in correct standard form.

$\dfrac{2}{8} = 0.25$; exponents: $3 - 5 = -2$.
$0.25 \times 10^{-2} = 2.5 \times 10^{-1} \times 10^{-2} = 2.5 \times 10^{-3}$

$2.5 \times 10^{-3}$

Try this (mixed): simplify $\dfrac{(4 \times 10^{6})(3 \times 10^{-2})}{6 \times 10^{3}}$.

Top: $(4 \times 3) \times 10^{6 + (-2)} = 12 \times 10^{4}$
Divide: $\dfrac{12}{6} = 2$; exponents: $4 - 3 = 1$.
$2 \times 10^{1} = 20$

$2 \times 10^{1}$ (or $20$)

End of lesson

Rounding & Sci Notation — HL · Mathslive.ie