Exponents & logs 1 · Standard form: writing very big and very small numbers

Module 4 of 46Lesson 1 of 12 · Exponents & logs 1 · Standard form: writing very big and very small numbers

Standard form, exponents & logarithms · Essential understanding 1

Standard form: writing very big and very small numbers

IB AA SL 1.1with Prof. Kitso

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

The big idea: Standard form writes any number as a × 10^k, where a is between 1 and 10 and k is a whole number. It lets us write huge and tiny numbers in a short, tidy way.

By the end of this lesson you can:

say whether a number is in standard form, and fix it if it is not
change a large or small number into standard form
change a number in standard form back into an ordinary number
compare and order numbers written in standard form
read standard form on the TI-Nspire screen (the “E” display)

💡 AA Paper 1 has no calculator

In AA, Paper 1 is calculator-free, so the by-hand working is what earns the marks there. Use the TI-Nspire steps for Paper 2, and to check your hand answers.

🧑🏾 Imagine this…

A grain of sand is about 0.000 45 m wide. The distance to the Sun is about 150 000 000 000 m. Count the zeros: it is easy to lose your place, and one missing zero changes the number completely. Scientists, engineers and even phone companies need a way to write such numbers short and safe. That way is standard form.

Standard form

a number written as a × 10^k with 1 ≤ a < 10 and k a whole number (positive, negative or zero). Also called scientific notation.

Coefficient (a)

the “front” number. It must have exactly one non-zero digit before the decimal point: 1.5, 7, 9.99 are fine; 12.5 and 0.8 are not.

Exponent (k)

the power of ten. It tells you how many places the decimal point has moved.

Power of ten

10³ = 1000, 10⁰ = 1, 10⁻² = 0.01. A negative power makes a number small, not negative.

★ How to change a number into standard form

Put the decimal point after the first non-zero digit. That gives you a.
Count how many places the decimal point moved. That count is the size of k.
Number bigger than 10 (point moved left) → k is positive. Number smaller than 1 (point moved right) → k is negative.
Number from 1 to 10 → k = 0. Example: 6.2 = 6.2 × 10⁰.
Going back: multiply by $10^k$ means move the decimal point k places right (k positive) or $|k|$ places left (k negative).

⚠️ Mistakes that lose marks

25 × 10³ is NOT standard form because 25 is not less than 10. Write it as 2.5 × 10⁴. The same goes for 0.62 × 10⁵ (the front number is less than 1).
A negative exponent does not give a negative number. 3 × 10⁻⁴ = 0.0003, which is positive.
Do not count the zeros you see; count how far the decimal point moves. 0.0072 → 7.2 × 10⁻³ (three jumps), not 10⁻².

💡 Check your sign of k with common sense

Ask: is my number big or small? A number like 45 000 000 is big, so k must be positive. A number like 0.000 08 is small, so k must be negative. If your answer disagrees with your common sense, recount.

🧪 Try it yourselfStandard form or not?

Check both rules: the front number is at least 1 and less than 10, and the power of ten is a whole number.

Worked examples

Example 1.1A large number into standard form

A mobile network has 2 460 000 customers. Write in the form a × 10^k where 1 ≤ a < 10 and k ∈ ℤ:

(a) the number of customers
(b) 580 000 000
(c) 7300

🧠 Think it through

Find the first non-zero digit: for 2 460 000 it is the 2. Put the decimal point right after it: 2.46.
Count how far the point moved from the end of 2 460 000 to just after the 2: 6 places.
The number is big, so k is positive: 2.46 × 10⁶. Do the same for (b) and (c).

✍️ By hand

2\,460\,000 = 2.46 \times 10^{6}

580\,000\,000 = 5.8 \times 10^{8}

(8 places)

7300 = 7.3 \times 10^{3}

(3 places)

🔢 On your TI-Nspire CX II

To check, type 2.46*10^6 and press enter ↵💻 Enter. The calculator gives back 2460000, so you got it right.

Typing a × 10^k turns it back into an ordinary number

✅ Answer (a) 2.46 × 10⁶ (b) 5.8 × 10⁸ (c) 7.3 × 10³

Example 1.2A small number into standard form

Write in standard form:

(a) 0.000 081
(b) 0.0302
(c) the mass of a dust grain, 0.000 000 75 kg

🧠 Think it through

Put the decimal point after the first non-zero digit. In 0.000 081 that is after the 8: 8.1.
The point moved to the right by 5 places, because the number is small. So k = −5.
Check with common sense: 10⁻⁵ = 0.000 01, and 8.1 of those is about 0.000 08. ✓

✍️ By hand

0.000\,081 = 8.1 \times 10^{-5}

0.0302 = 3.02 \times 10^{-2}

0.000\,000\,75 = 7.5 \times 10^{-7}

🔢 On your TI-Nspire CX II

Type 8.1*10^(-5) and press enter ↵💻 Enter. Use the (−)💻 type - key for the minus sign in the power, and brackets around the power.

Very small answers switch to E notation: 8.1E-5 means 8.1 × 10⁻⁵

✅ Answer (a) 8.1 × 10⁻⁵ (b) 3.02 × 10⁻² (c) 7.5 × 10⁻⁷ kg

Example 1.3From standard form back to an ordinary number

Write as ordinary numbers:

(a) 3.07 × 10⁴
(b) 8.6 × 10⁻³
(c) 4.15 × 10⁻⁵
(d) 5 × 10⁶

🧠 Think it through

Positive k: move the decimal point right k places, filling empty places with zeros.
Negative k: move the decimal point left |k| places, filling with zeros (and a 0 before the point).

✍️ By hand

3.07 \times 10^{4}

: move 4 places right → 30 700

8.6 \times 10^{-3}

: move 3 places left → 0.0086

4.15 \times 10^{-5}

: move 5 places left → 0.000 0415

5 \times 10^{6}

: move 6 places right → 5 000 000

🔢 On your TI-Nspire CX II

Type each one into a Calculator page to check. The calculator switches to E notation (4.15E-5) when a number is very small, so 4.15E-5 on the screen means 4.15 × 10⁻⁵.

✅ Answer (a) 30 700 (b) 0.0086 (c) 0.000 0415 (d) 5 000 000

Example 1.4Fixing numbers that are not in standard form

Each of these is almost in standard form. Write each one correctly:

(a) 25 × 10³
(b) 0.62 × 10⁵
(c) 480 × 10⁻⁶
(d) 0.007 × 10⁻²

🧠 Think it through

First turn the front number into standard form on its own.
Then add the two powers of ten: 10^m × 10ⁿ = 10^m+n.

✍️ By hand

25 \times 10^{3} = (2.5 \times 10^{1}) \times 10^{3} = 2.5 \times 10^{4}

0.62 \times 10^{5} = (6.2 \times 10^{-1}) \times 10^{5} = 6.2 \times 10^{4}

480 \times 10^{-6} = (4.8 \times 10^{2}) \times 10^{-6} = 4.8 \times 10^{-4}

0.007 \times 10^{-2} = (7 \times 10^{-3}) \times 10^{-2} = 7 \times 10^{-5}

🔢 On your TI-Nspire CX II

Check any one by typing the original and the new version. For example 480*10^(-6) and 4.8*10^(-4) should give the same value.

Same value, so the rewrite is correct

✅ Answer (a) 2.5 × 10⁴ (b) 6.2 × 10⁴ (c) 4.8 × 10⁻⁴ (d) 7 × 10⁻⁵

A big front number needs a bigger power, a small front number needs a smaller power. The value stays the same.

Example 1.5Ordering numbers in standard form

Put these in order, smallest first: 3.2 × 10⁵, 4.1 × 10⁻², 9.8 × 10⁴, 2.7 × 10⁵, 6 × 10⁻³.

🧠 Think it through

Compare the powers of ten first. A bigger power always wins, because the front numbers are all between 1 and 10.
Only when two numbers have the same power do you compare the front numbers.
Powers here: −2, −3, 4, 5, 5. Smallest power is −3, then −2, then 4, then the two 5s.

🔢 On your TI-Nspire CX II

To see them as ordinary numbers, type each in a Calculator page.
The two numbers with 10⁵: compare 2.7 and 3.2. So 2.7 × 10⁵ is smaller.

The ordinary numbers agree with the order

✅ Answer 6 × 10⁻³, 4.1 × 10⁻², 9.8 × 10⁴, 2.7 × 10⁵, 3.2 × 10⁵

Notice that 9.8 × 10⁴ has a bigger front number than 2.7 × 10⁵, but it is still the smaller number. The power decides first.

Challenge questions

⭐ Challenge 1.6Atoms and sandPaper 1 style

The width of an atom is 0.000 000 000 28 m. A grain of sand is 4.5 × 10⁻⁴ m wide.

(a) Write the width of the atom in standard form.
(b) Find how many atoms placed side by side would be as wide as the grain of sand. Give your answer in the form a × 10^k, with a to 3 significant figures.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Move the decimal point after the 2. Count the jumps to the right: 10.
$\text{Number of atoms} = \dfrac{\text{width of sand}}{\text{width of atom}}$ . Type it in with powers of ten.
Write the calculator’s answer in standard form and round the front number to 3 s.f.

✍️ By hand

0.000\,000\,000\,28 = 2.8 \times 10^{-10}

🔢 On your TI-Nspire CX II

Type 4.5*10^(-4)/(2.8*10^(-10)) and press enter ↵💻 Enter. Put brackets round the whole bottom line.

1 607 142.857… → 1.61 × 10⁶ (3 s.f.)

✅ Answer (a) 2.8 × 10⁻¹⁰ m (b) 1.61 × 10⁶ atoms

⭐ Challenge 1.7Small things in the labPaper 2 style

A science table gives these sizes: red blood cell 0.000 0075 m, virus 1.2 × 10⁻⁷ m, human hair 8 × 10⁻⁵ m, bacterium 2.5 × 10⁻⁶ m.

(a) Write the size of the red blood cell in standard form.
(b) Write all four sizes in order, smallest first.
(c) How many times wider is the hair than the virus? Give your answer in standard form to 3 significant figures.
(d) A line of 50 000 viruses is laid end to end. Find its length in metres in standard form, and say whether it is longer or shorter than the hair.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

(a) The point moves 6 places to the right, so k = −6.
(b) Compare powers first. Write the cell as 7.5 × 10⁻⁶ and the bacterium as 2.5 × 10⁻⁶: same power, so compare 2.5 and 7.5.
(c) “How many times” means divide: $\dfrac{\text{hair}}{\text{virus}}$ .
(d) Multiply 50 000 = 5 × 10⁴ by 1.2 × 10⁻⁷, then compare with 8 × 10⁻⁵.

✍️ By hand

(a)

0.000\,0075 = 7.5 \times 10^{-6}

(d)

5 \times 10^{4} \times 1.2 \times 10^{-7} = 6 \times 10^{-3}

🔢 On your TI-Nspire CX II

Type 8*10^(-5)/(1.2*10^(-7)) enter ↵💻 Enter for part (c).
Type 5*10^4*1.2*10^(-7) enter ↵💻 Enter for part (d), then compare 0.006 with 0.00008.

✅ Answer (a) 7.5 × 10⁻⁶ m (b) 1.2 × 10⁻⁷, 2.5 × 10⁻⁶, 7.5 × 10⁻⁶, 8 × 10⁻⁵ m (c) 6.67 × 10² times (d) 6 × 10⁻³ m; this is longer than the hair (8 × 10⁻⁵ m).

IB DP Mathematics: Analysis and Approaches (AA) — SL and HL Exponents & logs 1 · Standard form: writing very big and very small numbers

IB DP Mathematics: Analysis and Approaches (AA) — SL and HL

Standard form: writing very big and very small numbers

Worked examples

Challenge questions