C1.8; E1.8 - Calculating in standard form

C1.8; E1.8 - Calculating in standard form

You can multiply tiny and huge numbers without writing out all their zeros. You will use index laws for products and quotients, match scales for sums and differences, and check calculator displays. Core calculates with standard form on Paper 3 only; Extended also calculates with standard form on non-calculator Paper 2.

Multiplying and dividing standard form

Is (3×105)×(4×10−2)(3\times10^5)\times(4\times10^{-2}) closer to one thousand or one million? Separate the ordinary-number calculation from the powers of ten and the size becomes much easier to see.

You already know that multiplication is unchanged when you regroup its factors. Therefore, multiply the coefficients together, then multiply the powers of ten. For division, divide the coefficients and divide the powers of ten.

Products and quotients in standard form

For positive coefficients A,BA,B and integer indices m,nm,n,

(A×10m)(B×10n)=(AB)×10m+nA×10mB×10n=AB×10m−n\begin{aligned} (A\times10^m)(B\times10^n)&=(AB)\times10^{m+n}\\ \frac{A\times10^m}{B\times10^n}&=\frac AB\times10^{m-n} \end{aligned}

Here ABAB means A×BA\times B. The divisor must be nonzero. After calculating, adjust the result so its coefficient is at least 11 and smaller than 1010.

Not given in the List of formulas: learn these methods.

That final adjustment is called normalising. For example, 12×103=1.2×10412\times10^3=1.2\times10^4: dividing the coefficient by ten requires multiplying the scale by ten, so the value stays fixed.

Worked example 1: a product needing adjustment

Question. Work out (3×105)×(4×10−2)(3\times10^5)\times(4\times10^{-2}). Give your answer in standard form.

Plan. Multiply coefficients, add indices, then normalise.

(3×105)(4×10−2)=(3×4)×105+(−2)group matching factors=12×103multiply and add=1.2×104coefficient must be below ten\begin{aligned} (3\times10^5)(4\times10^{-2}) &=(3\times4)\times10^{5+(-2)} && \text{group matching factors}\\ &=12\times10^3 && \text{multiply and add}\\ &=1.2\times10^4 && \text{coefficient must be below ten} \end{aligned}

Answer. 1.2×1041.2\times10^4.

Check. In ordinary numbers, 300 000×0.04300\,000\times0.04 is four hundredths of 300 000300\,000, which is 12 00012\,000.

Keep both numbers, but change multiplication to division. Subtract the indices and divide the coefficients.

Worked example 2: a quotient needing adjustment

Question. Work out (3×105)÷(4×10−2)(3\times10^5)\div(4\times10^{-2}). Give your answer in standard form.

Plan. Divide coefficients, subtract indices, then normalise.

3×1054×10−2=34×105−(−2)separate coefficient and scale=0.75×107subtracting a negative adds=7.5×106coefficient must be at least one\begin{aligned} \frac{3\times10^5}{4\times10^{-2}} &=\frac34\times10^{5-(-2)} && \text{separate coefficient and scale}\\ &=0.75\times10^7 && \text{subtracting a negative adds}\\ &=7.5\times10^6 && \text{coefficient must be at least one} \end{aligned}

Answer. 7.5×1067.5\times10^6.

Check. Multiply back: 7 500 000×0.04=300 0007\,500\,000\times0.04=300\,000, the original dividend.

Watch out

In a quotient, 105÷10−2=105−(−2)=10710^5\div10^{-2}=10^{5-(-2)}=10^7, not 10310^3. Write the subtraction with brackets before simplifying the signs.

The opening product is 12 00012\,000, closer to one thousand than one million. The negative power in the second factor reduces the first number.

Adding and subtracting standard form

You can add three thousands to four thousands because the unit is the same. To add numbers in standard form, first express them with the same power of ten. Then add the coefficients, leaving that common scale unchanged.

For example, 3.5×105=0.35×1063.5\times10^5=0.35\times10^6. The intermediate coefficient 0.350.35 need not satisfy the standard-form condition: it is a useful equivalent expression. The final answer must satisfy the condition if standard form is requested.

A common scale for addition and subtraction

With the same power of ten,

(A×10n)+(B×10n)=(A+B)×10n(A×10n)−(B×10n)=(A−B)×10n\begin{aligned} (A\times10^n)+(B\times10^n)&=(A+B)\times10^n\\ (A\times10^n)-(B\times10^n)&=(A-B)\times10^n \end{aligned}

Match the powers first. Then add or subtract the coefficients; do not add or subtract the indices.

Not given in the List of formulas: learn these methods.

Choose a common scale that makes the arithmetic comfortable. Converting both numbers to ordinary notation is also a valid method.

Worked example 3: adding different scales

Question. Work out (4.2×106)+(3.5×105)(4.2\times10^6)+(3.5\times10^5). Give your answer in standard form.

Plan. Write both amounts as multiples of 10610^6, then add.

3.5×105=0.35×106keep the value while changing scale(4.2×106)+(3.5×105)=(4.2+0.35)×106add multiples of the same scale=4.55×106coefficient already valid\begin{aligned} 3.5\times10^5&=0.35\times10^6 && \text{keep the value while changing scale}\\ (4.2\times10^6)+(3.5\times10^5) &=(4.2+0.35)\times10^6 && \text{add multiples of the same scale}\\ &=4.55\times10^6 && \text{coefficient already valid} \end{aligned}

Answer. 4.55×1064.55\times10^6.

Check. 4 200 000+350 000=4 550 0004\,200\,000+350\,000=4\,550\,000.

Change only addition to subtraction. The common-scale step stays the same.

Worked example 4: subtracting different scales

Question. Work out (4.2×106)−(3.5×105)(4.2\times10^6)-(3.5\times10^5). Give your answer in standard form.

Plan. Use the same power 10610^6, then subtract the coefficients.

3.5×105=0.35×106match the scales(4.2×106)−(3.5×105)=(4.2−0.35)×106subtract like-sized units=3.85×106coefficient already valid\begin{aligned} 3.5\times10^5&=0.35\times10^6 && \text{match the scales}\\ (4.2\times10^6)-(3.5\times10^5) &=(4.2-0.35)\times10^6 && \text{subtract like-sized units}\\ &=3.85\times10^6 && \text{coefficient already valid} \end{aligned}

Answer. 3.85×1063.85\times10^6.

Check. Add back the subtracted value: 3.85×106+0.35×106=4.2×1063.85\times10^6+0.35\times10^6=4.2\times10^6.

Watch out

(4.2×106)+(3.5×105)=7.7×1011(4.2\times10^6)+(3.5\times10^5)=7.7\times10^{11} is wrong. Adding indices is a multiplication law. A sum needs the same scale first.

Match scales before subtracting this fresh pair.

Normalising and checking calculator results

A calculator may display a power of ten with a small raised index or with an E. A display such as 2.4 E−072.4\ \mathrm{E}{-07} means 2.4×10−72.4\times10^{-7}; the E does not mean an extra multiplication by a variable.

Use your calculator's scientific-notation entry key, often labelled EXP or ×10x\times10^x, to enter the coefficient and index. Use its negative-number key for a negative index. Alternatively, enter a bracketed product such as (7.2×10−4)(7.2\times10^{-4}). For a quotient, bracket the whole numerator and the whole denominator so the intended grouping is clear.

Before accepting a display, ask three questions: Is the sign sensible? Is the power of ten plausible? Is the coefficient in the required interval? Dividing a positive number by a positive number greater than 11 makes it smaller. Dividing by a positive number below 11 makes it larger.

Worked example 5: reading a small calculator result

Question. Calculate 7.2×10−43×103\dfrac{7.2\times10^{-4}}{3\times10^3}. Give your answer in standard form.

Plan. Enter the two complete numbers as a quotient, then check the display using index laws.

7.2×10−43×103=(7.2÷3)×10−4−3check coefficient and index=2.4×10−7normalised result2.4 E−07=2.4×10−7interpret a possible display\begin{aligned} \frac{7.2\times10^{-4}}{3\times10^3} &=(7.2\div3)\times10^{-4-3} && \text{check coefficient and index}\\ &=2.4\times10^{-7} && \text{normalised result}\\ 2.4\ \mathrm{E}{-07}&=2.4\times10^{-7} && \text{interpret a possible display} \end{aligned}

Answer. 2.4×10−72.4\times10^{-7}.

Check. In ordinary notation, 0.00000024×3000=0.000720.00000024\times3000=0.00072, recovering the numerator.

Change only the sign of the denominator's index. Now the divisor is below 11, so the quotient must be larger than the numerator.

Worked example 6: a small divisor changes the size

Question. Calculate 7.2×10−43×10−3\dfrac{7.2\times10^{-4}}{3\times10^{-3}}. Give your answer in standard form.

Plan. Enter the complete quotient and check the negative-index subtraction.

7.2×10−43×10−3=(7.2÷3)×10−4−(−3)subtract the denominator’s index=2.4×10−1simplify the index=0.24ordinary decimal display\begin{aligned} \frac{7.2\times10^{-4}}{3\times10^{-3}} &=(7.2\div3)\times10^{-4-(-3)} && \text{subtract the denominator's index}\\ &=2.4\times10^{-1} && \text{simplify the index}\\ &=0.24 && \text{ordinary decimal display} \end{aligned}

Answer. 2.4×10−12.4\times10^{-1}.

Check. 0.24×0.003=0.000720.24\times0.003=0.00072. The quotient is larger than 0.000720.00072, as expected when dividing by 0.0030.003.

A correct decimal display may still need rewriting to answer “in standard form”. If your written calculation gives 0.24×1000.24\times10^0, normalise it to 2.4×10−12.4\times10^{-1}. Moving the coefficient up by a factor of ten moves the power down by one.

Watch out

Do not copy 2.4 E−072.4\ \mathrm{E}{-07} as 2.4×(−7)2.4\times(-7), and do not drop the minus sign. Write 2.4×10−72.4\times10^{-7}. If this disagrees with your size check, inspect the entry rather than trusting the screen.

Use the scale to check the calculator's result.

Quick Recap

  • For products, multiply coefficients and add indices. For quotients, divide coefficients and subtract indices.
  • For sums and differences, match the powers of ten first.
  • Normalise the final coefficient to 1≤A<101\le A<10 when standard form is requested.
  • Read a calculator's E notation as a power of ten, and check its magnitude independently.

The opening product is 1.2×104=12 0001.2\times10^4=12\,000, much closer to one thousand than one million.