3.02a, 3.02b - Standard form

3.02a, 3.02b - Standard form

Standard form keeps very large and very small numbers readable without losing their place value. In 3.02a you will convert and compare these numbers; in 3.02b you will calculate with them, using a calculator and non-calculator methods.

Writing standard form

How can 0.0000720.000072 be written with a coefficient of 7.27.2 without changing its value? The missing information is a power of ten.

You already know that 103=100010^3=1000 and 10−3=0.00110^{-3}=0.001. So 7.2×1037.2\times10^3 is 72007200, while 7.2×10−37.2\times10^{-3} is 0.00720.0072. The sign of the exponent tells you whether to multiply or divide by a large power of ten; it does not make the answer negative.

Standard form

A positive number is in standard form when written as a×10na\times10^n, where 1≤a<101\le a<10 and nn is an integer. The number aa is the coefficient. For signed nonzero numbers, use 1≤∣a∣<101\le|a|<10; for example, −3.4×105-3.4\times10^5. Zero is written simply as 00: it cannot have a coefficient of magnitude at least 11.

The restriction makes the representation consistent. 72×10372\times10^3 has the right value for 7200072000, but its coefficient is too large for standard form. It becomes 7.2×1047.2\times10^4.

Worked example 1: a large number

Question. Write 7200072000 in standard form, then convert your answer back.

Plan. Choose a coefficient between 11 and 1010, then find the power of ten needed to restore the value.

72000÷7.2=10000find the scale factor10000=104write it as a power72000=7.2×104standard form\begin{aligned} 72000\div7.2&=10000 &&\text{find the scale factor}\\ 10000&=10^4 &&\text{write it as a power}\\ 72000&=7.2\times10^4 &&\text{standard form} \end{aligned}

Answer. 7.2×1047.2\times10^4; multiplying 7.27.2 by 1000010000 converts back to 7200072000.

Check. The exponent 44 places the leading digit in the ten-thousands position, as in the original number.

Now change only the size of the original number, keeping the significant digits. The power of ten must now shrink the coefficient.

Worked example 2: a small number

Question. Write 0.0000720.000072 in standard form, then convert your answer back.

Plan. Keep coefficient 7.27.2 and divide it by the appropriate power of ten.

0.000072=7.2100000restore the decimal place value=7.2×10−5use a reciprocal power7.2×10−5=7.2×0.00001=0.000072convert back\begin{aligned} 0.000072&=\frac{7.2}{100000} &&\text{restore the decimal place value}\\ &=7.2\times10^{-5} &&\text{use a reciprocal power}\\ 7.2\times10^{-5}&=7.2\times0.00001=0.000072 &&\text{convert back} \end{aligned}

Answer. 7.2×10−57.2\times10^{-5}.

Check. 7×10−5=0.000077\times10^{-5}=0.00007, close to the original number. A positive exponent would give a large number instead.

This answers the opening question. The coefficient is 7.27.2, and the factor 10−510^{-5} preserves the small value.

Watch out

7.2×10−57.2\times10^{-5} is positive. A negative exponent describes a reciprocal power of ten; a negative number needs a minus sign in its coefficient.

Use the coefficient to reconstruct the original decimal as a check.

Comparing very large and small numbers

For positive numbers already in standard form, compare the exponents first, then compare coefficients when the exponents match.

Why does that work? All numbers with exponent 55 lie from 100000100000 up to, but below, 10000001000000. All with exponent 66 start at 10000001000000. The coefficient restriction stops the ranges overlapping. This shortcut needs normalised coefficients and positive numbers.

Worked example 3: different exponents

Question. Put 4.2×1064.2\times10^6, 9.1×1059.1\times10^5 and 3.8×1063.8\times10^6 in ascending order.

Plan. Compare the powers of ten, then the two coefficients sharing a power.

105<106the first scale is smaller3.8<4.2compare equal-scale coefficients9.1×105<3.8×106<4.2×106\begin{aligned} 10^5&<10^6 &&\text{the first scale is smaller}\\ 3.8&<4.2 &&\text{compare equal-scale coefficients}\\ 9.1\times10^5&<3.8\times10^6<4.2\times10^6 \end{aligned}

Answer. 9.1×1059.1\times10^5, 3.8×1063.8\times10^6, 4.2×1064.2\times10^6.

Check. As ordinary numbers these are 910000910000, 38000003800000 and 42000004200000, in the same order.

Change only the exponents to small-number scales. Negative exponents still follow number order: −6<−5-6<-5.

Worked example 4: negative exponents

Question. Put 4.2×10−54.2\times10^{-5}, 9.1×10−69.1\times10^{-6} and 3.8×10−53.8\times10^{-5} in ascending order.

Plan. Treat the exponent −6-6 as a smaller positive scale than −5-5, then compare the equal-scale coefficients.

10−6=0.000001<0.00001=10−59.1×10−6<3.8×10−5<4.2×10−5\begin{aligned} 10^{-6}&=0.000001<0.00001=10^{-5}\\ 9.1\times10^{-6}&<3.8\times10^{-5}<4.2\times10^{-5} \end{aligned}

Answer. 9.1×10−69.1\times10^{-6}, 3.8×10−53.8\times10^{-5}, 4.2×10−54.2\times10^{-5}.

Check. 0.0000091<0.000038<0.0000420.0000091<0.000038<0.000042, agreeing with the exponent method.

If a coefficient is outside the permitted range, normalise it first: 42×10−6=4.2×10−542\times10^{-6}=4.2\times10^{-5}. For negative numbers compare their magnitudes and reverse the positive order; every negative number is below zero, and every positive number above it.

Watch out

A coefficient of 9.19.1 does not automatically beat one of 3.83.8. The scale matters: 9.1×10−6<3.8×10−59.1\times10^{-6}<3.8\times10^{-5}.

Normalise all three numbers before comparing them.

Calculator calculations

A calculator's scientific-notation entry key enters a power of ten as part of a number.

The key may be labelled EXP, EE or ×10x\times10^x. For example, entering coefficient 6.46.4, that key, then exponent 77 represents 6.4×1076.4\times10^7. Use the negative-number key for a negative exponent. The exact key sequence depends on the calculator; check its displayed expression before evaluating.

A display such as 3.2E103.2\text{E}10 means 3.2×10103.2\times10^{10}. The letter E is display shorthand here, not an extra number to multiply by. If using ordinary multiplication and powers instead, bracket each complete standard-form number and the complete denominator.

Worked example 5: entering a product

Question. Use a calculator to work out (6.4×107)(2×103)(6.4\times10^7)(2\times10^3). Give the answer in standard form.

Plan. Enter both complete numbers, multiply, then interpret the display.

(6.4×107)(2×103)=12.8×1010multiply coefficients and scales=1.28×1011normalise the coefficient\begin{aligned} (6.4\times10^7)(2\times10^3)&=12.8\times10^{10} &&\text{multiply coefficients and scales}\\ &=1.28\times10^{11} &&\text{normalise the coefficient} \end{aligned}

Answer. 1.28×10111.28\times10^{11}. A display of 1.28E111.28\text{E}11 represents this answer.

Check. 6.4×2=12.86.4\times2=12.8, and 107×103=101010^7\times10^3=10^{10}, independently confirming the calculator result.

Change only multiplication to division. The denominator must remain one complete number.

Worked example 6: entering a quotient

Question. Use a calculator to work out 6.4×1072×103\frac{6.4\times10^7}{2\times10^3}. Give the answer in standard form.

Plan. Enter the whole denominator in brackets, or use scientific-notation entry for it.

6.4×1072×103=6.42×107−3divide both parts=3.2×104coefficient already normalised\begin{aligned} \frac{6.4\times10^7}{2\times10^3}&=\frac{6.4}{2}\times10^{7-3} &&\text{divide both parts}\\ &=3.2\times10^4 &&\text{coefficient already normalised} \end{aligned}

Answer. 3.2×1043.2\times10^4, or ordinary display 3200032000.

Check. (3.2×104)(2×103)=6.4×107(3.2\times10^4)(2\times10^3)=6.4\times10^7, restoring the numerator.

The working above also shows how to verify a machine calculation without repeating the same key presses. On calculator papers you may still need to show working when asked.

Watch out

Typing 6.4×107÷2×1036.4\times10^7\div2\times10^3 as separate operations may multiply by 10310^3 after dividing by 22. Enter (6.4×107)÷(2×103)(6.4\times10^7)\div(2\times10^3) or use the scientific-notation key for each number.

Check the denominator and convert the display into standard form.

Non-calculator operations

Multiplication and division use index laws; addition and subtraction need a common power of ten.

For multiplication, multiply coefficients and add exponents. For division by a nonzero number, divide coefficients and subtract exponents. For a sum or difference, first write both numbers as multiples of the same power of ten, then combine the coefficients. Finally restore standard form if necessary.

Worked example 7: all four operations

Question. Let A=6×105A=6\times10^5 and B=2×104B=2\times10^4. Work out A+BA+B, A−BA-B, ABAB and A/BA/B in standard form without a calculator.

Plan. Use a common scale for addition/subtraction and index laws for multiplication/division.

Since B=0.2×105B=0.2\times10^5,

A+B=(6+0.2)×105=6.2×105A−B=(6−0.2)×105=5.8×105AB=12×109=1.2×1010add exponents, then normaliseA/B=3×105−4=3×101subtract exponents\begin{aligned} A+B&=(6+0.2)\times10^5=6.2\times10^5\\ A-B&=(6-0.2)\times10^5=5.8\times10^5\\ AB&=12\times10^9=1.2\times10^{10} &&\text{add exponents, then normalise}\\ A/B&=3\times10^{5-4}=3\times10^1 &&\text{subtract exponents} \end{aligned}

Answer. 6.2×1056.2\times10^5, 5.8×1055.8\times10^5, 1.2×10101.2\times10^{10} and 3×1013\times10^1, respectively.

Check. 600000+20000=620000600000+20000=620000 and 600000−20000=580000600000-20000=580000. Also (3×101)(2×104)=6×105(3\times10^1)(2\times10^4)=6\times10^5; dividing the product by BB likewise restores AA.

Now change only the exponent of BB. It becomes larger than AA, so the difference becomes negative.

Worked example 8: a change of scale

Question. Let A=6×105A=6\times10^5 and B=2×106B=2\times10^6. Find the same four results in standard form.

Plan. Use A=0.6×106A=0.6\times10^6 for the sum and difference, then apply the index laws.

A+B=(0.6+2)×106=2.6×106A−B=(0.6−2)×106=−1.4×106AB=12×1011=1.2×1012normalise after multiplyingA/B=3×10−1since 5−6=−1\begin{aligned} A+B&=(0.6+2)\times10^6=2.6\times10^6\\ A-B&=(0.6-2)\times10^6=-1.4\times10^6\\ AB&=12\times10^{11}=1.2\times10^{12} &&\text{normalise after multiplying}\\ A/B&=3\times10^{-1} &&\text{since }5-6=-1 \end{aligned}

Answer. 2.6×1062.6\times10^6, −1.4×106-1.4\times10^6, 1.2×10121.2\times10^{12} and 3×10−13\times10^{-1}.

Check. 600000−2000000=−1400000600000-2000000=-1400000. Also 0.3×2000000=6000000.3\times2000000=600000, checking the quotient, and (1.2×1012)/(2×106)=6×105(1.2\times10^{12})/(2\times10^6)=6\times10^5, checking the product.

The useful connection is that standard form separates two jobs: coefficients hold the significant digits, and powers of ten hold the scale. Changing one exponent by 22 multiplies that input by 100100, without changing its significant digits.

Watch out

Do not add exponents in a sum: 6×105+2×104≠8×1096\times10^5+2\times10^4\ne8\times10^9. Adding exponents is a multiplication rule. For addition, first match the powers of ten.

Treat the operations separately, and normalise each result.

Quick Recap

  • Standard form separates a normalised coefficient from an integer power of ten.
  • For positive normalised numbers, compare exponents before coefficients.
  • Read calculator E displays as powers of ten and bracket complete denominators.
  • Multiply/divide using index laws; add/subtract using a common power of ten.

The opening decimal is 7.2×10−57.2\times10^{-5}: compact notation keeps its value exactly.