Number, Year 9: standard form and powers

Standard form

HintIt is the agreed way scientists write very large and very small measurements.

The whyThe Earth is about 1.5 × 10⁸ km from the Sun. Written this way a huge number takes only a few characters, and the power of ten tells you its size at a glance.

7.3 × 10⁶

HintPut the decimal point after the first digit, then count how many places the digits have moved.

The why7.3 has to be multiplied by 10 six times to get back to 7 300 000, so the power is 6. The power counts place-value jumps, not zeros: this number has only five zeros.

306000

HintMultiplying by ten five times moves every digit five places to the left.

The why3.06 × 10 = 30.6, and four more multiplications by 10 give 306 000. Two of the five places are taken up by the digits 0 and 6, so only three zeros are added.

1.2 × 10⁵

HintCompare the powers of ten before you look at the numbers in front of them.

The why9.8 × 10⁴ is 98 000 and 1.2 × 10⁵ is 120 000. In standard form the power of ten decides the size first; the front numbers only matter when the powers are equal.

1000

HintEach extra power of ten makes a number ten times bigger; count how many extra there are.

The whyThe front numbers match, so only the powers differ, by 8 − 5 = 3. Three extra powers of ten means 10 × 10 × 10 times larger.

So that every number can be written in only one way

Hint45 × 10⁵, 4.5 × 10⁶ and 0.45 × 10⁷ all have the same value. What would go wrong if all three were allowed?

The whyWithout the rule one number could be written in endless ways, and you would have to convert before comparing. With it, each number has a single standard form, and the power of ten alone tells you roughly how big it is.

5.7 × 10⁻⁴

HintCount how many places the digits must move for the first non-zero digit to reach the units column.

The why5.7 must be divided by 10 four times to give 0.00057, so the power is −4. The negative power counts the places moved, which is one more than the number of zeros after the decimal point.

0.0064

HintA negative power of ten tells you to divide by ten that many times.

The why6.4 ÷ 10 = 0.64, dividing again gives 0.064, and a third time gives 0.0064. Each division moves the digits one place to the right.

7 × 10⁻⁵

HintAsk which of the two has been divided by ten more times.

The why7 × 10⁻⁵ is 0.00007 and 2 × 10⁻³ is 0.002. With negative powers, the further below zero the power is, the smaller the number: −5 is less than −3.

10⁻³

HintA thousand is three tens multiplied together; dividing by them instead changes the sign of the power.

The why10⁻³ means 1 ÷ 10 ÷ 10 ÷ 10, which is 1/1000 or 0.001. In the same way 10⁻¹ is one tenth and 10⁻² is one hundredth.

dividing

HintPositive powers of ten make a number bigger; negative ones do the opposite job.

The whyGoing down the powers of ten, each step is ten times smaller: 10² = 100, 10¹ = 10, 10⁰ = 1, 10⁻¹ = 0.1. A negative power simply carries on that pattern.

1

HintFollow the pattern 7³, 7², 7¹ downwards: what do you divide by at each step?

The whyEach step down the powers divides by 7: 343, 49, 7, and then 7 ÷ 7. The same argument works for every base except 0, so any non-zero number to the power 0 is 1.

1/25

HintWork out the positive power first, then ask what the minus sign tells you to do with it.

The why5² = 25, and a negative power means one divided by that, so 5⁻² = 1/25. Going down the pattern 25, 5, 1, 1/5, 1/25 gives the same answer.

1; 1/3; 1/9

HintKeep doing to each value exactly what took you from 9 to 3.

The whyNegative and zero powers are not new rules to memorise. They are what you get by carrying on the divide-by-the-base pattern past the power 1.

reciprocal

HintIt means one divided by the number, and the same word describes turning a fraction upside down.

The why4⁻¹ = 1/4 and 10⁻¹ = 1/10. For a fraction, taking the reciprocal turns it upside down, so (2/3)⁻¹ = 3/2.

2⁻⁴

HintFirst find which positive power of 2 makes 16.

The why16 = 2 × 2 × 2 × 2 = 2⁴, and one over a power is written with a negative index, so 1/16 = 2⁻⁴.