Number, Year 10: standard form, recurring decimals and error intervals

6 × 10⁹

HintDeal with the front numbers and the powers of ten separately.

The whyMultiply the front numbers: 3 × 2 = 6. Multiply the powers of ten by adding the indices: 10⁴ × 10⁵ = 10⁹. The order of a multiplication can be rearranged, which is why the two parts can be handled apart.

6

HintMaking the front number ten times smaller has to be balanced in the power of ten.

The why48 = 4.8 × 10, so 48 × 10⁵ = 4.8 × 10 × 10⁵ = 4.8 × 10⁶. The value has not changed (4 800 000 both ways); only how it is shared between the front number and the power.

2.5 × 10⁴

HintDivide the front numbers, divide the powers of ten, then check the front number is at least 1.

The why2 ÷ 8 = 0.25 and 10⁸ ÷ 10³ = 10⁵, giving 0.25 × 10⁵. That is not standard form, because 0.25 is less than 1. Make the front number ten times bigger and the power one smaller: 2.5 × 10⁴ (which is 25 000).

3.4 × 10⁵

HintThe two numbers have different powers of ten, so the front numbers cannot simply be combined.

The whyWrite both with the same power of ten, or as ordinary numbers: 300 000 + 40 000 = 340 000 = 3.4 × 10⁵. Adding the front numbers only works when the powers of ten already match.

42000000

HintThe power says how many places the digits move, not how many zeros to write.

The why4.2 × 10⁷ means 4.2 multiplied by 10 seven times. The digits move seven places, and one of those places is filled by the 2, so six zeros follow: 42 000 000. Other calculators show the same thing as 4.2 ×10 with a small raised 7.

A recurring decimal

HintThe word describes something that keeps coming back, like a dream you have again and again.

The why1/3 = 0.3333… and 3/11 = 0.272727… are examples. A decimal that stops, such as 0.375, is called terminating. Every fraction gives one kind or the other.

0.166666

HintOnly the digit under the dot repeats.

The whyA dot above a single digit means that digit repeats for ever, and digits with no dot appear once. So the 1 appears once and the 6 goes on: 0.16666… This is the decimal for 1/6.

last

HintThe two dots work like a pair of bookends.

The whyFor 0.142857142857… the dots go above the 1 and the 7, marking where the repeating block starts and stops. Dots above the 3 and the 2 of 0.312 would mean 0.312312312…

0.181818…, with the digits 1 and 8 recurring

HintCarry out 2 ÷ 11 as a short division and watch for a remainder you have already had.

The why20 ÷ 11 = 1 remainder 9, then 90 ÷ 11 = 8 remainder 2, and the remainder 2 is where the division began, so the digits 1, 8 repeat for ever. In dot notation there is a dot above the 1 and a dot above the 8.

5/12

HintLook at the prime factors of each denominator.

The whyA fraction in its simplest form terminates only if its denominator has no prime factors other than 2 and 5, the factors of 10. 8 = 2 × 2 × 2 and 20 = 2 × 2 × 5 pass, but 12 = 2 × 2 × 3 contains a 3. So 5/12 = 0.41666…, while 3/8 = 0.375 and 7/20 = 0.35.

7.35 ≤ L < 7.45

HintGo half of one tenth below the stated value and half of one tenth above it.

The whyRounding to 1 decimal place means to the nearest 0.1, so the true value can be up to 0.05 away on either side. Anything from 7.35 up to, but not including, 7.45 rounds to 7.4.

3.6 ≤ n < 3.7

HintChopping digits off can never make a positive number bigger.

The whyTruncating simply removes the digits after the first decimal place, so 3.6, 3.65 and 3.699 all become 3.6. The smallest possible value is 3.6 itself and the number must stay below 3.7.

Truncating the number

HintA related word describes a cone with its top sliced off.

The whyTruncated to 1 decimal place, 4.78 becomes 4.7, whereas rounding would give 4.8. Truncating always gives a value that is equal to or below the original positive number.

Because 7.45 itself would round up to 7.5

HintTry rounding the top value of the interval to 1 decimal place.

The whyA 5 in the next place rounds up, so 7.45 belongs with 7.5. The length can be 7.449 or 7.4499, as close to 7.45 as you like, but never equal to it. The lower end, 7.35, does round to 7.4, so it is included.

245

HintHalve the size of the measuring step, then take it away.

The whyTo the nearest 10 g, the true mass can be up to 5 g either side of 250 g. The least it can be is 250 − 5 = 245 g, and it must be less than 255 g. These limits of accuracy are the two ends of the error interval 245 ≤ m < 255.