Converting to and from standard form

Picture a map that shows ten times more with every notch you zoom out. One notch, and your street has become your town. Every notch adds another zero. Six of them, and your street has become the whole planet. Written out in full, numbers that size are rows of zeros. Miscount one zero and your answer is ten times too big. Standard form writes the notch count instead of the zeros.

First, what standard form is, and why it has a rule about the number at the front.

Start with powers of ten. Ten to the power three is ten times ten times ten. That is one thousand - a one with three zeros.

The index is counting something real. Multiply by ten, and every digit shifts one place to the left. So the index is a count of shifts.

Standard form is built on that. A number, times a power of ten. Three thousand is three times ten to the power three.

And there is one rule. The front number must be at least one, and less than ten. One digit before the decimal point. Never two.

Here is why. Without it, one number has dozens of writings. Thirty-six times ten to the power four. Three point six times ten to the power five. Nought point three six times ten to the power six. All the same number. The rule picks one, so any two numbers can be sized up by their power of ten alone.

Your turn, and it is a gentle one. Those three writings again. Which one is in standard form? Thirty-six times ten to the power four, three point six times ten to the power five, or nought point three six times ten to the power six.

Have a think. I'll wait.

The answer is the middle one. Three point six times ten to the power five.

Three point six is between one and ten, so it passes. Thirty-six is too big - that is two digits before the point. Nought point three six is too small - it is under one.

That middle option is the one examiners report students missing. This line is from a report on a Foundation paper. It is picked up mid-sentence, so it starts in the middle.

to give thirty-six before the power of ten, rather than the three point six it needs to be for standard form.

Thirty-six is not bad arithmetic. The number is right. It is just not in standard form yet.

So, line one of something worth remembering. One digit before the point.

Now the method for turning an ordinary number into standard form. Big numbers first.

There is a reason to carry a method rather than a memory of what standard form looks like. Another examiner report puts it like this.

Those who could not remember how to put a number into standard form offered the number written as words.

Written as words. Spelled out in full. Those students knew the number - they had nothing to fall back on. So here is something to fall back on.

Write fifty-two million. Five, two, then six zeros. Now find the decimal point. In a whole number it sits at the far right, after the last zero, even though nobody writes it in. Move it left until exactly one digit is in front of it. Hop, hop, hop. It lands between the five and the two, and that gives five point two. Count the hops. Seven. That count is the power of ten. And fifty-two million is far bigger than ten, so it is a positive seven. Five point two times ten to the power seven.

Then the check, every time. Is the front number between one and ten? Five point two. Yes. Finished.

So the line grows. One digit before the point. Count the hops.

Your turn. Write six million three hundred thousand in standard form.

Pause here and work it through. I'll wait.

The answer is six point three times ten to the power six.

The point starts at the far right. Hop it left until one digit is in front - six point three. Six hops. A big number, so a positive six. And six point three is between one and ten.

Small numbers next. Same method, with one thing reversed.

Write nought point nought nought nought four three in standard form.

The point is already written, and the digit in front of it is a nought. So hop it the other way - right this time - until one digit is in front. It lands between the four and the three. Four point three.

Count the hops. One, two, three, four.

And here is the reversed bit. The number is smaller than one, so the power is negative. Four point three times ten to the power minus four.

Why negative? Hopping right made the number bigger - ten times bigger, four times over. The power of ten has to undo that, so it divides instead of multiplying. That is what a minus index does, and the video on zero and negative indices builds it.

The check again. Four point three is between one and ten. Yes.

One thing that minus does not mean. Four point three times ten to the power minus four is still a positive number. A tiny one, but positive. The minus is about size, not sign.

And now the line is finished. One digit before the point. Count the hops. Big goes up, small goes down.

Your turn. Write nought point nought nought nought eight one in standard form.

Have a think. I'll wait.

The answer is eight point one times ten to the power minus four.

Hop right until one digit is in front - eight point one. Four hops. The number is under one, so the power is minus four. And eight point one is between one and ten.

One direction left. Standard form back into an ordinary number.

Write two point nine times ten to the power six as an ordinary number.

Same hops, run backwards. The power is a positive six, so the number is getting bigger, and the point moves right six places. Two point nine. Hop right, filling in zeros as you go.

Two million, nine hundred thousand.

There is no front-number check here - the answer is not in standard form any more. So check the size instead. Ten to the power six is a million, and two point nine million came out. That fits.

One more. Write five point one times ten to the power minus three as an ordinary number.

Have a think. I'll wait.

The answer is nought point nought nought five one.

Minus three, so hop left three times. Five point one. Nought point five one. Nought point nought five one. Nought point nought nought five one. A negative power, so the number shrinks.

Now the exam side of this, and a mistake examiners write about year after year.

This is from a report on a Foundation paper.

In part b the vast majority of students worked out the answer as one hundred and forty-five thousand but many did not know how to convert it to standard form.

Read that again. The vast majority got the number right. The maths was done. What was missing was the last step.

So treat the conversion as its own step, on its own line. When a question says give your answer in standard form, that is a separate job at the end. It is easy to forget because it is not the hard part.

And when you do that last step in your head, under pressure, that is exactly when the front number comes out as thirty-six instead of three point six. So the check earns its keep. Between one and ten, or it is not finished.

So, back to the shape of it.

Standard form is a number from one up to but not including ten, times a power of ten. The power counts how far the decimal point hopped. Big numbers give a positive power. Numbers under one give a negative power. To go back the other way, run the same hops in reverse. And every answer gets the same last check. Front number between one and ten, or it is not finished yet.

The whole method in one line. One digit before the point. Count the hops. Big goes up, small goes down.

Next in the chapter: calculating with numbers in standard form.

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