Standard form

 Numbers In Standard Form 

Standard form means the regular way to write out a number, like 4,300, as opposed to scientific notation, which is 4.3 x 10³, or in expanded form, like 4,000 + 300. You can convert numbers large and small to standard form through some steps. Standard form is a way of writing down very large or very small numbers readily. 10³ = 1000, so 4 × 10³ = 4000. Therefore, 4000 can be written as 4 × 10³. This idea can be made use of in writing even larger numbers down readily in standard form.

Therefore, lets take a big number such as 8,143,260. Now if we wish to convert this into standard form we want it to be just 8 before a decimal point, all these other numbers after a decimal and then we're going to multiply it by 10 to an exponent. Therefore, in this regard at the moment the decimal is over here. And we want it to be over here. Therefore we have to move it over 1,2,3,4,5,6 places. So we have 8.143,26 and you don't have to say the zero, times 10 to the 6th power. Now it's the 6th due to the fact that it is moved over 6 places.

📍 Rules for converting to standard form

• It must have one (1) whole number before the decimal. E.g 

a.bcd X 10³ ✔️. 

ab.cd X 10² ❌ 

• When a number to be converted begins with zero (0), the exponential will be negative. E.g 

0. 0054 ==> 5.4 X 10¯³ ✔️.  

0.0054==>      5.4 X 10³  ❌

• To place decimal, count from the initial decimal to where it should be placed. E.g 1234.56

To convert this, move the decimal to be in between the first and second whole number I.e 1 & 2 then count how many times it was moved (we moved three times(3)), so the exponential will be 10³

Note :

In the case of a whole number (453280) the decimal is at the end (i.e 453280.0). So to convert to standard form move the decimal to be in between the first and second whole number (i.e 4.53280) then count how many times it was moved (5 times). Therefore it is represented in standard form as 4.53280 X 10^5.

🙄Let's look at a few more examples

1️⃣ Write 5326.6 in standard form:

It's written as 5.3266 × 10³, because 5326.6 = 5.3266 × 1000 = 5.3266 × 10³

Standard Form of an Equation

The "Standard Form" of an equation is:

(some expression) = 0

Put in another way, "= 0" is on the right, and everything else is on the left.

Example: Put x² = 7 into Standard Form

Answer: x² - 7 = 0

Standard Form of a Polynomial

The "Standard Form" for writing down a polynomial is to put the terms with the highest degree first (such as the "2" in x² if there is one variable).

Example: Put this in Standard Form:

3x² - 7 + 4x³ + x^6

The highest degree is 6, therefore that goes first, then 3, 2 and then the constant last:

x^6 + 4x³ + 3x² -7

Standard Form of a Linear Equation

The "Standard Form" for writing down a Linear Equation is

Ax + By = C

A is not supposed to be negative, A and B oughtn't to be both zero, and A, B and C ought to be integers.

Example: Put this in Standard Form:

y = 3x + 2

Move 3x to the left hand side:

-3x + y = 2

Multiply all by -1:

3x - y = -2

Observe: A=3, B=-1, C=-2

This form:

Ax + By + C = 0 is sometimes known as "Standard Form", but is more correctly known as the "General Form".

Standard Form of a Quadratic Equation

The "Standard Form" for writing down a Quadratic Equation is ax² + bx + c = 0 (a not equal to zero)

Example: Put this in Standard Form:

x(x-1) = 3

Expand "x(x-1)":

x² - x = 3

Move 3 to left:

x² - x - 3 = 0

Observe that: a =1, b = -1, c = -3

More Example:

Write 81 900 000 000 000 in standard form: 81 900 000 000 000 = 8.19 × 10¹³

It’s 10¹³ due to the fact that the decimal point has been moved 13 places to the left to get the number to be 8.19

More Example:

Write 0.000 001 2 in standard form:

0.000 001 2 = 1.2 × 10^¯6

It’s 10^¯6 due to the fact that the decimal point has been moved 6 places to the right to obtain the number to be 1.2

👉 Practice questions