NUMERIC PROCESS 2 (STANDARD FORMS)

NUMERIC PROCESS 2 (STANDARD FORMS)

A number written with one digit to the left of the decimal point and multiplied by 10 raised to some power is said to be written in Standard Form. Thus: 5837 is written as 5.837×10^3 in standard form, and 0.0415 is written as 4.15×10⁻² in standard form. When a number is written in standard form, the first factor is called the Mantissa and the second factor is called the EXPONENT. Thus the number 5.8 × 10³ has a mantissa of 5.8 and an exponent of 10³.

(i) Numbers having the same exponent can be added or subtracted in standard form by adding or subtracting the mantissa and keeping the exponent

the same.

 Thus:

2.3 × 10⁴ + 3.7 × 10⁴

= (2.3 + 3.7) × 10⁴ = 6.0 × 10⁴

and 5.9 × 10⁻² − 4.6 × 10⁻²

= (5.9 − 4.6) × 10⁻² = 1.3 × 10⁻²

When the numbers have different exponents, one way of adding or subtracting the numbers is to express one of the numbers in non-standard form, so that both numbers have the same exponent.

Thus:

2.3 × 10⁴ + 3.7 × 10³

= 2.3 × 10⁴ + 0.37 × 10⁴

= (2.3 + 0.37) × 104 = 2.67 × 104

Alternatively,

2.3 × 10⁴ + 3.7 × 10³

= 23 000 + 3700 = 26 700

= 2.67 × 10⁴

(ii) The laws of indices are used when multiplying or dividing numbers given in standard form. For example,

(2.5 × 10³) × (5 × 10²)

= (2.5 × 10⁵) × (10³⁺²)

= 12.5 × 10⁵ or 1.25 × 10⁶

Similarly,

6 × 10⁴

1.5 × 10²

=

 6

1.5× (10⁴⁻²) = 4 × 10²

2.5 Worked problems on standard

Problem  

Express in standard form:

(a) 38.71 (b) 3746 (c) 0.0124

For a number to be in standard form, it is expressed with only one digit to the left of the decimal point. Thus:

(a) 38.71 must be divided by 10 to achieve one digit to the left of the decimal point and it must also be multiplied by 10 to maintain the equality, i.e.

38.71 =

38.71    ×10 = 3.871×10 in standard form

  10       

(b) 3746=

3746

1000    ×1000=3.746×10³ in standard form

(c) 0.0124=0.0124× 100

100

1.24

100     =1.24×10⁻² in standard form

Problem 15. Express the following numbers, which are in standard form, as decimal numbers:

(a) 1.725×10−2 (b) 5.491×104 (c) 9.84×100

(a) 1.725×10⁻² =

1.725

100         =0.01725

(b) 5.491×10⁴ =5.491×10 000= 54 910

(c) 9.84×10⁰ =9.84×1=9.84 (since 10⁰ =1)