To determine the whole number and the rational number to the nearest half between which [tex]\(\sqrt{10}\)[/tex] lies, let's first understand the value of [tex]\(\sqrt{10}\)[/tex].
The value of [tex]\(\sqrt{10}\)[/tex] is approximately [tex]\(3.1622776601683795\)[/tex].
Step-by-step solution:
1. Identifying the Whole Numbers:
- The largest whole number less than or equal to [tex]\(3.1622776601683795\)[/tex] is [tex]\(3\)[/tex].
- The smallest whole number greater than or equal to [tex]\(3.1622776601683795\)[/tex] is [tex]\(4\)[/tex].
- So, [tex]\(\sqrt{10}\)[/tex] lies between the whole numbers [tex]\(3\)[/tex] and [tex]\(4\)[/tex].
2. Identifying the Rational Numbers to the Nearest Half:
- The rational numbers to the nearest half around [tex]\(3.1622776601683795\)[/tex] are calculated as follows:
- The number line has rational points at whole numbers and halves, i.e., [tex]\(3\)[/tex], [tex]\(3.5\)[/tex], and [tex]\(4\)[/tex].
- To find the rational number to the nearest half lower than [tex]\(\sqrt{10}\)[/tex], we compare [tex]\(3.1622776601683795\)[/tex] with [tex]\(3 + 0.5 = 3.5\)[/tex]. Since [tex]\(3.5\)[/tex] is greater than [tex]\(3.1622776601683795\)[/tex], the nearest lower half is [tex]\(3\)[/tex].
- The rational number to the nearest half greater than [tex]\(\sqrt{10}\)[/tex] is [tex]\(3.5\)[/tex], as it is the next rational number after [tex]\(3.1622776601683795\)[/tex] when stepping up by 0.5.
So, [tex]\(\sqrt{10}\)[/tex] is between the whole numbers [tex]\(3\)[/tex] and [tex]\(4\)[/tex] and is between the rational numbers [tex]\(3\)[/tex] and [tex]\(3.5\)[/tex].
In conclusion, on a number line, [tex]\(\sqrt{10}\)[/tex] is between [tex]\( \boxed{3} \)[/tex] and [tex]\( \boxed{3.5} \)[/tex].
